\\
\\ The rank 28 unimodular lattices with no norm 1 vector
\\
\\ by Bill Allombert and Gaetan Chenevier, 13/11/2020
\\
\\
\\ This is a continuation of the work "Unimodular hunting" by the second author
\\ (classification of rank n unimodular lattices with n=26 or 27), available at
\\ http://gaetan.chenevier.perso.math.cnrs.fr/pub.html
\\ We refer to this work for the notations [d, x, t, 1/ord, rs, 1/g] we use for
\\ unimodular lattices.
\\
\\ We found 357003 unimodular lattices of rank 28 with no norm 1 vector. Adding
\\ the 17059 rank 28 unimodular lattices with a norm 1 vector previously found,
\\ there are thus exactly 374062 unimodular lattices of rank 28.
\\
\\ The full list of the 357003 lattices above may be downloaded at the address:
\\
\\ http://gaetan.chenevier.perso.math.cnrs.fr/unimodular_lattices/L28
\\
\\ This file is 47.1 MB and PARI/GP readable. We found it convenient to split it
\\ according to the following invariant, that proved useful in our computations:
\\ If L is an integral lattice, we denote by i(L) the largest integer i such
\\ that the root lattice A_{i-1} embeds into L. The number n(i) of rank 28
\\ unimodular lattices with no norm 1 vector and with i(L) =i is then given by:
\\
\\ i [ 1 2 3 4 5 6 7 8 9 10 11 12 13 14]
\\ n(i) [38 20560 121684 126661 55585 20919 6712 2935 960 516 168 142 45 35]
\\
\\ i [15 16 17 18 19 20 21 22 23 24 25 26 27 28]
\\ n(i) [ 8 20 3 3 1 5 1 0 0 1 0 0 0 1]
\\
\\ The full list of rank 28 unimodular lattices L with no norm 1 vector and with
\\ i(L)=k may be downloaded at the address:
\\
\\ http://gaetan.chenevier.perso.math.cnrs.fr/unimodular_lattices/L28_ik
\\
\\ (Warning: for k=2, 3, 4, 5 and 6, they are 2.7, 15.9, 16.8, 7.4 and 2.8 MB).
\\
\\ Notes:
\\ The lattices L with i(L)=1 are exactly those with no roots. There are 38 of
\\ them, which had been previously found by Bacher and Venkov.
\\ The lattices L with i(L)=2 are those with root system of the form nA1 for
\\ some (nonzero) integer n. The number # of lattices we found with root
\\ system nA1 is given by the following table:
\\
\\ n [ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 16 20 28]
\\ # [42 188 327 821 1190 2157 2620 3520 3096 2982 1675 1264 337 242 80 16 3]
\\
\\ The lattices L with i(L)=3 are those with root system of the form nA1 mA2
\\ for some integers n and m, with m nonzero, and so on. (Recall that we
\\ have i(D_n)=n for n>3, i(E_6)=6, i(E_7)=8 and i(E_8)=9.)
\\ As in the case of rank n unimodular lattices with n<28, it turns out that
\\ a rank 28 unimodular lattice is uniquely determined (up to isomorphism)
\\ by the configuration of its vectors v with v.v <=3. See the companion
\\ file tools_L28.gp for the invariant we used ( function BV(gram) ).
\\ For more explanations, see our forthcoming paper "Unimodular hunting II".
\\
\\ Our computations made a crucial use of the PARI/GP Calculator (version 2.13),
\\ and in particular of its parallel programming functions. Also, we used the
\\ following ressources:
\\
\\ -- the platform PlaFRIM (about 54 years of CPU time).
\\ Acknowledgment: Experiments presented in this work were carried out using
\\ the PlaFRIM experimental testbed, supported by Inria, CNRS (LABRI and IMB),
\\ Universite de Bordeaux, Bordeaux INP and Conseil Regional d'Aquitaine (see
\\ https://www.plafrim.fr/).
\\
\\ -- the cluster cinaps in the LMO (about 18 years of CPU time).
\\ See https://cinaps.math.u-psud.fr/ganglia/ .
\\
\\ We warmly thank PlaFRIM and the LMO for sharing their machines.
\\
\\ For a few helpful GP scripts to play with the lists above, including an
\\ automatized proof that they are complete, see the companion tool file
\\ and its tutorial
\\
\\ http://gaetan.chenevier.perso.math.cnrs.fr/unimodular_lattices/tools_L28.gp
\\ http://gaetan.chenevier.perso.math.cnrs.fr/unimodular_lattices/tuto_L28.txt
\\