\\
\\ The rank n unimodular lattices with no roots, for 1<=n<=29
\\
\\      by Bill Allombert and Gaetan Chenevier, 29/09/2020
\\
\\ This is a manual for the file http://gaetan.chenevier.perso.math.cnrs.fr/
\\ unimodular_lattices/unimodular_lattices_no_roots.gp
\\
\\ Each lattice there is given as a d-neighbor of the standard lattice Z^n, 
\\ and more precisely in the following form 
\\
\\	 	[d, x, t, 1/ord]
\\
\\ where:
\\
\\ -- d is an integer,
\\
\\ -- x=[x_1; x_2; ...; x_n] is an increasing sequence of integers with x_1=1, 
\\   x_n <=d/2 and x.x = 0 mod d,
\\
\\ -- t is an extra parameter equal to 0 or 1, only needed for d even. 
\\ 
\\ These [d, x, t] determine a rank n unimodular lattice L defined by 
\\
\\    		L = M + Z w
\\  
\\ with M = { v in Z^n  |  v.x = 0 mod d}  and  w = x/d - [a; 0; ...; 0]
\\ with a =  x.x * (d+1)/2d (case d odd), a = x.x/2d + t*d/2 (case d even)
\\
\\ (We always have L \cap Z^n = M, and x.x = 0 mod 2d for d even)
\\ 
\\ -- ord is the order of the isometry group O(L) of the unimodular lattice L
\\   (computed essentially using the Plesken-Souvignier algorithm)
\\
\\ For n=1,...,29, the vector Un contains representatives for the isometry
\\ classes of rank n unimodular lattices with no roots. 
\\
\\ Notes :
\\   The lattice U23[1] is the shorter Leech lattice of Conway and Sloane, 
\\    and U24[2] is the Leech lattice
\\   The lattices U24[1], U26[1] and U27[3] were found by Borcherds and Conway
\\   The other lattices in U27, and the 38 lattices in U28, were found by Bacher
\\    and Venkov (we give them in the order chosen by these authors) 
\\   We found 10092 unimodular lattices of rank 29 and without roots. The
\\    statistics for the order of their automorphism groups are as follows:
\\     
\\    [   2    4 6   8 12 16 20 24 32 36 40 48 60 64 72 80 96 120 128 144]
\\    [8081 1465 6 293 28 91  1 21 32  1  3 15  1 12  1  1 11   1   2   1]
\\
\\    [160 192 232 256 288 320 384 768 864 960 1024 1536 2400 2592 3072 5184]
\\    [  2   2   1   2   1   1   1   2   1   1    2    2    1    1    1    1] 
\\
\\    [6144 18432 24000]
\\    [   1     1     1] 
\\
\\    Among these 10092 lattices, there are exactly 105 lattices which are
\\    exceptional in the sense of Bacher and Venkov, i.e. with a characteristic
\\    vector of norm 5. The statistics for the order of the automorphism groups
\\    of the exceptional lattices are as follows:
\\
\\    [ 2  4  8 12 16 20 32 40 48 64 80 96 192 960 1024 2400 24000]
\\    [20 31 24  2 10  1  3  3  3  1  1  1   1   1    1    1     1]
\\
\\ This file is PARI/GP readable. For a few helpful GP scripts to play with the
\\ lists below, including an automatized proof that they are complete, see the
\\ companion file and its tutorial
\\
\\   http://gaetan.chenevier.perso.math.cnrs.fr/unimodular_lattices/tools_unimodular_lattices_no_roots.gp
\\   http://gaetan.chenevier.perso.math.cnrs.fr/unimodular_lattices/tutorial_unimodular_lattices_no_roots.txt
\\
\\ 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/).
\\