\\
\\ The rank n unimodular lattices with no norm 1 vector, for 1<=n<=27
\\
\\ 		by Gaetan Chenevier, 14/07/2020
\\
\\ slightly modifed on 10/2024 in order to fit notations in the associated
\\ new preprint [UH] "Unimodular Hunting" (24/10/2024)
\\
\\
\\ Each lattice below is given as a d-neighbor of the standard lattice Z^n, 
\\ and more precisely in the following form 
\\
\\	(*) 	[d, x, t, 1/ord, rs, 1/g]
\\
\\ where:
\\
\\ -- d is an integer,
\\ -- x=[x_1; x_2; ...; x_n] is a nondecreasing 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 x'/d
\\  
\\ with M = { v in Z^n  |  v.x = 0 mod d}  and  x' = 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. This 
\\ lattice L is denoted N_d(x) in [UH] for d odd, N_d(x; t) for d even.
\\ 
\\ The last 3 components of (*) follow from the first 3 but are given for 
\\ convenience: 
\\
\\ -- ord is the order of the isometry group O(L) of the unimodular lattice L
\\   (computed essentially using the Plesken-Souvignier algorithm), 
\\ -- rs is the root system of L, 
\\ -- g is the order of the quotient of O(L) by the Weyl group of L 
\\
\\ (in other words, 1/g is the reduced mass of L and 1/ord is the mass of L)
\\
\\ For n=1,...,27 the vector Nn contains representatives for the isometry
\\ classes of rank n unimodular lattices without norm 1 vector. For n = 0 mod 8
\\ the vectors NnI and NnII select respectively the odd and even elements of Nn.
\\
\\ Notes:
\\   The lattices in Nn were determined by Kneser for n<=16 (with contributions
\\     by Euler, Lagrange, Gauss, Hermite, Smith, Korokin-Zolotarev, Gosset...)
\\   N8II[1] is the E8 lattice
\\   The lattices in N24II were classified by Niemeier (Niemeier lattices)
\\   N24II[24] is Leech lattice, and this neighbor form was found by Thompson
\\   The lattices in Nn for 17<=n<=23 were determined by Conway and Sloane
\\   The lattices in N24I and N25 were determined by Borcherds   
\\   Bacher gave some neighbor forms for all the lattices mentionned above
\\   Our main contribution is the determination of N26 and N27. In particular,
\\   they have respectively 1901 and 14493 elements.
\\   We made crucial use of the computation by King of the mass of the 32-dim
\\     even unimodular lattices with given root system (and eventually provide
\\     independent checks of King's computations).
\\   For explanations, see the paper [UH] "Unimodular Hunting".
\\
\\ We warmly thank the LMO for sharing the machine pascaline that we used in our
\\ computations (24 CPU, 65Gb Memory)
\\
\\ This file is PARI/GP readable. For a few helpful GP scripts to play with the
\\ lists below, e.g. to give concrete gram matrices of each lattice, and for an
\\ automatized proof that they are complete, see the companion file and 
\\ its tutorial below
\\
\\   http://gaetan.chenevier.perso.math.cnrs.fr/unimodular_lattices/tools_unimodular_lattices.gp
\\   http://gaetan.chenevier.perso.math.cnrs.fr/unimodular_lattices/tutorial_unimodular_lattices.txt
\\