Unimodular Hunting

By Gaëtan Chenevier
28/05/2025

This website is a companion to the paper Unimodular Hunting by the author.

In this paper, we develop a method initiated by Bacher and Venkov, based on a study of the Kneser neighbors of the standard lattice Zn, to determine the isometry classes of unimodular integral euclidean lattices of rank 26 and 27. There are respectively 2566 and 17059 such lattices (up to isometry).

  • (14/07/2020) Here is a list of all unimodular integral lattices of rank <=27 given in neighbor form.
    This is a PARI/GP readable file (2.2 MB).
    See here for an explanation of the format of the lattices, and for some historical notes.

  • (14/07/2020) See also this companion file for a few helpful gp scripts to play with the list above,
    including an automatized proof that our lists of lattices are indeed complete (tutorial).

  • We also provide a list of Gram matrices for all the unimodular integral lattices of rank <=27 without norm 1 vectors: dim8_even, dim12, dim14, dim15, dim16_odd, dim16_even, dim17, dim18, dim19, dim20, dim21, dim22, dim23, dim24_odd, dim24_even (Niemeier lattices), dim25, dim26, dim27 (gzipped, 3.5 MB).

    See here for the format of these lists.