Odd unimodular lattices of rank 29 with no norm 1 vectors

Complete list here and here (see the main page for format details).

• 38592290 isometry classes (2721152 exceptional), mass 9683137883598841522700149306218386019856601/65188542827444074570459172044800000000.
• 11085 possible root systems, see here for statistics of reduced masses for each root system, and here for similar information restricted to exceptional lattices.
• Isometry classes distinguished by invariant: BV.
• Classified in: G. Chenevier & O. Taïbi, Unimodular lattices of rank 29 and related even genera of small determinant, arXiv preprint (2026). The 10092 lattices with no vectors of norm \(\leq 2\) were found by Allombert & Chenevier in Unimodular hunting II, Forum Math. Sigma 13 (2025). The masses root system by root system were first computed by King in A mass formula for unimodular lattices with no roots, Math. Comp. 72 (242), 839–863 (2003).
• Construction(s): Pairs of orthogonal roots in rank 29 unimodular lattices correspond to mod 2 vectors with norm 2 mod 4 in rank 27 unimodular lattices. Rank 29 unimodular lattices with no norm 1 vectors and root systems empty, \({\bf A}_1\) or \({\bf A}_2\) determined as cyclic neighbors of the standard lattice \({\rm I}_{29}\).
• Extra information: There are 10092 lattices with no roots, including 8081 with trivial (\(=\mathbb{Z}/2\)) isometry groups (Allombert-Chenevier); their isometry groups are either of order \(p^a\cdot q^b\) (for all but 5 of them), or have the following orders (and at most one non-abelian composition factor): 60 (none), 120 (none), 960 (\({\rm A}_5\)), 2400 (\({\rm A}_5\)) and 24000 (none).