Even lattices of rank 28 and determinant 5

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

• 2738211 isometry classes, mass 7735272734222256159335405970238469607373283/3655847529600758161712079490252800000000.
• 10674 possible root systems, see here for statistics of reduced masses for each root system.
• Isometry classes distinguished by invariant: \(\mathrm{BV}_{28,5}\).
• Classified in: G. Chenevier & O. Taïbi, Unimodular lattices of rank 29 and related even genera of small determinant, arXiv preprint (2026).
• Construction(s): Exceptional vectors in rank 29 unimodular lattices (unique orbit). Embeddings of \({\rm A}_4\) in rank 32 even unimodular lattices.
• Extra information: There are 115 lattices with no roots, including 20 with trivial (\(=\mathbb{Z}/2\)) isometry groups; their isometry groups are either of order \(p^a\cdot q^b\) (for all but 9 of them), or have the following orders (and at most one non-abelian composition factor): 120 (\({\rm A}_5\)), 240 (\({\rm A}_5\)), 960 (\({\rm A}_5\)), 2400 (\({\rm A}_5\)), 5760 (\({\rm A}_5\)), 7680 (\({\rm A}_5\)), 24000 (none), 3317760 (\({\rm U}_4(2)\)), 37440000 (\({\rm S}_4(5)\)).