Even lattices of rank 27 and determinant 10

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

• 24545511 isometry classes, mass 70578754793256507668758678067481511896176437/3107470400160644437455267566714880000000.
• 13960 possible root systems, see here for statistics of reduced masses for each root system.
• Isometry classes distinguished by invariant: \(\mathrm{BV}_{27,5}\).
• Classified in: G. Chenevier & O. Taïbi, Unimodular lattices of rank 29 and related even genera of small determinant, arXiv preprint (2026).
• Construction(s): Roots in rank 28 even lattices of determinant 5. Embeddings of \({\rm A}_1 \perp {\rm A}_4\) in rank 32 even unimodular lattices.
• Extra information: There are 656 lattices with no roots, including 233 with trivial (\(=\mathbb{Z}/2\)) isometry groups; their isometry groups are either of order \(p^a \cdot q^b\) (for all but 4 of them), or have the following orders (and exactly one non-abelian composition factor): 18144 (\({\rm L}_2(8)\)), 22464 (\({\rm L}_3(3)\)), 1376256 (\({\rm L}_3(2)\)) and 1268047872 (\({}^3{\rm D}_4(2)\)).