Even lattices of rank 27 and determinant 6

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• 285825 isometry classes, mass 184361388591800313635423567792726086296697/6214940800321288874910535133429760000000.
• 6770 possible root systems, see here for statistics of reduced masses for each root system.
• Isometry classes distinguished by invariant: \(\mathrm{BV}_{27,3}\).
• Classified in: G. Chenevier & O. Taïbi, Unimodular lattices of rank 29 and related even genera of small determinant, arXiv preprint (2026).
• Construction(s): Norm 6/5 vectors in duals of rank 28 even lattices of determinant 5. Embeddings of \({\rm A}_5\) in rank 32 even unimodular lattices.
• Extra information: There are 9 lattices with no roots; their isometry groups have the following orders (and at most one non-abelian composition factor): 48 (none), 48 (none), 48 (none), 120 (\({\rm A}_5\)), 240 (\({\rm A}_5\)), 1152 (none), 4608 (none), 5760 (\({\rm A}_5\)) and 18720000 (\({\rm S}_4(5)\)).