Even lattices of rank 26 and determinant 7

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• 44955 isometry classes, mass 16203189657383115756971349096767953177/32369483335006712890159037153280000000.
• 5119 possible root systems, see here for statistics of reduced masses for each root system.
• Isometry classes distinguished by invariant: \(\mathrm{BV}_{26,7}\).
• Classified in: G. Chenevier & O. Taïbi, Unimodular lattices of rank 29 and related even genera of small determinant, arXiv preprint (2026).
• Construction(s): Norm 7/6 vectors in duals of rank 27 even lattices of determinant 6. Norm 7/10 vectors in duals of rank 27 even lattices of determinant 10. Embeddings of \({\rm A}_6\) in rank 32 even unimodular lattices.
• Extra information: There are 3 lattices with no roots; their isometry groups have order 4608, 1376256 and 2005590343680, and are respectively solvable (order \(2^9 \cdot 3^2\)), of the form \(\mathbb{Z}/2 \times ((\mathbb{Z}/4)^4 : ((\mathbb{Z}/2)^4 : {\rm L}_3(2)))\), or of the form \(\mathbb{Z}/2 \times ( (\mathbb{Z}/2)^{12} : {\rm M}_{24} )\).