Even lattices of rank 25 and determinant 6

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• 2827 isometry classes, mass 93404856100273993087460069509909307/258955866680053703121272297226240000000.
• 1630 possible root systems, see here for statistics of reduced masses for each root system.
• Isometry classes distinguished by invariant: \(\mathrm{BV}_{25,3}\).
• Classified in: R. Borcherds, The Leech lattice and other lattices, PhD thesis, University of Cambridge (1984) (slightly corrected 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 26 even lattices of determinant 3. Norm -6 vectors in the rank 26 even unimodular Lorentzian lattice.
• Extra information: There are 3 lattices with no roots, including \({\rm Leech} \perp \langle\,6\,\rangle\); their isometry groups have order 277136640, 1983066624000 and 16631107226173440000, and are respectively of the form \(\mathbb{Z}/2 \times ((\mathbb{Z}/3)^6 : (\mathbb{Z}/2 \cdot {\rm M}_{12}))\), an extension of \({\rm Co}_3\) by a group of order 4 and \({\rm Co}_0 \times \mathbb{Z}/2\).