Even lattices of rank 26 and determinant 3

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

• 678 isometry classes, mass 16521194172507546167260484745149/1313385629822757581342040391680000000.
• 587 possible root systems, see here for statistics of reduced masses for each root system.
• Isometry classes distinguished by invariant: \(\mathrm{BV}_{26,3}\).
• Classified in: R. Borcherds, The Leech lattice and other lattices, PhD thesis, University of Cambridge (1984) (a few indeterminacies settled by Mégarbané).
• Construction(s): Roots in rank 26 even lattices of determinant 3 correspond to orbits of norm -6 vectors in the rank 26 even unimodular Lorentzian lattice. Exceptional vectors in rank 27 unimodular lattices (unique orbit). Norm 3/10 vectors in duals of rank 27 even lattices of determinant 10. Embeddings of \({\rm E}_6\) in rank 32 even unimodular lattices.
• Extra information: The 24 lattices from \(\#655\) to \(\#678\) have the form \({\rm Niemeier} \perp {\rm A}_2\). Lattice \(\#554\) is the unique lattice with no roots, and the orthogonal of 1 in the Euclidean exceptional Jordan algebra over \(\mathbb{Z}\) with empty root system; its isometry group has order 1268047872 and is of the form \((\mathbb{Z}/2:{}^3{\rm D}_4(2)\,):\mathbb{Z}/3\).