Odd unimodular lattices of rank 24 with no norm 1 vectors

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

• 156 isometry classes (0 exceptional), mass 194238286903184339707/5901018623998487785635840000.
• 149 possible root systems, see here for statistics of reduced masses for each root system.
• Isometry classes distinguished by invariant: BV.
• Classified in: R. Borcherds, The 24 dimensional Odd Unimodular lattices, Chapter 17 in Conway & Sloane's SPLAG.
• Construction(s): Norm 1 vectors in rank 25 unimodular lattices.
• Extra information: First dimension with non-isometric unimodular lattices with same root system (happens for 7 root systems, such as \(3{\bf D}_8\), with two lattices in each case). Lattice \(\#156\) is the unique lattice with no root (odd Leech lattice); its isometry group is \((\mathbb{Z}/2)^{12} : {\rm M}_{24}\).