Odd unimodular lattices of rank 23 with no norm 1 vectors

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

• 49 isometry classes (48 exceptional), mass 21569773276937492389/28590262351867673365708800000.
• 49 possible root systems, see here for statistics of reduced masses for each root system, and here for similar information restricted to exceptional lattices.
• Isometry classes distinguished by invariant: root system.
• Classified in: J. H. Conway & N. J. A. Sloane, The unimodular lattices of dimension up to 23, Pure and Applied Math. Comm. 35, 763–812 (1982).
• Construction(s): The even parts of rank 23 unimodular lattices correspond to norm 4 vectors in Niemeier lattices.
• Extra information: Lattice \(\#49\) is the unique lattice with no root (shorter Leech lattice); its isometry group is \(\mathbb{Z}/2 \times {\rm Co}_2\).