Odd unimodular lattices of rank 27 with no norm 1 vectors

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

• 14493 isometry classes (557 exceptional), mass 18471746857358122138056975582390629/121385562506275173338096389324800000.
• 2797 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: BV.
• Classified in: G. Chenevier, Unimodular hunting, Algebraic Geometry 12 (6), 769–812 (2025). The 3 lattices without roots were determined by Bacher and Venkov (some were known to Borcherds), and more generally those with root systems \(n\,{\bf A}_1\) with \(n \leq 4\), in Réseaux entiers unimodulaires sans racine en dimension 27 et 28, Enseign. Math. 37 212–267 (2001). Masses root system by root system first computed by King in A mass formula for unimodular lattices with no roots, Math. Comp. 72 (242), 839–863 (2003).
• Construction(s): Determined by Chenevier as cyclic neighbors of the standard lattice \({\rm I}_{27}\).
• Extra information: There are 3 lattices with no roots (Bacher-Venkov); their isometry groups are \(\mathbb{Z}/2 \times (( (\mathbb{Z}/2)^5 : {\rm A}_5) : \mathbb{Z}/2)\), \(\mathbb{Z}/2 \times ( (\mathbb{Z}/2)^6 \cdot {\rm O}_5(3))\) and \(\mathbb{Z}/2 \times ({}^3{\rm D}_4(2) : \mathbb{Z}/3)\).