Odd unimodular lattices of rank 28 with no norm 1 vectors

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• 357003 isometry classes (16381 exceptional), mass 1722914776839913679032185321786744287148737/16573175467523436999761427022479360000000.
• 4722 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: B. Allombert & G. Chenevier, Unimodular hunting II, Forum Math. Sigma 13 (2025). The 38 lattices with no vectors of norm \(\leq 2\) were found by Bacher and Venkov in Réseaux entiers unimodulaires sans racine en dimension 27 et 28, Enseign. Math. 37 212–267 (2001). The masses root system by root system were 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 Allombert & Chenevier as cyclic neighbors of the standard lattice \({\rm I}_{28}\).
• Extra information: There are 38 lattices with no roots (Bacher-Venkov); their isometry groups are either of order \(p^a \cdot q^b\) (for 28 of them), or have the following orders (and a unique non-abelian composition factor): 7680 (\({\rm A}_5\)), 7680 (\({\rm A}_5\)), 15360 (\({\rm A}_5\)), 96768 (\({\rm L}_2(8)\)), 116480 (\({\rm Sz}(8)\)), 344064 (\({\rm L}_3(2)\)), 3317760 (\({\rm U}_4(2)\)), 9676800 (\({\rm J}_2\)), 696729600 (\({\rm O}_8^+(2)\)), 18341406720 (\({\rm S}_6(3)\)).