On automorphisms of extremal even unimodular lattices of dimension 48

2012/12/04 by Gabriele Nebe, Nebe, Gabriele
Computer Science · Mathematics · #11H31 (Secondary) #11H56 (Primary) 11H06 #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1212.0865

openalex publication_date 2012/12/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The automorphism groups of the three known extremal even unimodular lattices of dimension 48 and the one of dimension 72 are computed using the classification of finite simple groups. Restrictions on the possible automorphisms of 48-dimensional extremal lattices are obtained. We classify all extremal lattices of dimension 48 having an automorphism of order m with φ(m) > 24. In particular the lattice P_48n is the unique extremal 48-dimensional lattice that arises as an ideal lattice over a cyclotomic number field.

Citations

• The Leech lattice and other lattices

Related

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• A fourth extremal even unimodular lattice of dimension 48
• A note on Quebbemann's extremal lattices of rank 64
• Automorphisms of modular lattices
• An even unimodular 72-dimensional lattice of minimum 8