## The known extremal strongly modular lattices of level 2,3,5,6,7,11,14,15,23

This table was originally created by Michael Juergens during his dissertation project as: Extremal lattices It is maintained and updated here by Gabriele Nebe.

Keywords: modular lattices, tables, minimal norm, quadratic forms

Part of the Catalogue of Lattices which is a joint project of Gabriele Nebe, RWTH Aachen university (nebe@math.rwth-aachen.de) and Neil J. A. Sloane, (njasloane@gmail.com).

## Modular Lattices

A strongly N-modular lattice is an even lattice of even level N which is similar to all its partial dual lattices. Extremal strongly modular lattices are those lattices of square free level N with divisor sum dividing 24 for which the minimum is as high as the theory of modular forms allows it to be, for a precise definition see Quebbemann, H.-G. Atkin-Lehner eigenforms and strongly modular lattices. Enseign. Math. (2) 43 (1997).
2 3 5 6a 6b 7 11 14 15 23
2 - 1 - - - 1 1 - - 1
4 1 1 1 1 1 1 1 1 1 1
6 - 1 - - - 1 1 - - -
8 1 2 1 1 1 1 1 2 -
10 - 3 - - - 4 2 - - -
12 3 1 4 6 4 0 0 1 3 (5) -
14 - 1 - - - 1 ? - - -
16 1 6 1 8 ≥ 18 - ? ≥ 1 -
18 - 37 - - - 0 ? - - -
20 3 ≥ 100 ≥ 72 ? ≥ 13 ≥ 1 - ? ? -
22 - ≥ 1000 - - - ? ? - - -
24 ≥ 8 ≥ 1 ≥ 1 ≥ 2 0 - ? ? -
26 - ≥ 6 - - - ? - - - -
28 ≥ 24 ≥ 9 ≥ 1 ? ? ? - ? ? -
30 - ≥ 2 - - - - - - - -
32 ≥ 7 ≥ 33 ? ? ? - - - -
34 - ≥ 100 - - - ? - - - -
36 ≥ 3 ? ? ? ? - - - - -
38 - ? - - - ? - - - -
40 ≥ 6 ≥ 1 ? ? ? - - - -
42 - ? - - - - - - - -
44 ≥ 1 ? ? ? ? - - - - -
46 - ? - - - ? - - - -
48 ≥ 6 ? ? ? - - - - -

## Note That Unimodular Lattices Are In a Separate File

This table is based on the work of many people, including C. Bachoc, G. Nebe, H.-G. Quebbemann, E. M. Rains, R Scharlau, N. J. A. Sloane, B. B. Venkov