Modular Lattices, Together With A Table of the Best Such Lattices

 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 modular lattice is an integral lattice which is similar to its dual. If the similarity factor is N, the lattice is called N-modular. For example, the root lattice A2 is 3-modular.

If a lattice is N-modular its entry should indicate this by saying:
%PROPERTIES
INTEGRAL=1
MODULAR=N

Some of these entries have an additional field, %SIMILARITY, which contains an integral matrix S such that

S A S^t = N A (where A is a Gram matrix for L)
A^(-1) S in GL_n(Z).

The first condition says that S is a similarity of L that multiplies norms by N; the second says that S takes L* to L. Thus S exists if and only if the lattice is N-modular.

See the footnotes file for references.

Note That Unimodular Lattices Are In a Separate File

Table of Highest Minimal Norms of Modular Lattices

The table gives the highest possible minimal norm of a strongly N-modular (even or odd) lattice. The column headed "+" refers to lattices that are rationally equivalent to the direct sum of the appropriate number of copies of the two-dimensional lattice with Gram matrix
          (   1       0  )
          (   0       N  )
while the column headed "-" refers to the other genera of lattices.

See the footnotes file for references. Any unmarked upper bounds are from [RaSl98]. "NA" means "Not Applicable", or in other words no lattice is possible.

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

Many gaps remain! Please send any improvements to (nebe@math.rwth-aachen.de). Also, we need explicit Gram matrices for the lattices indicated by the footnotes B, C and Q!

Dim N = 2 N = 3 N = 5 N = 6 N = 7 N = 11 N = 14 N = 15
  -       + -       + -       +   -        + -       + -       +
2 1b 2      1b 2b     1b NA 2b 2b      3b 3b    NA NA
4 2 1d      2d 2      2d 2       2 3 2d      4 NA    4 4      4
6 1S 2d       2 2d       3 NA 4 4      4 4     NA NA
8 2d 2d       2d 2d       4 2d       4 4B 4      6 NA    6 4d      6
10 2 2d       3 3      3S NA 4B 6      -6 -6    NA NA
12 2d 2d       4 4     -4 4    -4 -5g -6     -7 NA    8 -7     8
14 2d   4       3C -4     -4 NA 6 -8     -8 -8    NA NA
16 4 2-3      4 -4      6 -4      6 6 -8     -9S NA    -10 -8    10
18 2-3S 4B     4C -5     -5S NA 6 -10     -9 ?     NA NA
20 4, 4, 4 2-4     4B -6     -6 -6     -6 8 -10    -10 NA    -12 -10   -12
22 2-4 4     4C -6     -6 NA -8 -12    -11 ?     NA NA
24 4 3-4      6 -6      8 -6      8 8-10 12-12 NA   8-14 -12   -14
26 2-4 6N,B  4C-5S -7     -8 NA -10 -12    -12 ?     NA NA
28 4 3-5S   6N,B 8     -8 -8     -8 -10 -13    -16 NA     -16 ?     -16
30 2-4 6     4C-6 -8     -8 NA -11 -14    -16   ?      NA   NA
32 6Q,B 3-6     6B -8    -10   -12 -15    -18 NA    -18  
34 -5 6B     4C-6 -9    -10   -12      
36 6B 3-6   6-8 -10   -10   -12      
38 -6 4-8     4-8 -10   -10   -14      
40 6B -7     8 -10   -12   -14      
42 -6 -8     -8 -11   -12   -14      
44 6B -8     -8 -12   -12   -15      
46 -6 -8     -8 -12   -12   -16      
48 8B -8 8 -10 -12   12-14   -16 18    
50                
52                
54                
56                
58                
60                
62                
64             12            
72           16-20           24-?    

Further Examples

The table contains many other examples of modular lattices, some of which are listed here:

LATTICE CATALOGUE HOME PAGE | ABBREVIATIONS