An updated table of extremal modular lattices has been established by Michael Juergens during his dissertation project: Extremal 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)**.

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

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.

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 | 7-8 6-8 | -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-? |

**Unimodular lattices are in a separate file**- All 2-d integral lattices are modular.

For example, the lattice [3,1;1,5] is 14-modular. -
The 3-d
**"Mean-centered cuboidal" lattice**, although similar to its dual, is not integral and is therefore not modular. - Some 2-modular lattices:
The 16-d
**Barnes-Wall lattice BW16**,**the odd Barnes-Wall lattice**; - Some 3-modular lattices:
- The 2-d root lattice
**A2** - The 10-d
**shorter Coxeter-Todd lattice** - The 12-d
**odd Coxeter-Todd lattice** - The complete list of
**even**3-modular lattices in 12 dimensions. Their names are determined by their (modular) root lattices.**The Coxeter-Todd lattice K12****A1^6.sqrt(3)A1^6****A2^3.sqrt(3)A2^3****A3^2.sqrt(3)A3^2****A6.sqrt(3)A6****D6.sqrt(D6)****E6.sqrt(3)E6****G2.A5.sqrt(3)A5****G2^2.D4.sqrt(3)D4****G2^6**

- The 2-d root lattice
- Two 15-modular lattices, one in
**dimension 12 with minimal norm 6**, another in**dimension 16 with minimal norm 8**.