The Lattice L20
An entry from 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)
Last modified Fri Jul 18 13:17:54 CEST 2014
INDEX FILE |
ABBREVIATIONS
Contents of this file
NAME
DIMENSION
GRAM
DIVISORS
MINIMAL_NORM
KISSING_NUMBER
DENSITY
HERMITE_NUMBER
GROUP_ORDER
GROUP_NAME
GROUP_GENERATORS
PROPERTIES
REFERENCES
NOTES
LAST_LINE
-
NAME
L20
-
DIMENSION
20
-
GRAM
20 0
8
2 8
-1 0 8
0 -2 0 8
2 -1 -2 4 8
-2 0 -1 -1 -1 8
-2 -1 0 -1 -2 0 8
0 -1 0 -1 -2 -2 -1 8
0 0 -1 -2 -1 0 0 -1 8
1 1 0 2 2 0 0 0 0 8
1 4 2 -2 -1 -2 -2 2 -2 0 8
-4 -2 2 0 0 0 -1 -1 1 0 -1 8
4 0 -1 0 1 -2 2 0 -1 -2 1 -4 8
-4 1 0 -1 0 -1 0 0 -1 -1 2 2 0 8
-2 -2 -4 4 2 0 2 -1 -2 0 -2 0 0 -1 8
4 1 -2 2 2 2 -2 -2 -1 -1 -1 -4 4 -2 1 8
2 -1 -2 -1 0 0 -1 -2 -1 -2 -2 -1 0 -2 2 2 8
-2 0 0 -2 -2 -1 1 0 0 -1 2 1 -1 0 2 -2 2 8
0 0 -1 -2 -1 0 0 2 4 -4 -2 -1 0 -1 -2 0 0 1 8
4 1 -1 -2 1 -2 1 -4 0 -2 1 -1 4 0 0 2 4 2 -1 8
-
DIVISORS
7^10
-
MINIMAL_NORM
8
-
KISSING_NUMBER
6160
-
DENSITY
-
HERMITE_NUMBER
3.02
-
GROUP_ORDER
2^9 * 3^2 * 5 * 7 * 11
-
GROUP_NAME
2.M22.2
-
GROUP_GENERATORS
2
20 20
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
-13 4 -4 5 2 -1 0 6 1 -2 -2 -1 0 -7 -9 2 0 0 -5 8
-9 2 -3 3 1 -1 1 4 0 0 0 0 0 -5 -6 2 1 -1 -2 5
0 1 -1 1 1 0 1 1 1 -1 0 0 0 0 -2 0 1 1 -2 -1
-8 4 -5 5 4 -1 3 6 2 -5 -2 0 -1 -5 -11 2 1 3 -8 3
5 -2 2 -2 -1 1 0 -2 0 1 1 0 -1 3 4 0 -1 0 2 -2
13 -3 3 -3 -2 1 1 -5 0 0 1 1 -1 7 7 -1 0 1 3 -8
19 -7 8 -9 -6 2 -4 -11 -3 6 3 1 2 11 18 -4 -1 -2 11 -10
-3 1 -1 1 1 0 0 2 1 -1 -1 0 0 -2 -2 0 0 0 -2 2
3 0 0 0 0 0 1 0 1 -2 0 0 -1 1 0 0 0 1 -2 -2
-12 4 -4 5 3 -1 1 6 1 -3 -2 -1 0 -7 -10 2 1 1 -6 6
-21 7 -7 8 5 -2 2 11 2 -4 -3 -1 0 -12 -16 3 1 1 -9 11
9 -2 2 -2 -1 1 1 -4 0 0 1 1 0 5 4 -1 1 1 2 -6
-19 7 -7 8 5 -2 2 10 2 -5 -4 -1 0 -11 -17 3 1 2 -10 9
9 -2 2 -2 -1 1 0 -4 0 0 1 0 0 5 5 -2 0 1 2 -6
6 -2 2 -2 -1 1 0 -3 0 1 1 0 0 4 4 -1 0 0 2 -3
-4 0 -1 0 0 -1 -1 0 -1 1 0 -1 0 -2 -1 0 -1 -1 1 3
5 -3 2 -2 -2 0 -1 -4 -1 2 2 0 0 3 5 -1 0 -1 4 -3
11 -5 5 -6 -4 1 -3 -7 -2 5 2 1 1 7 12 -2 -1 -2 8 -5
-17 6 -7 8 5 -2 3 9 2 -5 -3 -1 -1 -10 -16 3 1 2 -9 9
20 20
-26 9 -9 11 7 -2 3 15 3 -6 -5 -2 -1 -15 -22 4 1 2 -13 15
-9 4 -4 4 3 -1 1 5 1 -3 -3 -1 0 -6 -9 1 0 2 -6 4
0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0
4 -2 3 -3 -2 1 -2 -2 -1 3 1 0 0 3 7 -1 -1 -2 4 0
3 -2 2 -2 -2 1 -1 -2 -1 3 2 0 0 2 5 -1 0 -2 4 0
0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0
-3 1 -1 1 1 0 0 1 0 -1 0 0 1 -2 -2 0 0 0 -1 1
0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
7 -2 3 -3 -2 1 -1 -4 -1 2 1 1 1 4 6 -2 0 -1 4 -4
0 -1 1 -1 -1 0 -1 -1 -1 2 1 0 0 0 2 0 -1 -1 2 1
-14 4 -5 5 2 -2 0 6 0 -1 -2 -1 0 -8 -10 2 0 0 -4 8
33 -12 12 -14 -9 3 -4 -18 -4 8 6 2 0 19 28 -5 -2 -3 17 -17
-25 8 -9 10 6 -2 2 13 2 -5 -4 -2 0 -14 -20 3 1 1 -11 14
8 -3 1 -3 -2 0 0 -5 -1 1 2 1 0 4 5 -1 0 0 3 -6
6 -2 3 -3 -2 1 -1 -3 -1 2 1 0 0 4 7 -1 -1 -1 4 -2
-25 9 -9 10 7 -2 3 14 3 -6 -5 -2 -1 -14 -21 4 1 2 -13 14
-5 3 -3 4 3 0 3 5 2 -4 -2 0 -1 -3 -8 2 1 3 -6 2
10 -3 3 -3 -2 1 0 -5 -1 2 2 1 0 6 7 -1 0 0 5 -6
9 -2 3 -3 -1 1 0 -4 0 1 1 1 1 5 6 -2 1 0 3 -6
-17 6 -7 8 5 -1 3 10 2 -5 -3 -1 -1 -10 -16 3 1 2 -9 9
-
PROPERTIES
INTEGRAL=1
MODULAR=7
-
REFERENCES
J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker,
R. A. Wilson, ATLAS of finite groups. Oxford University Press 1985, p. 39.
W. Plesken, G. Nebe, Finite rational matrix groups.
AMS-Memoirs, vol. 116, No 556 (1995), p. 67.
-
NOTES
The 7-modular lattice of 2.M22.2
This lattice is unimodular hermitian over Z[ (1+sqrt(-7))/2]
- i.e. it has the structure of a Kleinian lattice.
The group order as a real lattice is 1774080 =
[ <2, 9>, <3, 2>, <5, 1>, <7, 1>, <11, 1> ] = |2.M_22.2|
The Gram matrix as obtained from codes over Z4 is:
[ 8 4 -1 1 -1 4 0 -1 -1 -4 -1 4 1 -1 -3 -3 -4 3 -4 6]
[ 4 8 -2 -2 -1 4 0 -1 0 -4 -1 4 -1 -1 0 -3 -4 5 -1 6]
[ -1 -2 8 4 4 -1 0 -1 0 2 4 -1 2 4 -2 3 1 -1 0 -5]
[ 1 -2 4 8 0 1 4 -2 0 1 2 -3 4 3 -1 1 0 -5 1 -5]
[ -1 -1 4 0 8 -3 -4 -3 0 2 4 1 1 1 0 2 1 -1 1 -5]
[ 4 4 -1 1 -3 8 4 2 -1 -4 -3 0 -1 0 -3 -2 -4 1 -5 6]
[ 0 0 0 4 -4 4 8 1 0 0 0 -4 2 2 -1 -1 -1 -4 -1 0]
[ -1 -1 -1 -2 -3 2 1 8 -3 1 -3 0 -2 0 -2 0 0 0 -4 5]
[ -1 0 0 0 0 -1 0 -3 8 0 0 -1 1 4 2 1 4 1 5 -5]
[ -4 -4 2 1 2 -4 0 1 0 8 2 -4 3 1 2 3 4 -5 2 -6]
[ -1 -1 4 2 4 -3 0 -3 0 2 8 1 4 2 2 2 1 -3 2 -5]
[ 4 4 -1 -3 1 0 -4 0 -1 -4 1 8 -1 -2 0 0 -3 5 -1 6]
[ 1 -1 2 4 1 -1 2 -2 1 3 4 -1 8 2 3 2 1 -6 2 -5]
[ -1 -1 4 3 1 0 2 0 4 1 2 -2 2 8 -2 0 4 -1 4 -5]
[ -3 0 -2 -1 0 -3 -1 -2 2 2 2 0 3 -2 8 3 1 -3 3 -4]
[ -3 -3 3 1 2 -2 -1 0 1 3 2 0 2 0 3 8 2 -4 2 -5]
[ -4 -4 1 0 1 -4 -1 0 4 4 1 -3 1 4 1 2 8 -3 6 -7]
[ 3 5 -1 -5 -1 1 -4 0 1 -5 -3 5 -6 -1 -3 -4 -3 12 -2 8]
[ -4 -1 0 1 1 -5 -1 -4 5 2 2 -1 2 4 3 2 6 -2 12 -8]
[ 6 6 -5 -5 -5 6 0 5 -5 -6 -5 6 -5 -5 -4 -5 -7 8 -8 16]
(but that was LLL-ed)
-
LAST_LINE
Haftungsausschluss/Disclaimer
Gabriele Nebe