The Lattice L80
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 13:18:14 CEST 2014
INDEX FILE |
ABBREVIATIONS
Contents of this file
NAME
DIMENSION
GRAM
DET
MINIMAL_NORM
KISSING_NUMBER
DENSITY
HERMITE_NUMBER
GROUP_ORDER
GROUP_NAME
GROUP_GENERATORS
PROPERTIES
REFERENCES
NOTES
LAST_LINE
-
NAME
L80
-
DIMENSION
80
-
GRAM
80 0
8
-4 8
-4 4 8
-4 2 0 8
-4 2 4 0 8
2 -4 -4 -1 -4 8
-2 4 4 0 0 0 8
-4 0 4 1 2 0 0 8
-4 2 0 1 4 0 0 2 8
2 2 2 0 -2 0 4 -2 -2 8
-4 4 0 2 1 0 0 0 3 0 8
-4 0 4 2 2 0 0 4 0 0 0 8
2 0 -2 -3 1 -2 -3 -3 -1 -1 2 -2 8
-4 2 3 4 4 -3 1 1 2 -1 0 3 -1 8
2 -4 0 -3 0 0 0 0 -3 -2 -4 0 0 0 8
-4 1 3 0 4 -2 -1 4 4 -1 2 2 -1 0 -1 8
-3 4 1 0 3 -1 2 -1 4 1 4 -1 0 1 -3 3 8
-4 0 0 2 4 -2 -2 0 4 -3 2 2 0 3 -1 3 3 8
2 1 2 -2 2 -4 1 -1 0 1 -2 -2 0 0 1 1 1 0 8
-3 4 0 1 2 -2 2 0 4 0 3 -1 2 1 -2 2 4 1 0 8
0 -2 2 0 -1 2 2 2 -2 2 -2 2 -2 1 2 0 -2 -2 -2 -2 8
2 -1 -2 -2 1 -1 -2 -3 1 -2 -1 -3 1 -1 0 0 1 2 3 -1 -1 8
2 -2 0 -2 2 0 0 -2 0 2 -1 0 2 1 2 0 0 0 1 0 3 2 8
-2 3 4 0 3 -3 2 1 0 2 2 2 1 1 0 1 1 0 2 1 -2 -3 1 8
0 0 2 -1 3 -2 0 -1 0 2 0 2 1 0 0 2 1 2 2 0 -2 -1 1 4 8
-2 -2 0 2 0 2 -1 2 0 0 0 3 -2 0 -1 2 0 1 -3 -1 0 -3 -2 -1 1 8
-2 3 3 1 -1 -1 2 2 -2 0 1 2 -1 0 0 0 -1 -2 -1 0 -1 -4 -4 3 1 0 8
0 -2 1 1 0 2 0 1 0 2 0 2 -3 1 0 1 0 0 0 -3 1 -1 2 1 2 3 0 8
-2 -2 0 1 0 0 -2 2 0 -2 1 2 1 1 0 1 -1 2 -1 -1 2 0 -1 -1 0 2 0 0 8
0 1 -2 2 0 0 -2 -2 2 0 3 -1 0 0 -2 0 1 1 0 0 -1 3 1 -1 0 -1 -1 2 1 8
-2 1 -1 0 3 -2 -2 1 3 -3 1 0 2 1 0 1 2 2 -1 4 -1 2 1 -1 -1 -1 -2 -2 0 2 10
-4 1 4 2 2 -1 2 5 0 0 1 3 -1 3 0 2 0 0 -1 1 4 -4 0 2 0 1 2 0 4 -2 0 10
2 0 -2 1 -2 1 -2 0 0 0 1 -2 1 0 -2 -2 -1 -2 -1 0 1 0 -1 -2 -3 -1 0 0 0 2 0 -1 8
-4 -1 2 0 4 0 1 3 4 -3 0 2 -2 2 1 4 2 4 1 1 1 0 0 1 2 2 0 2 3 -1 1 2 -1 10
-4 1 -1 4 2 -1 -3 0 3 -3 4 2 0 3 -2 2 2 5 -1 1 -4 0 -2 1 0 2 1 1 2 2 0 0 0 1 10
2 1 0 -1 -3 0 1 -2 -4 1 0 -1 1 -2 1 -2 -1 -3 0 -1 -1 0 -2 0 1 -1 3 0 -2 0 -2 -3 1 -2 -2 8
0 -2 1 2 0 1 1 2 -1 1 -3 2 -4 1 1 0 -2 0 0 -2 3 0 2 -1 -1 2 -1 2 0 0 -1 2 -1 0 -1 -2 8
-3 2 0 2 3 -2 -2 2 3 -3 3 0 1 1 -2 2 2 3 0 2 -1 2 0 1 -2 -2 0 -2 1 2 4 2 1 1 3 -3 1 8
-2 0 2 -2 1 1 2 1 0 0 0 2 0 0 1 2 2 1 -1 0 3 1 1 -1 0 1 0 0 2 -1 1 2 -2 2 -1 0 0 0 8
-4 4 2 1 4 -4 0 1 3 -1 3 1 3 3 -2 3 3 2 0 5 0 1 1 0 -1 0 -1 -3 1 1 5 3 0 0 1 -2 0 4 2 10
-4 2 4 -1 4 -2 1 3 3 -1 1 3 0 1 -1 5 2 2 1 2 0 -1 -1 1 2 2 1 -1 2 -2 0 2 -3 4 1 -2 -1 0 2 3 10
-2 3 3 1 0 0 2 1 -1 3 3 2 -1 0 -3 0 2 0 -2 -1 0 -3 -2 2 1 2 3 2 0 0 -1 1 0 -1 1 1 0 0 0 -1 2 10
-4 0 2 1 4 1 1 4 5 -1 1 2 -2 3 0 4 2 2 -2 2 2 -2 1 0 0 2 -1 1 1 -1 1 3 -1 4 1 -4 1 1 1 2 5 0 10
-4 4 2 3 0 -3 1 0 -1 -1 3 2 0 1 -2 0 1 2 0 1 -2 0 -3 2 0 0 3 -2 2 0 0 1 -2 0 3 1 -1 2 0 2 3 2 -2 10
-3 3 0 1 3 -2 1 0 5 0 2 -1 0 0 -2 3 4 3 0 5 -3 0 0 1 0 0 -2 -3 -3 0 3 -1 -2 0 2 -3 -1 2 0 3 1 -1 1 -1 12
-1 2 2 2 -2 0 2 1 -1 3 1 2 -2 0 -2 0 0 -1 0 0 1 -1 -2 0 0 1 2 1 1 1 -2 0 1 -1 0 2 1 -1 1 0 0 3 -2 2 0 8
-4 3 3 1 1 0 3 3 3 0 2 2 -3 1 -2 3 3 1 1 2 0 0 -1 0 -2 1 0 0 0 -1 -1 1 -2 2 1 -1 1 1 1 2 5 1 4 4 1 2 10
1 0 -1 -1 1 1 0 -1 0 1 2 -1 2 -1 0 0 2 -1 1 1 1 0 2 0 1 -1 -1 1 0 2 3 1 1 0 -1 0 -1 1 2 1 0 0 1 -2 -1 -2 -2 10
0 2 0 0 1 -1 2 -2 1 0 1 -3 1 2 -1 -1 2 0 2 2 1 1 1 0 -1 -2 -1 -2 0 0 0 2 2 1 -1 -1 0 1 0 3 1 0 0 0 0 -1 0 2 10
1 0 -2 -2 0 2 -1 -1 2 -1 1 -2 0 -3 -2 1 2 0 0 0 -2 2 -1 -1 1 0 0 0 -1 2 1 -2 0 1 -1 0 -1 1 0 0 1 -1 1 -1 0 -2 1 1 0 8
-2 1 -1 2 -1 2 0 0 0 0 4 0 1 -1 -2 0 0 0 -4 0 0 -1 -1 0 -2 2 1 1 0 1 -1 0 1 -1 2 1 0 2 1 1 -2 2 -1 0 1 2 -1 -1 -1 0 10
2 -2 -2 1 -1 1 -2 -2 -2 0 -1 -1 1 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 3 0 1 -1 -1 -2 -2 2 0 0 2 -1 -3 -2 -2 -2 -1 -2 -2 0 0 -3 0 1 0 1 10
2 1 0 -2 -3 -1 0 -2 -4 1 0 0 3 -2 0 -2 -1 -2 0 0 -2 -2 -2 1 2 0 4 -1 0 -2 -2 -1 0 -2 -1 4 -3 -3 0 -1 0 2 -4 2 -3 2 -1 -1 -1 0 0 1 10
-2 2 2 0 -1 0 2 1 0 1 2 2 0 0 -1 1 1 0 0 1 -1 -2 -2 1 0 1 2 0 0 -2 -3 0 -1 0 1 2 -2 -2 1 0 2 2 0 2 0 3 4 -2 -1 -2 1 0 4 8
-1 -2 2 0 1 1 0 4 0 -2 -2 3 -2 1 1 0 -2 1 0 -2 2 -1 0 0 0 0 0 0 1 -2 0 2 -1 2 -2 -1 2 0 1 -2 1 0 2 -1 0 1 2 0 0 -1 -2 1 -1 1 10
-1 1 2 2 2 -1 0 0 -1 1 0 3 -1 2 -1 -1 0 1 0 -2 0 0 1 2 1 1 0 2 -1 1 0 0 0 0 1 0 2 1 -1 0 0 2 0 2 -2 -1 0 0 -1 0 1 -1 -1 -1 -1 8
0 -1 1 0 1 -1 1 0 -1 -1 -2 -1 -2 1 2 -1 -1 1 1 -2 -1 0 -1 1 2 1 0 1 -1 -2 -1 0 -1 2 -1 1 1 -2 -1 -2 0 1 -1 1 0 0 -1 -3 2 0 -1 2 0 -1 1 1 10
0 -1 -1 -2 1 -1 -3 1 2 -4 -1 -1 2 0 0 1 -1 2 0 0 -2 3 -2 -2 -1 -1 -1 -3 2 0 2 -1 0 0 1 -1 -2 1 1 1 1 -2 1 -1 0 -1 0 -3 -1 2 -1 -1 0 0 1 -2 0 10
0 1 2 1 -2 0 3 0 -3 2 -2 1 -2 0 0 -1 -2 -3 0 -1 1 -3 -2 0 1 2 3 0 0 -2 -4 2 -2 -1 -2 2 1 -4 0 -1 1 1 -1 1 -2 2 1 -2 0 -1 -1 1 4 3 1 -1 0 -2 10
3 -4 -2 -2 -1 0 -4 0 -2 -3 -2 -1 1 -1 3 -1 -2 1 1 -3 -2 0 0 0 0 0 1 2 0 -1 -1 -2 1 1 1 0 0 0 -3 -4 -2 -1 -1 -1 -3 -3 -2 0 -2 0 -2 2 2 -1 2 0 0 0 -2 12
-4 0 0 5 1 0 0 1 2 -1 1 2 -3 3 -1 1 1 4 -2 0 -1 0 -2 -1 -1 3 -1 1 1 0 -1 -1 -1 1 5 -1 2 1 0 0 0 0 3 1 2 1 2 -2 -2 -2 2 0 -3 2 1 1 -1 0 1 0 12
-4 2 0 4 2 -2 1 -1 1 -2 3 -1 0 2 -1 1 2 3 0 2 -1 2 -1 -1 0 0 0 -1 2 2 1 2 -2 1 3 0 0 2 1 3 1 -1 1 4 1 0 2 1 1 0 1 -1 0 1 -1 0 0 0 3 -3 5 12
-1 1 1 -2 -1 -1 -1 0 -2 0 3 1 2 -2 1 2 2 1 -1 0 -1 -1 -2 1 1 0 3 0 1 -1 0 0 -1 -1 2 2 -4 0 2 0 2 3 0 3 -1 0 1 1 -2 1 1 -1 3 2 -2 0 0 1 -3 2 -2 -1 12
0 0 2 -2 3 -2 -1 0 1 0 0 0 0 0 0 3 2 2 3 -1 0 4 1 0 1 0 -2 0 1 1 0 -1 0 2 1 -1 0 1 1 1 1 0 0 -1 0 1 1 -1 1 0 -1 0 -1 1 0 0 -1 2 0 0 2 1 -1 10
0 0 2 0 3 -1 2 -1 1 3 0 1 -1 2 0 1 3 2 4 0 -1 -1 3 4 4 0 -1 3 -1 -1 -3 1 -2 2 2 -2 1 -1 -1 -2 0 0 1 -1 2 -1 1 2 0 -1 -3 1 -1 0 1 1 0 -4 0 2 1 0 -1 1 12
1 -2 1 -1 2 -1 0 -1 -2 -1 -2 0 1 1 3 0 -2 0 2 -2 2 2 2 -1 2 0 -1 0 2 0 -1 0 -1 2 -2 1 1 -2 1 0 2 -1 1 -1 -4 -1 -1 1 2 0 -1 1 0 -1 1 1 2 1 1 -1 -1 2 -1 2 -1 10
0 -1 -2 0 0 2 -1 -1 0 0 2 -1 1 -2 -1 1 2 1 -2 0 1 1 1 -2 0 2 -2 1 1 2 1 0 0 0 0 0 1 1 2 1 1 2 1 0 0 0 0 3 0 2 2 -1 -1 -2 -1 0 -1 -1 -1 -1 0 2 1 -1 0 0 8
-1 -1 1 2 -1 2 0 2 0 1 1 2 -3 0 -2 1 -1 0 0 -3 2 0 -1 0 1 2 1 3 3 1 -3 2 1 2 1 0 2 0 1 -1 2 1 2 2 -4 3 2 1 1 1 0 0 -1 0 2 1 1 -1 1 -1 1 1 0 0 0 1 1 10
0 0 -2 1 -3 1 -1 -1 -2 -2 1 -1 1 -1 0 -2 -2 -1 -2 -1 -1 -1 -3 0 -1 -1 2 -1 0 -1 -1 -1 1 -1 1 3 -3 -1 -1 -2 -3 0 -4 1 0 1 -3 -2 0 -1 3 2 2 1 0 -2 1 0 1 0 0 0 1 -2 -3 -1 -1 -1 10
-3 0 -1 3 1 2 -2 2 3 -3 2 2 -2 2 -2 2 2 3 -2 0 0 1 -2 -2 -2 1 0 0 1 1 1 0 2 2 4 -1 0 3 2 1 0 -1 3 1 -1 0 2 1 -1 1 1 0 -2 1 1 1 -2 2 -2 0 4 2 0 1 -1 -1 0 3 -1 10
-3 2 -1 4 2 -3 -2 1 3 -4 2 -2 1 2 -2 2 1 2 1 3 -4 1 -2 1 -2 -1 1 -2 0 1 2 0 0 1 4 -1 -2 4 -2 2 0 -3 0 2 5 -1 1 -2 1 0 2 2 -2 -1 0 -1 0 2 -2 0 2 4 -1 -1 0 -1 -1 -2 2 1 14
0 1 0 -2 1 -2 0 -1 0 -1 1 -1 3 0 1 0 2 1 2 2 -3 1 0 2 1 -1 1 -1 1 0 1 0 -1 0 1 1 -1 1 1 2 0 -1 -2 2 0 -1 0 0 1 0 0 -1 1 0 -2 0 0 1 -2 2 -1 0 2 0 1 0 0 -2 -1 -1 2 8
3 -3 -2 -4 -2 2 -1 0 -1 -2 -3 -1 1 -2 3 -2 -2 -1 0 -1 1 1 0 -3 -2 -1 -1 -1 0 -1 2 -1 1 0 -3 0 -1 -1 2 -1 0 -2 -1 -2 -1 -1 -1 1 1 1 -2 -1 0 -1 2 -2 1 2 -1 1 -3 -3 1 -1 -3 0 1 -1 0 -2 -1 1 10
-2 2 0 1 -1 2 3 0 2 0 1 1 -3 1 0 0 2 0 -1 2 0 -2 -2 0 -1 0 3 0 -1 0 -1 1 0 2 2 0 0 0 0 0 1 1 2 0 0 0 2 0 1 1 0 -2 1 2 -2 0 -2 -2 2 0 1 0 0 -1 0 -2 -1 0 0 2 -2 0 -1 10
0 2 2 -2 0 -2 3 0 0 0 -2 -1 0 0 1 0 0 -1 3 1 -1 0 -1 1 0 -2 2 -1 -1 -2 -1 0 -2 1 -2 1 -1 -1 1 0 2 -1 -1 1 1 0 1 -1 1 0 -1 -1 1 1 0 -1 1 0 1 0 -2 -1 1 -2 0 0 -2 -2 1 -3 2 2 2 1 8
1 -2 -1 2 -3 3 2 1 -1 3 -2 0 -3 0 1 -1 -1 -3 -2 0 5 -2 1 -3 -3 2 -2 1 0 -1 -1 1 2 0 -3 0 3 -2 0 -1 0 0 2 -2 0 2 1 1 1 -2 0 0 -3 -1 0 0 -1 -4 1 -2 2 -1 -3 -1 -1 0 2 2 -1 -1 -2 -3 1 1 -1 12
1 -1 -3 1 0 0 -3 -1 0 -1 0 0 3 0 -1 -2 -1 1 -2 1 0 1 1 -2 -2 0 -2 -3 1 1 4 -1 2 -2 0 -2 1 3 0 3 -2 -1 0 -1 1 0 -2 1 0 0 2 0 0 -1 1 1 -2 2 -2 0 1 0 -1 0 -3 0 1 -1 -1 2 0 0 1 -2 -3 0 10
2 -3 0 -1 -3 1 -1 1 -2 0 -2 0 -2 -2 2 0 -4 -1 1 -4 0 -1 -2 -1 1 2 1 3 2 0 -5 -1 0 0 0 1 0 -4 -1 -5 0 1 -1 -1 -4 1 -1 -3 -2 -2 0 0 2 2 1 -1 2 1 3 2 0 -2 1 0 -1 2 -1 2 1 -2 -2 -1 2 -1 1 -1 -3 12
1 1 1 0 -2 -1 2 0 -1 1 -2 -2 -3 0 1 -1 0 -2 2 -1 2 1 -1 -2 -3 -2 0 -1 -1 0 0 0 1 -1 -2 1 0 -1 0 -1 -1 -1 -2 0 0 0 1 0 2 -1 -3 -1 -1 0 0 -1 0 -1 1 0 0 0 -1 1 -1 -1 -2 -1 0 -1 0 -1 1 1 1 3 -2 0 10
2 -3 -2 1 0 -1 -4 -2 -2 -1 -1 0 2 0 0 -1 -3 2 1 -3 -1 3 1 0 1 0 -2 0 2 1 -1 -2 0 -1 2 0 1 1 -2 -1 -2 -1 -3 1 -3 -1 -3 -2 -1 -1 1 2 -1 -2 -1 2 1 2 -2 1 1 0 -1 2 0 3 0 0 1 0 2 1 -1 -4 -2 -2 2 2 -2 10
-
DET
1
-
MINIMAL_NORM
8
-
KISSING_NUMBER
1250172000
-
DENSITY
-
HERMITE_NUMBER
8
-
GROUP_ORDER
2^12 * 3^4 * 5^2 * 7^2 * 11
-
GROUP_NAME
2.M22.2 otimes(sqrt(-7)) 2.Alt7
-
GROUP_GENERATORS
6
80 80
1 2 1 1 0 2 -1 1 0 1 -1 -1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
-6 1 1 -1 0 3 -4 -1 -1 2 -2 -2 0 0 0 -1 -1 0 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
-2 1 -1 0 0 1 0 0 0 0 -1 0 1 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
-1 -1 -1 -2 0 -1 0 0 -1 0 1 0 -2 1 -1 0 -1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
0 -1 1 1 -1 -1 1 -1 2 -1 0 1 1 -1 0 0 1 -1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
2 0 -4 -2 2 -2 2 1 -2 0 2 1 -2 1 -1 1 -1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
-2 1 -4 -2 2 0 0 1 -2 1 0 0 -1 1 -1 0 -1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
-1 0 -2 -1 0 -1 1 0 0 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
-1 -1 0 0 0 -1 0 -1 1 0 0 1 0 -1 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
-1 2 -1 0 1 2 -1 1 -1 1 -1 -1 0 0 0 0 -1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
-3 0 0 -1 0 1 -2 -1 -1 1 0 -1 -1 0 0 0 -1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
2 -1 -1 0 0 -2 3 0 0 -2 1 1 0 0 -1 1 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
0 0 3 2 -1 1 -1 -1 1 0 -1 -1 1 -1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
-1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
3 -1 -1 1 0 -2 3 1 1 -1 1 1 1 0 0 0 1 0 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
0 -2 1 1 -1 -1 1 -1 2 -1 0 1 1 -1 0 0 1 -1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
-3 0 2 0 0 2 -3 -1 0 1 -1 -1 0 -1 0 -1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
2 -2 2 1 -1 -2 1 -1 2 -2 1 1 0 -1 0 0 1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
-1 2 2 1 -1 3 -2 0 1 1 -2 -1 2 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
-3 -1 2 0 0 1 -2 -1 0 1 -1 -1 0 -1 0 -1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
2 0 -3 0 1 -1 3 2 -1 0 1 1 0 1 0 1 0 0 -1 -1 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
4 1 5 4 -3 1 -1 1 3 -1 0 0 4 -1 2 -1 2 -1 -1 0 1 2 -1 0 1 0 0 0 -1 0 0 0 -1 1 1 -1 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
1 1 -5 0 0 -2 2 6 -1 1 1 0 2 1 0 -1 1 0 0 -1 2 2 0 0 1 1 1 -1 -2 1 -1 -1 -1 1 1 -1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
-5 1 -5 -3 1 0 -1 2 -2 2 -1 -1 -1 1 -1 -1 -1 1 2 0 1 0 1 0 0 1 1 -1 -1 1 -1 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
1 -1 -1 0 0 -1 3 -1 0 -2 0 1 -1 0 -1 2 0 0 0 -1 -1 -1 1 0 0 0 0 0 1 0 0 0 1 -1 -1 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
6 -3 5 3 -1 -2 3 -5 3 -4 2 3 0 -2 0 2 1 -1 -2 -1 -2 -1 -1 0 -1 -1 -1 1 1 -1 1 1 1 0 -1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
-13 2 -5 -6 1 3 -6 2 -4 5 -3 -4 -2 2 -1 -3 -2 1 4 1 1 -1 2 -1 1 1 1 -1 -1 1 -1 0 0 0 0 1 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
-5 2 -7 -4 1 0 -1 4 -3 2 0 -1 -1 2 -1 -1 -1 1 2 0 2 0 1 0 1 1 1 -1 -1 1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
-5 3 2 0 -3 5 -3 2 -1 2 -3 -3 2 1 2 -1 0 2 2 0 0 1 2 -1 2 2 1 -1 -1 1 0 0 0 0 0 0 -1 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
-10 5 3 0 -3 8 -8 2 0 5 -4 -5 3 0 3 -3 -1 2 4 1 1 2 2 -1 2 2 2 -1 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
-11 2 -1 -3 1 3 -8 1 -1 6 -2 -3 -1 0 1 -4 -1 1 3 2 0 1 2 -1 0 1 2 -1 0 0 -1 0 0 0 -1 1 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
0 0 -1 0 0 0 1 1 -1 0 0 0 -1 0 0 0 0 1 0 -1 -1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
-2 0 6 2 -2 4 -2 -3 1 1 -2 -1 2 -1 2 1 0 0 0 0 -2 0 -1 0 0 -1 -1 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
-14 1 -8 -7 1 2 -3 3 -4 5 -3 -2 -1 3 -1 -1 -1 1 4 0 0 0 3 -1 1 2 2 -2 -1 2 -1 0 1 -1 -1 1 -1 1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
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-2 4 8 9 -7 2 -4 13 5 5 -1 -6 8 -4 6 -9 1 3 -1 -1 4 6 2 -1 2 4 1 -3 -4 0 -2 -3 -6 2 5 -1 -2 0 0 1 2 -1 -2 -4 -3 2 0 0 2 2 -3 2 -1 -1 -1 1 0 -1 0 0 -2 1 0 0 1 -1 0
-2 1 1 -1 1 0 0 0 0 1 1 -1 -1
82 -18 -60 -10 63 -50 68 -8 -12 -17 40 52 -35 -6 -24 23 -6 -3 -13 -3 2 -22 -15 2 -33 -14 -17 30 21 -5 6 -2 0 -12 -5 19 0 13 8 -6 4 -9 11 14 9 -4 -14 -11 7 3 -4 2 9 0 -1 -6 -1 5 3 2 1 -2 1 0 -2 1 -2
0 2 0 2 1 -2 1 -2 1 -1 2 -1 -1
13 -12 12 1 -4 -6 11 -15 3 -13 1 4 -6 1 -4 9 2 -6 -7 -4 -11 -6 1 1 1 -5 -6 -1 3 -1 3 4 6 1 -2 -1 2 0 0 1 0 1 0 2 1 2 0 2 -1 0 2 -1 -1 2 0 0 -2 0 0 0 1 0 1 -1 -1 0 0
0 0 0 1 0 0 0 -1 0 0 0 0 1
-92 29 62 14 -56 54 -99 5 36 38 -40 -56 33 0 35 -39 2 2 14 8 3 33 13 1 24 19 24 -33 -26 -4 -11 10 -6 18 4 -15 4 -20 -4 6 -5 10 -19 -15 -14 7 13 12 -6 1 -1 -2 -11 0 3 6 -1 -7 -5 0 0 4 2 -1 1 0 3
-4 -2 -1 -1 -2 1 -3 0 -1 1 1 -2 -1
16 -20 -6 -7 7 -22 17 -1 -8 -5 10 6 -15 7 -11 9 2 -1 -15 -1 -12 -2 2 3 -7 -1 1 -6 1 2 -1 -1 1 2 -2 -3 -3 -2 -1 -1 2 -4 -6 -6 -5 3 3 3 1 -1 0 -3 -2 0 1 0 -1 -2 -1 -2 0 1 1 -2 -2 -1 1
0 1 0 0 0 -1 0 0 0 0 0 -2 0
-61 25 27 4 -30 32 -64 4 23 26 -23 -32 21 -1 24 -27 -2 1 13 8 7 20 5 -1 14 12 14 -15 -17 -3 -9 8 -5 10 1 -8 3 -11 -1 2 -4 8 -10 -8 -7 3 7 5 -4 1 -2 0 -5 -1 2 4 -1 -4 -3 0 1 2 1 0 1 1 2
-1 -1 0 0 -1 1 -2 0 -1 1 0 0 0
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PROPERTIES
Unimodular=1
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REFERENCES
C. Bachoc, G. Nebe: Extremal lattices of minimum 8 related to the Mathieu group M22, submitted
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NOTES
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LAST_LINE
Haftungsausschluss/Disclaimer
Gabriele Nebe