The Lattice M23
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 May 17 15:11:42 CEST 2019
INDEX FILE |
ABBREVIATIONS
Contents of this file
NAME
DIMENSION
DET
MINIMAL_NORM
KISSING_NUMBER
GRAM
GROUP_ORDER
GROUP_NAME
GROUP_GENERATORS
PROPERTIES
NOTES
REFERENCES
LAST_LINE
-
NAME
M23
-
DIMENSION
23
-
DET
6
-
MINIMAL_NORM
4
-
KISSING_NUMBER
75900
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GRAM
23 23
4 2 1 -2 1 -2 -1 -2 -2 2 -1 -2 1 0 -1 -1 1 -2 -1 -2 -1 -2 -2
2 4 2 0 2 -2 -1 0 -2 2 0 -2 -1 -1 1 1 -1 -2 1 0 -1 0 0
1 2 4 0 2 0 -2 1 -2 1 1 -2 0 1 1 1 -2 -2 -1 -1 -2 -1 1
-2 0 0 4 0 0 -1 2 2 -2 0 2 0 -1 2 1 -1 0 1 2 1 2 2
1 2 2 0 4 -1 0 0 -2 1 -1 -2 -1 -1 2 2 -2 -1 0 -1 0 0 0
-2 -2 0 0 -1 4 1 0 1 -2 2 1 1 2 -1 1 0 1 -1 1 -1 0 1
-1 -1 -2 -1 0 1 4 -1 0 -1 -1 0 0 0 -1 1 1 1 0 1 1 0 0
-2 0 1 2 0 0 -1 4 1 0 1 1 -1 -1 2 1 -2 1 0 1 1 2 1
-2 -2 -2 2 -2 1 0 1 4 -2 1 2 0 -1 0 -1 0 2 0 1 1 2 1
2 2 1 -2 1 -2 -1 0 -2 4 0 -2 -1 -1 0 -1 -1 -1 0 -1 0 -1 -2
-1 0 1 0 -1 2 -1 1 1 0 4 0 0 1 0 0 -1 0 0 0 -1 0 1
-2 -2 -2 2 -2 1 0 1 2 -2 0 4 1 0 0 0 1 1 1 2 1 1 0
1 -1 0 0 -1 1 0 -1 0 -1 0 1 4 1 -2 -1 2 -1 -2 0 -1 -2 0
0 -1 1 -1 -1 2 0 -1 -1 -1 1 0 1 4 -1 0 1 -1 -1 0 -1 -2 0
-1 1 1 2 2 -1 -1 2 0 0 0 0 -2 -1 4 2 -2 0 1 0 1 2 1
-1 1 1 1 2 1 1 1 -1 -1 0 0 -1 0 2 4 -1 0 1 1 0 1 1
1 -1 -2 -1 -2 0 1 -2 0 -1 -1 1 2 1 -2 -1 4 0 0 0 0 -1 -1
-2 -2 -2 0 -1 1 1 1 2 -1 0 1 -1 -1 0 0 0 4 0 0 1 2 0
-1 1 -1 1 0 -1 0 0 0 0 0 1 -2 -1 1 1 0 0 4 1 1 1 0
-2 0 -1 2 -1 1 1 1 1 -1 0 2 0 0 0 1 0 0 1 4 1 1 1
-1 -1 -2 1 0 -1 1 1 1 0 -1 1 -1 -1 1 0 0 1 1 1 4 1 -1
-2 0 -1 2 0 0 0 2 2 -1 0 1 -2 -2 2 1 -1 2 1 1 1 4 1
-2 0 1 2 0 1 0 1 1 -2 1 0 0 0 1 1 -1 0 0 1 -1 1 4
-
GROUP_ORDER
991533312000 = 2^11 3^7 5^3 7 11 23
-
GROUP_NAME
2.Co3
-
GROUP_GENERATORS
3
23 23
2 -3 3 3 -1 -1 2 -2 -2 1 2 1 -2 -1 0 0 1 1 -1 1 0 1 0
1 -2 2 3 -1 0 1 -2 -2 1 1 1 -2 -1 0 0 1 1 -1 0 0 0 0
-1 1 -2 0 0 0 -1 0 -1 -1 0 0 -1 0 0 0 0 0 -1 0 0 -1 0
-6 5 -1 -6 -1 0 -3 0 3 -2 -3 -1 3 0 1 1 -1 -1 0 -1 1 -1 -1
0 0 0 1 0 1 0 0 -1 0 0 0 -1 0 0 -1 1 0 0 0 0 0 0
3 -2 -2 2 2 0 1 2 -1 1 1 0 -1 1 -1 -1 0 0 1 0 -1 0 1
4 -3 -1 3 2 1 1 2 -1 2 1 0 -1 1 -1 -1 1 0 1 0 -1 1 1
-3 3 -3 -3 0 0 -2 1 2 -1 -2 -1 2 1 0 1 -1 -1 0 -1 0 -1 0
-4 3 0 -6 -1 -1 -2 0 3 -2 -2 -1 4 0 1 1 -2 -1 1 0 1 0 -1
2 -2 2 4 -1 0 2 -2 -2 1 2 1 -2 -1 0 0 1 1 -1 0 0 0 0
0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 -1 0
-2 2 -1 -3 0 0 -1 1 3 0 -2 -1 2 1 0 1 -1 -1 1 -1 0 0 0
0 0 -1 0 0 0 0 1 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0
-1 1 -2 -1 1 0 -1 1 0 -1 0 0 -1 0 0 0 0 -1 -1 0 0 -1 0
-5 4 0 -3 -1 0 -2 -1 2 -1 -2 0 1 0 0 1 0 -1 -1 -1 1 -1 -1
2 -1 -1 3 1 1 1 1 -1 2 0 1 -2 1 -1 -1 1 0 0 -1 -1 0 1
1 -1 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0
-1 1 -1 -2 0 0 -1 1 1 -1 -1 -1 2 0 0 0 -1 -1 1 0 0 0 0
-1 1 1 0 0 1 0 -1 1 1 -1 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 -1 0 0 0
-3 2 0 -3 0 0 -1 0 2 -1 -1 -1 2 0 1 0 0 -1 0 0 0 0 -1
0 0 1 -1 0 0 0 0 1 0 -1 0 1 0 0 0 0 0 1 0 0 0 0
-2 2 -2 -2 0 0 -2 1 0 -2 -1 -1 1 0 0 0 -1 0 0 0 0 -1 -1
23 23
4 -3 0 4 1 0 2 0 0 3 1 0 -1 2 -1 0 0 1 1 0 -1 1 1
2 -1 0 2 1 0 1 0 0 2 0 0 0 2 -1 0 0 1 1 0 0 1 1
1 0 1 0 0 0 0 -1 0 1 0 1 0 1 0 0 0 1 0 0 1 1 1
-3 3 -1 -5 -1 0 -2 1 1 -3 -2 -1 3 0 1 0 -2 -1 1 0 1 0 -1
0 1 -1 -2 1 1 -1 1 1 -1 -1 -1 1 1 1 -1 0 0 1 1 0 0 0
-2 2 -1 -3 -1 1 -2 0 0 -2 -1 0 0 -2 1 0 0 -1 -1 0 1 -1 0
1 -1 -1 1 1 0 0 1 -1 -1 1 -1 -1 -1 0 -1 1 0 0 1 -1 -1 0
-2 2 0 -3 -1 0 -1 0 0 -1 -1 0 2 0 1 0 -1 0 0 0 1 0 0
-2 1 -1 -2 -1 0 -1 1 0 -2 -1 -1 1 -1 1 0 -1 -1 0 0 0 -1 -1
2 -2 2 3 1 0 2 -1 0 3 1 1 -1 1 -1 0 1 1 0 0 0 1 1
-1 0 0 0 -1 0 -1 -1 -1 0 0 0 0 -1 0 1 0 0 -1 0 1 0 0
-5 5 -2 -6 -2 1 -3 1 2 -3 -3 -1 3 -1 1 1 -2 -2 0 -1 1 -1 -1
1 -1 0 0 -1 0 0 0 0 0 0 -1 1 0 0 0 -1 0 1 0 0 1 0
3 -2 1 3 0 0 1 -1 -2 1 2 2 -3 -1 -1 0 1 1 -1 0 0 0 1
-4 4 -2 -5 0 1 -3 1 1 -3 -2 -1 2 0 1 0 -1 -1 0 0 1 -1 -1
-3 4 -3 -5 0 1 -3 1 1 -3 -2 -1 2 0 1 0 -1 -1 0 0 1 -1 0
1 -1 -1 3 0 0 1 0 -1 1 1 0 -1 0 -1 0 0 0 0 -1 -1 0 0
-2 2 -2 -1 1 0 -1 1 1 0 -1 0 1 1 -1 1 0 -1 0 -1 0 0 0
-2 2 0 -3 -1 0 -1 0 0 -2 -1 0 1 -1 0 0 -1 -1 0 0 1 0 0
2 -2 0 1 1 0 1 1 -1 0 1 0 0 0 0 -1 0 0 1 1 -1 0 0
-4 3 -2 -4 0 0 -2 1 1 -2 -2 -1 2 0 1 0 -1 -1 0 0 0 -1 -1
-2 1 1 -1 -1 -1 -1 -1 -1 -2 0 0 0 -1 0 0 0 0 -1 0 1 0 -1
23 23
2 -1 2 0 0 0 1 -1 -1 -1 1 1 -1 -1 1 -1 0 1 0 1 0 0 0
6 -5 3 6 0 0 4 -1 -4 1 3 1 -3 -1 0 -2 1 2 0 1 -1 1 0
4 -3 2 3 0 1 2 0 -2 1 1 1 -2 -1 0 -1 1 1 0 0 0 0 1
1 -2 0 2 0 -1 1 0 0 2 0 0 0 1 -1 1 0 0 0 0 -1 1 0
5 -4 1 6 1 0 3 0 -3 2 2 1 -3 0 -1 -1 1 2 0 0 -1 0 1
-2 2 -2 -3 0 1 -2 2 2 -1 -2 -2 2 0 1 0 0 -1 1 0 0 -1 0
0 1 -1 1 1 1 0 1 0 0 0 0 -1 0 0 -1 1 0 0 0 0 -1 0
3 -3 0 4 1 0 2 1 -1 3 1 1 -2 1 -2 0 1 0 0 -1 -1 1 1
-1 0 -2 -1 1 -1 -1 1 2 1 -1 -1 2 2 -1 1 -1 -1 1 0 -1 1 0
2 -2 2 2 0 -1 2 -1 -2 0 2 1 -1 0 0 -1 0 1 0 0 0 1 0
3 -2 -1 1 1 1 1 2 0 1 0 -1 1 1 0 -1 0 0 2 0 -1 1 1
-5 3 -3 -5 0 -1 -3 1 3 -1 -2 -2 3 1 0 2 -1 -2 0 0 0 0 -1
-3 3 0 -5 0 0 -2 0 3 -1 -2 0 2 0 1 1 0 -1 0 0 1 0 0
-2 3 -1 -4 -1 2 -2 1 2 -2 -2 -1 1 -1 2 0 0 -1 0 0 1 -2 0
6 -6 1 8 1 -1 4 0 -3 4 3 1 -3 1 -2 0 1 2 0 0 -2 1 1
3 -2 0 4 0 1 2 1 -2 1 1 0 -2 -1 0 -1 1 1 0 0 -1 -1 0
-4 4 0 -4 -1 0 -2 -1 2 -2 -1 0 1 -1 1 1 0 -1 -1 0 1 -1 -1
-2 2 -3 -1 0 0 -2 1 1 0 -1 -1 1 1 -1 1 -1 -1 0 -1 0 0 0
-1 1 -1 0 -1 0 0 0 -1 -1 0 -1 1 0 0 0 -1 0 0 0 0 0 -1
-2 1 0 -1 -1 0 0 0 1 0 -1 -1 1 0 1 0 0 -1 0 0 0 0 -1
1 0 -2 2 1 0 1 1 0 2 0 1 -1 2 -1 0 0 0 0 -1 -1 0 1
3 -4 0 6 1 -1 2 0 -2 3 2 0 -2 1 -2 0 1 1 0 0 -2 1 0
3 -3 2 4 0 0 2 -1 -1 3 1 1 -1 0 -1 0 1 1 0 0 0 1 1
-
PROPERTIES
INTEGRAL =1
-
NOTES
Orthogonal of norm 6 vector in Leech
-
REFERENCES
-
LAST_LINE
Haftungsausschluss/Disclaimer
Gabriele Nebe