The Lattice L40
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 13:18:07 CEST 2014
INDEX FILE |
ABBREVIATIONS
Contents of this file
NAME
DIMENSION
GRAM
DIVISORS
MINIMAL_NORM
KISSING_NUMBER
HERMITE_NUMBER
GROUP_ORDER
GROUP_NAME
GROUP_GENERATORS
PROPERTIES
REFERENCES
NOTES
LAST_LINE
-
NAME
L40
-
DIMENSION
40
-
GRAM
40 40
8 4 -2 2 1 0 -4 4 4 1 -4 -4 -4 -4 0 -2 0 0 -4 4 4 2 -1 1 2 0 -2 2 2 2 -2 -2 -2 -2 0 -1 0 0 -2 2
4 8 0 2 2 3 -4 4 4 3 -4 -4 -4 -4 -1 0 4 -4 -2 4 2 4 0 1 1 3 -2 2 2 3 -2 -2 -2 -2 -2 0 2 -2 -1 2
-2 0 8 2 4 -2 2 0 -2 3 4 0 3 4 0 -1 2 -1 4 -2 -1 0 4 1 2 -1 1 0 -1 0 2 0 0 2 0 -2 1 1 2 -1
2 2 2 8 4 2 -4 2 4 2 0 -4 0 -1 -3 -1 1 -4 2 0 1 1 1 4 2 1 -2 1 2 1 0 -2 0 -2 -3 -2 -1 -2 1 0
1 2 4 4 8 -1 -2 2 2 2 2 -2 0 0 0 -4 0 -2 3 -2 -1 1 2 2 4 1 -1 1 1 1 1 -1 0 0 0 -2 0 -1 3 -1
0 3 -2 2 -1 8 -4 0 2 2 -4 -2 0 -2 -3 4 1 -4 0 2 0 0 -1 1 -2 4 -2 0 1 1 -2 -1 0 -1 -3 2 -1 -2 0 1
-4 -4 2 -4 -2 -4 8 -4 -4 -3 4 4 4 4 2 0 -1 2 0 -4 -2 -2 1 -2 -1 -2 4 -2 -2 -3 2 2 2 2 1 0 1 1 0 -2
4 4 0 2 2 0 -4 8 4 3 -4 -4 -4 -4 2 -2 2 -1 0 4 2 2 0 1 1 0 -2 4 2 3 -2 -2 -2 -2 1 -1 1 1 0 2
4 4 -2 4 2 2 -4 4 8 -1 -4 -4 -4 -4 -1 -2 0 -4 -1 0 2 2 -1 2 1 1 -2 2 4 1 -2 -2 -2 -2 -2 -1 0 -2 1 0
1 3 3 2 2 2 -3 3 -1 8 -1 -1 1 1 0 2 3 -1 2 3 -1 0 3 1 1 1 0 0 -2 4 1 1 2 2 0 1 0 1 1 0
-4 -4 4 0 2 -4 4 -4 -4 -1 8 4 4 4 -1 -2 0 2 2 -4 -2 -2 2 0 1 -2 2 -2 -2 -2 4 2 2 2 1 -1 0 1 1 -2
-4 -4 0 -4 -2 -2 4 -4 -4 -1 4 8 4 4 1 1 -2 4 0 -4 -2 -2 0 -2 -1 -1 2 -2 -2 -2 2 4 2 2 2 2 -1 2 0 -2
-4 -4 3 0 0 0 4 -4 -4 1 4 4 8 4 0 2 -2 0 2 -4 -2 -2 3 0 0 0 2 -2 -2 -1 2 2 4 2 0 1 -1 0 1 -2
-4 -4 4 -1 0 -2 4 -4 -4 1 4 4 4 8 2 2 0 2 4 -4 -2 -2 2 1 0 -1 2 -2 -2 -1 2 2 2 4 1 1 0 1 2 -2
0 -1 0 -3 0 -3 2 2 -1 0 -1 1 0 2 8 1 -1 3 0 0 0 1 0 0 0 0 1 1 1 0 -2 -1 0 1 4 2 1 0 0 0
-2 0 -1 -1 -4 4 0 -2 -2 2 -2 1 2 2 1 8 2 -1 0 2 -1 0 1 1 -2 2 0 -1 -1 1 -1 -1 1 1 -1 4 1 -2 0 1
0 4 2 1 0 1 -1 2 0 3 0 -2 -2 0 -1 2 8 -2 0 4 0 2 1 2 0 2 -2 1 0 3 0 -1 -1 0 -2 1 4 -1 0 2
0 -4 -1 -4 -2 -4 2 -1 -4 -1 2 4 0 2 3 -1 -2 8 -1 0 0 -2 -2 -2 -1 -2 1 -2 -2 -2 1 2 0 1 3 1 -1 4 -2 0
-4 -2 4 2 3 0 0 0 -1 2 2 0 2 4 0 0 0 -1 8 -3 -2 -1 2 1 0 0 0 0 -2 1 1 0 1 2 0 0 0 1 4 0
4 4 -2 0 -2 2 -4 4 0 3 -4 -4 -4 -4 0 2 4 0 -3 8 2 2 -1 0 -1 1 -2 2 0 3 -2 -2 -2 -2 0 1 2 0 -3 4
4 2 -1 1 -1 0 -2 2 2 -1 -2 -2 -2 -2 0 -1 0 0 -2 2 8 4 -2 2 1 0 -4 4 4 1 -4 -4 -4 -4 0 -2 0 0 -4 4
2 4 0 1 1 0 -2 2 2 0 -2 -2 -2 -2 1 0 2 -2 -1 2 4 8 0 2 2 3 -4 4 4 3 -4 -4 -4 -4 -1 0 4 -4 -2 4
-1 0 4 1 2 -1 1 0 -1 3 2 0 3 2 0 1 1 -2 2 -1 -2 0 8 2 4 -2 2 0 -2 3 4 0 3 4 0 -1 2 -1 4 -2
1 1 1 4 2 1 -2 1 2 1 0 -2 0 1 0 1 2 -2 1 0 2 2 2 8 4 2 -4 2 4 2 0 -4 0 -1 -3 -1 1 -4 2 0
2 1 2 2 4 -2 -1 1 1 1 1 -1 0 0 0 -2 0 -1 0 -1 1 2 4 4 8 -1 -2 2 2 2 2 -2 0 0 0 -4 0 -2 3 -2
0 3 -1 1 1 4 -2 0 1 1 -2 -1 0 -1 0 2 2 -2 0 1 0 3 -2 2 -1 8 -4 0 2 2 -4 -2 0 -2 -3 4 1 -4 0 2
-2 -2 1 -2 -1 -2 4 -2 -2 0 2 2 2 2 1 0 -2 1 0 -2 -4 -4 2 -4 -2 -4 8 -4 -4 -3 4 4 4 4 2 0 -1 2 0 -4
2 2 0 1 1 0 -2 4 2 0 -2 -2 -2 -2 1 -1 1 -2 0 2 4 4 0 2 2 0 -4 8 4 3 -4 -4 -4 -4 2 -2 2 -1 0 4
2 2 -1 2 1 1 -2 2 4 -2 -2 -2 -2 -2 1 -1 0 -2 -2 0 4 4 -2 4 2 2 -4 4 8 -1 -4 -4 -4 -4 -1 -2 0 -4 -1 0
2 3 0 1 1 1 -3 3 1 4 -2 -2 -1 -1 0 1 3 -2 1 3 1 3 3 2 2 2 -3 3 -1 8 -1 -1 1 1 0 2 3 -1 2 3
-2 -2 2 0 1 -2 2 -2 -2 1 4 2 2 2 -2 -1 0 1 1 -2 -4 -4 4 0 2 -4 4 -4 -4 -1 8 4 4 4 -1 -2 0 2 2 -4
-2 -2 0 -2 -1 -1 2 -2 -2 1 2 4 2 2 -1 -1 -1 2 0 -2 -4 -4 0 -4 -2 -2 4 -4 -4 -1 4 8 4 4 1 1 -2 4 0 -4
-2 -2 0 0 0 0 2 -2 -2 2 2 2 4 2 0 1 -1 0 1 -2 -4 -4 3 0 0 0 4 -4 -4 1 4 4 8 4 0 2 -2 0 2 -4
-2 -2 2 -2 0 -1 2 -2 -2 2 2 2 2 4 1 1 0 1 2 -2 -4 -4 4 -1 0 -2 4 -4 -4 1 4 4 4 8 2 2 0 2 4 -4
0 -2 0 -3 0 -3 1 1 -2 0 1 2 0 1 4 -1 -2 3 0 0 0 -1 0 -3 0 -3 2 2 -1 0 -1 1 0 2 8 1 -1 3 0 0
-1 0 -2 -2 -2 2 0 -1 -1 1 -1 2 1 1 2 4 1 1 0 1 -2 0 -1 -1 -4 4 0 -2 -2 2 -2 1 2 2 1 8 2 -1 0 2
0 2 1 -1 0 -1 1 1 0 0 0 -1 -1 0 1 1 4 -1 0 2 0 4 2 1 0 1 -1 2 0 3 0 -2 -2 0 -1 2 8 -2 0 4
0 -2 1 -2 -1 -2 1 1 -2 1 1 2 0 1 0 -2 -1 4 1 0 0 -4 -1 -4 -2 -4 2 -1 -4 -1 2 4 0 2 3 -1 -2 8 -1 0
-2 -1 2 1 3 0 0 0 1 1 1 0 1 2 0 0 0 -2 4 -3 -4 -2 4 2 3 0 0 0 -1 2 2 0 2 4 0 0 0 -1 8 -3
2 2 -1 0 -1 1 -2 2 0 0 -2 -2 -2 -2 0 1 2 0 0 4 4 4 -2 0 -2 2 -4 4 0 3 -4 -4 -4 -4 0 2 4 0 -3 8
-
DIVISORS
3^20
-
MINIMAL_NORM
8
-
KISSING_NUMBER
360360
-
HERMITE_NUMBER
4.6188
-
GROUP_ORDER
2^10 * 3^3 * 5 * 7 * 11
-
GROUP_NAME
2.M22.2 otimes(sqrt(-7)) tilde(S)_3
-
GROUP_GENERATORS
4
40 40
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40 40
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 -2 0 -2 2 -2 -2 1 -1 1 1 0 0 0 0 0 -2
0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 -2 -2 0 0 -2 -1 0 2 1 0 0 0 0 0 1
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0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -3 2 -2 -2 2 -2 0 3 2 0 0 -3 2 4 -4 2 -2 2 -2 1
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0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 2 -2 1 0 -1 1 0 -1 -1 1 1 2 -2 -2 1 -1 0 -2 1 0
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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 -2 0 -1 1 0 0 0 0 1 0 0 -1 -1 0 1 1 -1 -1 -2
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 -2 0 1 1 0 0 0 0 0 2 1
1 2 0 0 -2 0 -2 2 -2 -2 1 -1 1 1 0 0 0 0 0 -2 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
-1 2 0 0 0 -2 -2 0 0 -2 -1 0 2 1 0 0 0 0 0 1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
-1 1 -1 -2 1 -2 -1 2 1 0 1 -2 2 2 -2 2 -1 0 -1 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
-1 2 -2 -1 1 -1 -2 4 -1 -2 1 -3 2 4 -2 2 0 1 -2 -2 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1 -1 -1 1 -2 -2 3 -2 -1 0 -2 1 2 -2 2 0 0 -2 -3 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
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-1 -1 0 0 0 0 1 -2 2 2 -1 1 0 0 0 -2 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0
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2 -2 1 0 -1 1 0 -1 -1 1 1 2 -2 -2 2 -1 0 -2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0
1 -1 0 0 0 1 1 -3 2 1 0 2 0 -2 2 -1 1 0 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0
-2 2 -1 -2 2 -2 0 1 3 -2 0 -1 3 3 -2 1 -1 1 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0
1 -1 1 0 -1 1 1 0 0 1 0 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 -1 0 0
1 -2 0 -1 1 0 0 0 0 1 0 0 -1 -1 0 1 1 -1 0 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0
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40 40
2 -3 1 1 0 2 1 -2 -1 1 -1 2 -2 -3 2 -1 2 -1 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
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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 0 1 1 0 0 0 0 -1 1 -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 -1 1 0 -1 0 -1 0 0 2 0 0 -1 2 1 -1 0 -1 1 0 2
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 -2 1 1 -1 1 0 -2 -2 1 -1 2 -2 -3 2 -1 2 -1 1 -2
40 40
0 0 0 1 -1 1 1 0 -1 1 0 0 -1 -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
1 -1 1 1 -1 1 1 -1 0 1 0 1 -1 -2 1 -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
1 -1 1 1 -1 0 0 -2 0 1 -1 1 -1 -2 1 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
2 -2 2 1 -1 2 1 -2 -1 1 -1 2 -2 -3 2 -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
1 -1 1 1 -1 1 1 -2 0 1 -1 1 -1 -2 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
1 -2 1 0 0 1 1 -2 1 1 0 2 -1 -2 1 -1 0 -1 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
-1 1 -1 0 1 -1 -1 1 0 -1 0 -1 1 2 -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
2 -1 1 1 -2 1 0 -1 -2 1 0 1 -2 -3 2 -1 1 -1 1 -3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
2 -2 1 1 -1 2 1 -1 -2 1 0 1 -2 -3 2 -1 1 -1 1 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 -1 1 0 -1 0 0 -2 1 1 -1 1 0 -2 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
-1 1 0 0 1 -1 0 0 1 -1 -1 0 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
-2 1 -1 -1 2 -2 -1 1 1 -1 -1 -1 2 2 -2 1 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 -1 0 0 1 0 0 -1 1 0 -1 1 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 -1 -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
1 1 -1 0 -1 0 -1 1 -2 0 1 -1 0 0 0 0 1 0 0 -3 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 -1 0 1 0 1 0 1 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
1 0 1 1 -1 0 0 -1 0 0 0 1 0 -2 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
-2 3 -2 -1 1 -2 -1 2 0 -1 0 -2 2 3 -2 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
1 -1 1 0 -1 0 -1 -2 0 0 -1 1 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 1 0 0 -1 0 0 1 0 0 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 0 0 0 1 -1 1 1 0 -1 1 0 0 -1 -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 1 -1 1 1 -1 1 1 -1 0 1 0 1 -1 -2 1 -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 1 -1 1 1 -1 0 0 -2 0 1 -1 1 -1 -2 1 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 2 -2 2 1 -1 2 1 -2 -1 1 -1 2 -2 -3 2 -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 1 -1 1 1 -1 1 1 -2 0 1 -1 1 -1 -2 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 1 -2 1 0 0 1 1 -2 1 1 0 2 -1 -2 1 -1 0 -1 1 2
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 -1 0 1 -1 -1 1 0 -1 0 -1 1 2 -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 2 -1 1 1 -2 1 0 -1 -2 1 0 1 -2 -3 2 -1 1 -1 1 -3
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 -2 1 1 -1 2 1 -1 -2 1 0 1 -2 -3 2 -1 1 -1 1 -2
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 1 0 -1 0 0 -2 1 1 -1 1 0 -2 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 -1 1 0 0 1 -1 0 0 1 -1 -1 0 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 -2 1 -1 -1 2 -2 -1 1 1 -1 -1 -1 2 2 -2 1 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 -1 0 0 1 0 0 -1 1 0 -1 1 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 -1 -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 1 1 -1 0 -1 0 -1 1 -2 0 1 -1 0 0 0 0 1 0 0 -3
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 -1 0 1 0 1 0 1 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 1 0 1 1 -1 0 0 -1 0 0 0 1 0 -2 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 -2 3 -2 -1 1 -2 -1 2 0 -1 0 -2 2 3 -2 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 1 -1 1 0 -1 0 -1 -2 0 0 -1 1 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 1 0 0 -1 0 0 1 0 0 1 0 0 0 0 0 -1 0 0 0
-
PROPERTIES
INTEGRAL=1
MODULAR=3
-
REFERENCES
C. Bachoc, G. Nebe: Extremal lattices of minimum 8 related to the Mathieu group M22, Crelle 1998.
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NOTES
Extremal 3-modular lattice.
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LAST_LINE
Haftungsausschluss/Disclaimer
Gabriele Nebe