The Lattice L_32,6
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 13:18:01 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
L_32,6
-
DIMENSION
32
-
GRAM
32 0
6
-3 6
2 1 6
2 1 2 6
-1 -1 1 -3 6
-3 0 -3 -2 2 6
-2 0 0 0 2 3 6
0 2 0 2 0 1 -1 6
-1 1 2 1 2 0 3 -1 6
2 1 3 1 -1 -2 -2 0 1 6
-1 -1 1 -1 3 0 1 -2 3 1 6
0 0 2 -1 3 -1 1 -2 2 -1 1 6
-2 -1 0 -2 1 0 1 -1 0 0 2 0 6
1 1 1 -1 2 0 1 0 0 -1 -1 3 -2 6
1 1 3 2 1 -1 0 0 2 1 2 2 -2 2 6
-2 2 -2 -2 0 2 -1 1 0 1 -1 0 0 0 -2 6
-2 2 -1 -2 1 0 1 0 1 -1 1 1 1 2 -1 2 6
1 0 -1 -1 1 1 -2 3 -1 0 -2 0 -2 1 -1 3 1 6
0 2 0 2 -1 0 1 0 0 -1 -1 0 -2 2 2 -2 0 -2 6
-2 0 -3 -1 0 3 2 -1 -1 -2 -1 0 -1 0 -1 0 0 -1 2 6
2 0 2 3 -1 -2 -2 2 -1 1 -1 -1 -1 -1 0 -2 -3 0 1 -2 6
-2 1 1 1 1 2 1 1 2 0 1 1 0 -1 2 0 -2 0 -1 0 0 6
-1 -1 -1 0 1 0 -1 0 -1 -2 2 1 1 0 2 -1 0 -1 1 1 0 0 6
0 0 2 2 0 -1 2 -1 1 0 0 2 0 0 0 -2 0 -1 0 1 1 1 -1 6
1 -1 -1 -2 1 -1 -1 -1 0 1 2 0 0 0 0 0 2 0 -1 1 -2 -2 0 -1 6
0 -2 0 1 -1 1 1 0 0 0 -1 -2 0 -2 0 -1 -3 -1 -1 1 0 1 0 0 -1 6
1 -1 -2 0 -2 1 1 0 -2 0 -1 -3 1 -1 -2 -1 0 -1 1 1 0 -2 -1 -1 1 0 6
2 -1 2 3 -1 -3 0 -1 1 1 2 1 1 0 2 -3 0 -2 1 -1 1 0 2 2 0 -1 0 6
0 -2 1 -2 1 1 1 -1 -1 -1 0 0 1 0 0 -1 -1 0 -2 0 -1 1 0 0 -1 2 0 -1 6
2 1 3 2 0 -1 -1 1 0 3 1 -1 -2 1 3 -2 -2 -1 2 -1 3 0 0 0 0 0 0 1 0 6
-1 0 1 1 1 0 2 1 1 0 2 0 2 -1 1 -2 1 -1 -1 0 -1 1 1 2 1 1 1 2 0 0 6
2 -2 1 2 -2 -3 -2 -2 0 0 1 1 0 -1 2 -2 -1 -1 0 -2 1 0 2 1 -1 0 -1 3 0 0 0 6
-
DIVISORS
17^4
-
MINIMAL_NORM
6
-
KISSING_NUMBER
233376
-
HERMITE_NUMBER
4.21
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GROUP_ORDER
2^7 * 3^3 * 17
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GROUP_NAME
(SL_2(17) o tilde(S)_3 ).2
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GROUP_GENERATORS
3
32 32
2 3 -2 2 2 2 -2 -1 0 0 0 1 1 0 1 -1 1 -1 -1 0 1 -1 -1 0 0 1 0 0 1 -1 0 0
0 0 0 1 0 -1 2 -1 -1 1 0 -1 0 0 -1 -1 0 1 0 0 0 1 1 -1 0 -1 -1 -1 0 0 0 0
3 3 -3 4 2 3 -4 -3 1 1 0 0 2 1 1 -2 2 -1 -1 1 2 -1 -1 -1 -1 0 -1 -1 2 -2 1 -1
1 4 -1 1 2 2 -1 -1 0 0 0 0 0 0 -1 -1 -1 0 0 -1 0 -1 0 0 1 1 0 1 1 -1 0 1
-1 -2 1 -2 -3 -1 1 2 0 0 2 1 -1 1 -1 1 -1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0
-2 -2 1 -3 -1 -3 5 2 -2 1 0 0 -2 -1 -1 1 -1 1 1 -1 -1 1 1 0 1 0 0 1 -1 1 -1 1
2 1 0 -4 0 -3 3 1 -1 -1 1 -1 -1 -1 0 2 0 0 0 1 0 1 0 0 -1 1 0 1 -1 1 0 1
0 2 1 -1 -1 1 2 1 -1 0 1 1 -1 0 -3 -1 -2 1 1 -2 -1 0 1 0 2 1 0 1 0 0 -1 1
1 1 -1 -1 0 -1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0
2 2 -2 4 2 2 -2 -3 0 1 -1 0 2 0 1 -2 2 0 -1 0 1 0 0 -1 0 0 -1 -1 1 -1 0 -1
-1 -1 0 0 -1 0 -1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
-1 -2 0 1 -1 0 -2 0 1 0 1 0 0 1 1 1 0 -1 0 1 1 -1 -1 0 -1 -1 0 -1 1 -1 1 -1
0 -2 2 -2 -1 0 -3 1 2 -3 0 -1 0 0 1 2 0 -1 0 1 -1 0 -1 1 -1 0 1 1 -1 1 0 -1
2 1 -2 0 1 -2 3 -1 -2 1 0 0 0 -1 1 0 2 0 -1 1 2 1 0 -1 -1 0 -1 -1 0 0 0 0
1 2 -3 3 2 1 0 -3 -1 2 -1 0 1 0 1 -2 2 0 -1 0 2 0 0 -1 0 0 -1 -1 1 -1 0 0
-2 -3 1 1 0 -1 0 0 0 0 -1 -1 0 0 1 1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 -1
1 0 0 -1 0 -2 1 0 0 -1 0 -1 0 0 0 1 0 0 -1 1 0 1 0 0 -1 0 0 0 -1 1 0 0
-1 0 0 2 0 2 -1 0 0 1 0 1 0 1 -1 -1 -1 0 0 -1 0 -1 0 0 1 0 0 0 1 -1 0 0
0 1 0 -2 0 -3 4 1 -2 0 1 0 -1 -1 -1 0 -1 1 0 0 -1 1 1 0 0 0 0 0 -1 1 -1 1
-2 -2 2 -2 -1 -3 4 1 -1 0 0 -1 -1 0 -1 2 -2 1 1 0 -1 1 1 0 1 -1 0 0 -1 1 0 1
-1 1 1 1 -2 3 -2 1 1 0 2 2 0 1 -2 -2 -2 1 1 -1 -1 -2 0 0 1 0 0 0 1 -1 0 0
-1 0 -1 1 1 1 0 -1 0 2 -1 0 -1 0 -1 -1 0 0 1 -2 0 0 1 0 1 0 0 0 1 -1 0 0
-3 -2 1 1 -1 1 -1 0 1 0 -1 0 0 1 0 0 -1 0 0 0 0 -1 0 0 1 -1 0 0 0 0 0 0
1 2 0 1 0 2 -2 -1 1 0 1 0 0 1 -1 0 -1 0 0 0 0 -1 0 0 0 0 0 0 1 -1 1 0
0 -1 1 -1 -1 -1 1 0 0 -1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 -1 1 0 0
2 2 -1 -1 1 1 0 -1 0 0 -1 0 1 -1 1 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0
2 2 -1 -1 2 -1 2 0 -1 -1 -1 0 0 -2 0 0 1 0 -1 0 -1 1 0 0 0 1 0 1 -1 2 -1 1
1 2 -1 2 2 2 -4 -2 2 -1 -1 -1 1 1 1 0 0 -1 -1 1 1 -1 -1 0 0 0 0 0 1 -1 1 0
4 2 -3 2 2 2 -2 -3 1 1 -2 -1 2 0 2 -1 3 -1 -1 1 2 0 0 0 -1 0 -1 -1 1 -1 1 -1
2 3 -2 2 0 2 1 -2 -1 2 0 1 1 0 -1 -3 1 1 0 -1 1 0 1 -1 1 0 -2 -1 1 -1 0 0
1 2 1 -1 -1 2 -1 0 1 -1 1 0 0 1 -2 0 -1 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1
-1 1 -1 3 2 3 -5 -1 2 0 -1 0 1 1 1 -1 0 -1 -1 0 0 -2 -1 1 0 0 1 0 1 -1 0 0
32 32
-4 -2 1 2 1 0 -1 0 0 1 -1 0 -1 0 0 -1 0 0 -1 0 -1 -1 0 0 0 -1 1 0 0 0 -1 0
3 1 -1 -1 -1 -1 3 -1 -2 1 1 1 0 -1 0 -1 2 1 0 0 1 2 1 -1 -1 1 -1 -1 -1 0 0 -1
-2 -1 -1 3 0 1 1 -2 -1 3 -1 1 0 0 0 -2 1 1 0 -1 1 0 1 -1 1 -1 -1 -1 1 -1 0 -1
3 2 -3 2 2 1 1 -4 -1 2 -3 0 1 -2 2 -2 4 0 -2 0 2 1 1 -1 -1 0 -1 -1 0 0 0 -1
-3 -2 1 1 -1 0 -1 1 0 0 1 0 0 2 0 1 -2 0 1 0 0 -1 -1 0 1 -1 0 0 1 -1 1 0
3 1 -1 -2 -1 0 0 1 0 -1 1 0 1 0 1 1 0 -1 0 1 1 0 -1 0 -1 0 -1 0 0 0 1 0
3 2 -3 0 1 1 0 -1 0 1 -1 0 1 0 1 0 1 -1 0 0 2 0 0 0 0 0 -1 0 1 -1 1 0
2 2 -2 3 3 0 1 -3 -2 2 -2 0 1 -1 1 -2 3 0 -2 1 1 1 0 -2 -1 0 -1 -1 0 0 0 0
1 1 -3 2 0 0 1 -2 -1 2 0 1 1 0 1 -1 1 1 0 0 2 0 0 -1 0 0 -1 -1 1 -1 1 -1
-1 -1 -1 3 1 1 -1 -2 0 2 -1 0 0 0 0 -1 1 0 0 0 1 0 0 -1 0 -1 -1 -1 1 -1 1 -1
-2 -2 0 1 0 -1 0 0 0 0 0 -1 0 1 1 2 -1 0 1 0 1 0 -1 0 1 -1 0 0 1 -1 1 0
-3 -2 2 0 -3 1 -2 1 1 -1 2 1 0 2 -1 0 -2 1 1 -1 -1 -1 0 1 1 0 1 0 0 0 0 -1
1 2 -2 3 3 2 -2 -2 1 1 -2 -1 1 1 -1 0 0 -1 1 -1 1 0 0 0 2 0 -1 0 2 -1 1 1
-3 -2 2 -1 -2 -1 1 2 -1 0 2 1 -1 1 -1 -1 -1 1 0 0 -1 0 0 0 0 0 1 0 -1 0 -1 0
-2 -2 0 2 -1 0 1 -1 -1 1 0 1 0 0 1 -1 1 1 -1 0 1 0 0 -1 0 -1 0 -1 0 0 0 -1
0 0 2 -2 -3 0 0 2 0 -1 3 1 0 1 -2 0 -2 1 1 0 -1 0 0 0 0 1 0 0 -1 0 0 0
-1 0 2 -2 -1 -2 2 2 -1 -1 2 0 -1 1 -2 1 -2 1 1 0 -1 1 0 0 1 1 1 1 -1 0 -1 1
-1 1 2 -2 -1 -1 1 2 -1 -1 2 1 -1 0 -2 0 -2 1 0 0 -2 0 0 0 0 1 1 1 -1 1 -1 1
3 1 -1 -1 -1 0 2 -1 -1 0 0 0 0 -1 1 -1 2 0 -1 0 1 1 1 0 -1 0 -1 -1 -1 0 0 -1
2 0 0 -2 -1 2 -3 1 2 -2 0 0 0 0 1 1 0 -2 0 0 0 -1 -1 1 -1 0 0 1 0 0 1 -1
3 2 -2 1 2 0 1 -3 -1 1 -2 -1 1 -2 2 -1 3 0 -1 0 1 1 1 0 -1 0 -1 -1 0 0 0 -1
2 1 -3 2 0 1 -1 -2 0 1 0 1 2 0 2 -1 2 0 -1 1 2 0 -1 -1 -1 0 -1 -1 1 0 1 -1
0 0 1 0 -1 1 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 0 0 0
2 1 -1 0 1 2 -3 -2 2 -1 -2 -1 1 0 1 1 1 -1 0 0 1 0 0 1 0 0 0 1 1 0 1 -1
-5 -4 2 2 2 -1 -3 1 1 -1 -1 -1 -1 1 1 2 -1 -1 0 1 -1 -1 -2 0 0 -1 2 1 0 0 0 0
1 1 -2 2 2 2 -1 -2 1 1 -3 0 1 -1 1 -1 1 -1 -1 0 1 -1 0 -1 0 -1 -1 0 1 0 1 0
3 2 -2 0 4 -1 0 -1 0 0 -2 -2 0 -1 1 1 2 -2 -1 1 1 1 0 0 -1 0 0 0 0 0 0 1
0 1 -1 2 1 2 -1 -2 0 1 -1 0 0 0 0 -1 1 0 0 -1 1 0 0 0 1 0 0 0 1 -1 0 0
-1 1 -1 1 1 1 0 1 0 1 0 1 0 0 -1 -1 -1 0 0 -1 -1 -1 0 0 1 0 0 1 1 0 -1 1
0 -1 -2 2 2 -1 2 -2 -2 2 -2 0 0 -2 2 -1 3 0 -1 0 1 1 0 -1 -1 -1 -1 -1 0 0 0 -1
0 0 -1 2 3 0 0 -2 0 1 -3 -1 0 0 0 1 1 -1 0 0 1 1 0 -1 1 -1 0 0 1 0 1 1
-1 0 1 0 -1 0 0 0 1 -1 0 0 0 0 -1 0 -1 1 0 -1 -1 0 1 1 1 0 1 0 0 1 -1 0
32 32
0 0 0 0 0 -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 -1 0
-3 -3 2 1 -1 1 -3 0 2 -1 0 0 0 2 0 1 -1 0 0 1 0 -1 -1 0 0 -1 1 0 0 0 1 0
-2 -1 -1 3 1 2 -2 -1 1 1 -2 0 1 1 0 -1 0 0 0 -1 0 -1 0 0 1 -1 0 0 1 0 0 0
-1 -1 -1 4 3 0 -2 -2 0 0 -2 -1 1 1 1 1 1 -1 -1 1 1 0 -1 -1 0 -1 0 0 1 0 1 0
2 2 -1 -2 0 1 0 0 0 0 0 0 0 -1 0 0 0 0 1 -1 0 0 0 1 0 1 0 1 0 0 0 0
2 2 -2 0 2 -1 1 -1 -1 1 0 0 0 -1 0 0 1 0 0 0 0 1 0 0 -1 1 0 0 0 0 0 0
2 3 -3 3 4 1 -1 -3 -1 1 -2 0 1 -1 0 -1 2 0 0 -1 0 1 0 0 0 1 0 0 1 0 0 0
-1 -3 0 1 1 -1 -1 -1 0 0 -1 -1 0 0 2 2 1 -1 0 1 1 0 -1 0 -1 -1 0 0 0 0 1 -1
1 1 -1 1 2 2 -3 -2 1 -1 -2 -1 1 0 1 1 1 -1 0 0 1 0 -1 0 0 0 0 1 1 0 1 0
-2 -2 1 0 -1 1 -1 1 1 0 0 0 0 1 0 0 -1 0 0 0 0 -1 0 0 0 -1 0 0 0 0 0 0
4 4 -3 0 2 3 -2 -2 1 0 -2 0 1 -1 0 -1 1 -1 0 -1 1 0 0 0 0 1 -1 1 1 0 1 0
-1 0 1 -1 -2 0 1 1 0 0 1 0 0 1 -1 0 -1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 1
0 1 -2 2 2 2 0 -2 0 2 -3 0 0 -1 0 -2 1 0 0 -2 0 0 1 0 1 0 -1 0 1 0 0 0
0 0 2 -2 -3 0 0 2 0 -1 3 1 0 1 -1 0 -1 1 1 0 -1 0 0 1 0 1 1 0 -1 0 -1 0
0 1 0 -1 -1 1 0 1 0 0 1 1 0 1 -2 0 -1 0 1 -1 0 0 0 0 1 1 0 1 0 0 0 1
-3 -4 3 -2 -2 -2 0 2 1 -1 1 -1 -1 1 1 2 -1 0 0 2 0 0 -1 0 -1 -1 1 0 -1 0 0 0
2 0 0 -1 0 1 -3 -1 2 -2 -1 -1 1 0 2 1 1 -1 0 1 1 0 -1 1 -1 0 0 0 0 0 1 -1
-1 -3 2 -2 -2 -2 1 1 0 -1 1 -1 0 0 2 2 0 0 0 2 1 0 -1 0 -1 -1 0 0 -1 0 0 -1
0 1 1 -2 -1 0 0 2 0 -1 2 1 -1 1 -2 1 -2 0 1 -1 -1 0 0 1 1 1 1 1 0 0 0 1
2 2 1 -4 -2 -2 3 2 -1 -1 3 0 -1 0 -2 1 -2 1 1 0 -1 1 1 1 0 1 0 0 -1 0 0 1
-1 -1 -1 2 2 -1 1 -1 -1 1 -2 0 0 -1 1 0 1 0 -1 0 0 0 0 -1 0 -1 0 0 0 1 0 0
1 1 -2 2 1 1 0 -2 0 1 -1 0 1 0 0 0 1 0 0 0 1 0 0 -1 0 0 -1 0 1 0 1 0
1 1 1 -4 -2 -1 2 2 0 -1 1 0 -1 0 -2 1 -2 0 1 -1 -1 1 1 1 1 1 0 1 -1 1 0 1
1 1 -2 4 3 1 -1 -3 0 1 -3 -1 2 0 1 -1 2 0 -1 1 1 0 0 -1 0 -1 -1 -1 1 0 1 0
2 1 1 -5 -2 -1 1 2 0 -2 2 0 -1 -1 0 1 -1 0 1 0 0 0 0 1 -1 1 0 1 -1 0 0 0
-1 -1 0 -1 0 -2 3 1 -2 1 0 0 -1 -1 0 0 0 0 0 -1 -1 1 1 0 0 0 0 0 -1 0 -1 0
1 2 -2 1 3 -1 2 -1 -2 1 -1 1 -1 -2 -1 -1 1 0 0 -2 -1 1 1 0 0 1 0 0 0 1 -1 0
2 2 -1 2 1 2 -2 -2 1 -1 -1 -1 2 1 0 0 0 0 0 0 1 0 0 0 1 0 -1 0 1 0 1 0
1 1 -1 0 -1 0 3 0 -1 2 0 1 0 -1 -2 -2 0 1 1 -2 -1 1 2 0 1 0 -1 -1 0 0 -1 0
1 1 -1 0 0 1 0 0 0 1 0 1 0 0 -1 -1 0 0 0 -1 0 0 1 0 0 0 0 0 0 0 0 0
1 1 -3 3 3 2 -2 -3 0 1 -3 0 1 -1 1 -1 2 -1 0 -1 1 0 0 0 0 0 -1 0 1 0 1 -1
1 2 -2 2 2 1 0 -1 0 1 -2 0 1 0 0 -1 1 -1 -1 0 1 0 0 -1 1 0 -1 0 1 0 0 1
-
PROPERTIES
-
REFERENCES
T. Shioda, Mordell-Weil lattices and sphere packings,
Amer. J. Math. 113 (1991), 931-948.
G. Nebe, Some cyclo-quaternionic lattices, J. Alg. 199 (1998), 472-498.
-
NOTES
CycloQuaternionic lattice of type Delta6
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LAST_LINE
Haftungsausschluss/Disclaimer
Gabriele Nebe