The Lattice LAMBDA24 (The Leech lattice as a Hurwitzian lattice)
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 13:19:28 CEST 2014
INDEX FILE |
ABBREVIATIONS
Contents of this file
NAME
DIMENSION
DET
GRAM
MINIMAL_NORM
KISSING_NUMBER
HERMITIAN_GROUP_NAME
HERMITIAN_GROUP_ORDER
HERMITIAN_GROUP_GENERATORS
HERMITIAN_STRUCTURE
PROPERTIES
REFERENCES
NOTES
LAST_LINE
-
NAME
LAMBDA24 (The Leech lattice as a Hurwitzian lattice)
-
DIMENSION
24
-
DET
1
-
GRAM
24 24
4 0 0 2 -1 2 0 0 -1 -1 1 0 -2 -1 -1 -2 -2 -1 0 -1 -2 -1 0 -1
0 4 0 2 -2 -1 -1 -2 1 -1 1 0 1 -2 0 0 1 -2 1 0 1 -2 1 0
0 0 4 2 0 1 -1 1 -1 -1 -1 -2 1 0 -2 -1 0 -1 -2 -2 0 -1 -2 -2
2 2 2 4 -1 1 -2 -1 -1 -1 1 -1 0 -2 -1 -2 -1 -2 0 -2 -1 -2 0 -2
-1 -2 0 -1 4 0 0 2 -1 1 1 0 1 1 0 0 -1 2 0 0 0 1 0 0
2 -1 1 1 0 4 0 2 -1 -1 -1 -2 -1 1 -2 -1 -2 -1 -1 -2 -1 0 -1 -1
0 -1 -1 -2 0 0 4 2 -1 1 -1 0 0 2 1 2 0 1 -1 1 0 1 0 1
0 -2 1 -1 2 2 2 4 -1 1 -1 -1 1 2 -1 1 -1 1 -2 -1 0 1 -1 0
-1 1 -1 -1 -1 -1 -1 -1 4 0 0 2 0 -1 0 0 1 0 0 0 2 0 1 2
-1 -1 -1 -1 1 -1 1 1 0 4 0 2 1 0 1 1 0 1 -1 0 0 2 1 1
1 1 -1 1 1 -1 -1 -1 0 0 4 2 0 -1 0 -1 0 1 1 1 -1 -1 2 0
0 0 -2 -1 0 -2 0 -1 2 2 2 4 0 -1 1 0 1 1 0 1 0 1 2 2
-2 1 1 0 1 -1 0 1 0 1 0 0 4 0 0 2 2 0 -1 0 1 0 0 0
-1 -2 0 -2 1 1 2 2 -1 0 -1 -1 0 4 0 2 0 2 -1 1 0 1 -1 0
-1 0 -2 -1 0 -2 1 -1 0 1 0 1 0 0 4 2 1 1 2 2 0 1 1 1
-2 0 -1 -2 0 -1 2 1 0 1 -1 0 2 2 2 4 2 1 0 2 1 1 0 1
-2 1 0 -1 -1 -2 0 -1 1 0 0 1 2 0 1 2 4 0 0 2 1 0 1 1
-1 -2 -1 -2 2 -1 1 1 0 1 1 1 0 2 1 1 0 4 0 2 0 1 0 0
0 1 -2 0 0 -1 -1 -2 0 -1 1 0 -1 -1 2 0 0 0 4 2 -1 0 1 0
-1 0 -2 -2 0 -2 1 -1 0 0 1 1 0 1 2 2 2 2 2 4 0 1 1 1
-2 1 0 -1 0 -1 0 0 2 0 -1 0 1 0 0 1 1 0 -1 0 4 0 0 2
-1 -2 -1 -2 1 0 1 1 0 2 -1 1 0 1 1 1 0 1 0 1 0 4 0 2
0 1 -2 0 0 -1 0 -1 1 1 2 2 0 -1 1 0 1 0 1 1 0 0 4 2
-1 0 -2 -2 0 -1 1 0 2 1 0 2 0 0 1 1 1 0 0 1 2 2 2 4
-
MINIMAL_NORM
4
-
KISSING_NUMBER
196560
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HERMITIAN_GROUP_NAME
2.G_2(4)
-
HERMITIAN_GROUP_ORDER
2^13 * 3^3 * 5^2 * 7 * 13
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HERMITIAN_GROUP_GENERATORS
2
24 24
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 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 0 1
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 0 0 0 0 0 0 0 0 1 0 -1 0 0 1 1 0 0 -1 0 0 -1 0
0 0 0 0 0 0 0 0 1 1 1 -2 0 0 1 -1 0 0 -1 1 -1 -1 -1 2
0 0 0 0 0 0 0 0 -1 0 0 0 -1 -1 -1 1 1 1 1 -1 1 0 0 0
0 0 0 0 0 0 0 0 0 1 1 -1 -1 0 0 0 1 0 0 0 0 -1 -1 1
0 0 -1 0 0 0 0 0 1 1 0 -1 1 1 1 -2 0 -1 -1 1 0 0 0 -1
-1 -1 -1 2 0 0 0 0 -1 0 -1 1 0 0 -1 0 0 0 0 1 0 -1 -1 1
1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 -1 1 1 0 -1
0 -1 -1 1 0 0 0 0 0 0 -1 1 1 1 0 -1 0 -1 0 1 1 0 0 -1
1 0 1 -1 0 0 0 -1 0 1 1 -1 0 1 0 0 1 0 1 -1 0 0 0 0
1 1 0 -1 0 -1 -1 1 0 0 0 -1 -1 0 0 0 1 1 0 -1 0 0 0 0
0 1 1 -1 1 1 0 -1 -1 0 -1 1 -1 -1 -1 2 0 1 1 -1 0 0 0 0
1 1 1 -1 1 0 0 -1 0 1 0 -1 -1 0 -1 1 1 1 1 -1 0 0 0 0
2 0 2 -2 0 0 0 -1 0 1 0 0 1 1 0 0 0 0 2 -1 1 0 1 -1
2 2 0 -2 0 -1 -1 1 -1 0 0 0 -1 1 0 0 2 1 1 -3 1 1 0 -1
0 2 2 -2 1 1 0 -1 -1 -1 -1 2 -1 -1 0 2 -1 1 0 -1 0 1 1 -1
2 2 2 -2 1 0 0 -1 -1 0 -1 1 -1 0 -1 2 1 2 2 -3 1 1 1 -1
0 -2 -1 2 0 0 2 -1 2 2 2 -3 0 0 -1 0 2 0 1 -1 0 0 -1 0
1 1 1 0 2 1 1 -3 0 1 -1 -1 -1 -1 -1 2 1 2 0 -1 -1 -1 -1 2
1 0 2 -2 -1 1 0 -1 -1 1 0 1 1 0 0 0 0 1 2 -1 1 0 0 0
0 -1 1 1 1 2 2 -3 1 2 0 -1 0 -1 -1 1 1 1 1 0 0 -1 -1 1
24 24
-5 -2 -4 6 0 0 0 3 0 -3 -2 3 0 -1 0 -1 -2 -2 -3 5 -1 -1 -1 0
-2 -3 2 2 0 3 3 -3 1 1 1 1 1 -1 -1 1 -1 0 2 1 0 -2 -1 2
0 -4 -3 2 -3 -3 0 3 2 0 3 -3 2 2 1 -3 0 -3 0 1 2 1 0 -2
-4 -4 -2 3 -3 0 0 3 0 -2 1 2 2 0 1 -2 -3 -3 -1 4 1 -1 0 -1
1 1 -1 -2 -2 -3 -3 4 -1 -1 0 0 0 1 1 -1 -1 -1 -1 0 1 2 2 -3
-2 -2 -3 4 0 -1 1 2 1 -1 0 0 0 0 0 -1 0 -2 -1 2 0 0 -1 -1
2 1 0 0 2 -1 1 -2 1 1 0 -2 -1 0 -1 1 2 1 0 -2 -1 1 -1 1
1 -1 -2 1 -1 -3 0 2 1 0 1 -2 0 1 0 -1 1 -1 0 -1 1 2 0 -2
0 0 2 0 1 3 2 -3 0 1 -1 1 0 -1 -1 1 1 2 2 -1 0 -1 0 1
2 2 2 -4 -1 0 -1 -1 -2 0 0 1 0 0 0 1 0 2 1 -3 1 1 1 -1
-2 0 0 0 -2 0 -2 3 -1 -2 -1 3 1 0 1 -1 -3 -1 -1 3 0 0 1 -1
0 2 2 -2 0 2 -1 -1 -2 -1 -2 3 0 -1 0 1 -1 2 0 0 0 0 1 0
2 -1 2 -2 -2 -1 0 0 0 1 2 -1 1 1 0 0 0 0 2 -2 2 1 1 -1
3 2 0 -2 1 -2 0 0 1 1 1 -3 -1 1 0 0 2 0 0 -2 0 2 0 -1
1 3 3 -4 1 1 -1 -2 -2 0 -1 1 -1 -1 0 2 0 2 0 -2 -1 0 1 1
3 2 3 -3 1 0 1 -3 0 2 1 -2 -1 0 -1 2 2 2 2 -4 0 1 0 1
3 1 3 -3 0 0 0 -2 0 1 1 -1 0 1 0 1 1 1 2 -3 1 1 1 0
2 3 0 -3 0 -2 -2 2 0 0 0 -1 -1 1 1 -1 1 0 -1 -1 0 2 1 -2
-1 2 2 -1 2 2 0 -2 -1 0 -1 1 -2 -2 -1 3 0 2 0 -1 -2 -1 0 2
2 3 2 -2 2 0 0 -2 0 1 0 -1 -2 0 -1 2 2 2 1 -3 -1 1 0 1
1 -1 1 0 0 1 2 -2 1 2 1 -1 1 1 0 -1 1 0 2 -1 1 0 0 0
2 2 1 -2 1 0 0 -2 -1 1 0 -1 -1 0 -1 1 2 2 1 -3 0 1 0 0
0 1 2 -2 -1 1 -1 0 -2 -1 -1 3 0 0 0 1 -1 1 1 -1 0 0 1 0
1 1 2 -1 1 2 1 -3 -1 1 -1 1 0 0 -1 1 1 2 2 -2 0 0 0 1
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HERMITIAN_STRUCTURE
4
24 24
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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 1
24 24
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 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 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 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 -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 -2 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 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 -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 -2 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 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 -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 -2 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 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 -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 -2 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 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 -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 -2
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 -1
24 24
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 2 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
-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 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 2 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 -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 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 2 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 -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 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 2 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 -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 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 2 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 -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 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 2
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 -1 -1 0 1
24 24
-1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 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 -1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
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 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 -1 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 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 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 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 -1 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 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 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 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 -1 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 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 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 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 -1 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
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 -1 0 0 0
-
PROPERTIES
Unimodular=1
-
REFERENCES
Bachoc, Nebe: Classification of two genera of 32-dimensional lattices
over the Hurwitz order
J. Tits, Four presentations of Leech's lattice.
Finite simple groups II, Academic Press (1980), 303-307.
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NOTES
The unique structure of the Leech lattice over the Hurwitz order
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LAST_LINE
Haftungsausschluss/Disclaimer
Gabriele Nebe