The Lattice Shorter Leech
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 13:23:36 CEST 2014
INDEX FILE |
ABBREVIATIONS
Contents of this file
NAME
DIMENSION
GRAM
DIVISORS
MINIMAL_NORM
KISSING_NUMBER
GROUP_ORDER
GROUP_NAME
GROUP_GENERATORS
NOTES
REFERENCES
LAST_LINE
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NAME
Shorter Leech
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DIMENSION
23
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GRAM
23
3 1 -1 0 -1 0 0 1 0 0 0 -1 1 0 0
0 1 1 -1 1 0 0 -1
1 3 -1 0 -1 0 1 0 0 0 -1 -1 1 0 0
1 0 1 -1 1 1 1 1
-1 -1 3 1 1 1 0 -1 0 -1 1 0 -1 0 0
1 0 -1 1 -1 -1 0 1
0 0 1 3 1 0 1 -1 -1 -1 0 -1 -1 0 -1
1 0 0 0 -1 0 1 1
-1 -1 1 1 3 0 0 -1 -1 -1 0 0 -1 0 0
0 0 -1 0 -1 -1 1 1
0 0 1 0 0 3 0 1 0 0 1 0 -1 1 1
0 -1 0 -1 0 0 -1 1
0 1 0 1 0 0 3 0 0 -1 -1 0 0 1 0
1 0 1 0 -1 1 0 1
1 0 -1 -1 -1 1 0 3 0 1 0 1 0 0 0
0 -1 0 -1 1 1 -1 -1
0 0 0 -1 -1 0 0 0 3 0 0 1 1 0 1
0 0 1 1 0 1 -1 0
0 0 -1 -1 -1 0 -1 1 0 3 -1 0 0 0 -1
-1 0 0 -1 1 1 0 -1
0 -1 1 0 0 1 -1 0 0 -1 3 0 0 0 1
0 0 -1 1 -1 -1 -1 0
-1 -1 0 -1 0 0 0 1 1 0 0 3 -1 -1 1
0 -1 0 1 0 0 -1 0
1 1 -1 -1 -1 -1 0 0 1 0 0 -1 3 0 0
0 1 0 0 0 1 0 -1
0 0 0 0 0 1 1 0 0 0 0 -1 0 3 0
-1 1 1 -1 -1 0 -1 0
0 0 0 -1 0 1 0 0 1 -1 1 1 0 0 3
-1 0 1 0 0 -1 -1 1
0 1 1 1 0 0 1 0 0 -1 0 0 0 -1 -1
3 -1 -1 1 0 1 1 1
1 0 0 0 0 -1 0 -1 0 0 0 -1 1 1 0
-1 3 1 0 -1 -1 0 -1
1 1 -1 0 -1 0 1 0 1 0 -1 0 0 1 1
-1 1 3 -1 0 0 -1 0
-1 -1 1 0 0 -1 0 -1 1 -1 1 1 0 -1 0
1 0 -1 3 -1 0 0 0
1 1 -1 -1 -1 0 -1 1 0 1 -1 0 0 -1 0
0 -1 0 -1 3 0 1 0
0 1 -1 0 -1 0 1 1 1 1 -1 0 1 0 -1
1 -1 0 0 0 3 0 0
0 1 0 1 1 -1 0 -1 -1 0 -1 -1 0 -1 -1
1 0 -1 0 1 0 3 1
-1 1 1 1 1 1 1 -1 0 -1 0 0 -1 0 1
1 -1 0 0 0 0 1 3
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DIVISORS
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
-
MINIMAL_NORM
3
-
KISSING_NUMBER
4600
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GROUP_ORDER
84610842624000
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GROUP_NAME
C2 x Co2
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GROUP_GENERATORS
2
23
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 0
0 0 0 0 -1 0 -1 1 0 -1 -1 0 0 0 -1
-1 0 0 0 -1 0 0 1
0 0 0 0 -1 -1 -1 1 0 0 -1 -1 0 0 0
0 -1 0 0 -2 -1 0 1
0 0 0 0 0 0 0 0 0 0 1 0 0 0 -1
0 0 1 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1
-1 0 0 1 0 0 0 1
0 0 -1 0 -2 -1 0 1 2 -1 -1 -1 -2 -1 0
0 1 -3 -2 -2 -1 0 1
0 0 0 0 0 0 0 0 0 0 0 0 -1 0 1
0 1 -1 0 0 1 0 0
1 0 0 1 0 1 0 -1 -1 1 -1 2 2 1 0
0 -1 0 1 0 -1 0 0
1 0 1 -1 0 0 0 0 0 0 0 0 0 0 0
-1 -1 0 0 -1 0 1 0
0 0 0 0 1 0 0 0 -1 0 0 0 0 0 0
0 0 1 1 1 1 -1 0
0 1 0 1 0 0 0 0 1 0 -1 0 -1 0 1
0 1 -2 0 0 0 -1 -1
1 -1 0 0 0 0 0 -1 -1 1 0 1 1 1 1
1 -1 1 0 0 0 1 0
0 -1 0 0 0 0 0 0 -1 0 1 1 1 0 -1
0 0 1 0 1 0 0 1
0 0 0 0 1 0 1 -1 0 0 1 1 0 0 -1
0 1 0 0 2 1 -1 0
0 1 -1 1 -1 0 -1 1 1 0 -2 -1 -1 0 1
0 0 -2 0 -2 -1 0 0
1 0 1 0 0 -1 -1 0 -1 0 0 1 1 1 0
0 -1 1 0 0 0 0 1
0 0 0 0 0 0 0 0 0 0 0 1 1 0 -1
0 0 0 0 0 -1 0 1
0 1 0 1 0 0 0 0 0 0 -1 0 0 0 1
0 0 -1 0 0 0 -1 -1
-1 0 0 0 1 1 1 -1 0 0 1 0 0 0 0
0 1 0 0 2 1 0 -1
1 0 0 0 -1 0 0 0 0 1 -1 0 0 0 1
0 -1 -1 0 -2 -1 1 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 -1
0 0 0 0 0 0 0 0
23
-1 -1 -1 -1 -1 0 0 0 1 -2 0 -1 -2 -1 0
1 2 -1 -2 0 1 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 -1 1 0 0 0 1 1 1 -1 -1 0 0 0
0 0 -1 1 0 -2 -1 1
-1 0 -1 0 0 0 1 1 1 0 0 -1 -1 -1 0
0 1 -1 0 0 -1 0 0
0 0 0 0 0 0 0 0 -1 1 0 0 1 0 0
0 -1 1 1 0 -1 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 1 0
0 0 0 1 1 0 0 0
-1 0 0 0 1 0 1 0 0 1 1 0 0 0 0
1 0 1 0 1 0 0 -1
-1 0 0 0 0 1 0 0 0 -1 0 0 0 0 0
0 1 0 0 1 1 0 -1
1 0 0 0 0 0 -1 0 0 0 -1 1 1 1 -1
0 -1 0 0 -1 -1 0 1
0 1 1 1 1 0 -1 0 -1 0 0 1 1 1 1
0 0 1 1 1 1 0 -1
0 -1 0 -1 -1 0 1 0 1 -1 0 -1 -1 -1 -1
-1 1 -1 -1 0 0 0 1
0 0 0 1 1 1 0 -1 -1 1 0 1 2 1 0
0 -1 1 1 1 0 0 -1
0 0 0 -1 -1 0 0 0 1 -1 0 0 -1 -1 -1
0 1 -1 -2 -1 0 0 1
0 0 1 0 0 -1 0 0 -1 0 1 1 0 1 0
0 0 1 0 1 1 0 0
0 -1 0 -1 1 1 1 -2 -1 0 2 1 1 0 -2
0 0 2 0 2 1 0 0
0 0 -1 0 -1 0 0 2 2 0 -2 -2 -1 -1 0
0 0 -2 0 -2 -2 0 1
0 -1 0 0 0 0 0 -1 -1 0 1 1 0 0 0
1 1 1 -1 1 1 0 0
0 0 0 0 1 0 0 -1 -1 0 1 1 0 1 0
1 0 1 0 1 1 0 -1
1 -1 0 0 0 0 0 0 0 1 -1 0 1 0 0
0 -1 0 0 -1 -1 0 1
0 0 0 0 0 0 -1 0 0 -1 0 0 0 0 0
0 0 0 0 0 1 0 0
0 1 0 0 0 0 0 1 1 0 -1 0 0 0 0
0 0 -1 0 -1 -1 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 1 0 1 0 0 1 1 0 1 0 -1
0 -1 1 1 1 -1 0 0
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NOTES
The shorter Leech lattice O_23.
Also called F_1(vi) by Plesken, Q_23(4)^{+4} in LDL II (see references)
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REFERENCES
SPLAG p. 179.
Plesken, J. Algebra, vol. 97 (1985) pp. 286-312
Conway-Sloane, "Low-Dimensional Lattices II: Subgroups of GL(n,Z)",
Proc. Royal Soc. London, A 419 (1988), 29-68.
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LAST_LINE
Haftungsausschluss/Disclaimer
Gabriele Nebe