The Lattice LAMBDA(QR)
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 13:17:47 CEST 2014
INDEX FILE |
ABBREVIATIONS
Contents of this file
NAME
DIMENSION
GRAM
DET
MINIMAL_NORM
MINVECS
KISSING_NUMBER
DENSITY
HERMITE_NUMBER
GROUP_ORDER
GROUP_GENERATORS
BACHER_POLYNOMIALS
PROPERTIES
REFERENCES
NOTES
LAST_LINE
-
NAME
LAMBDA(QR)
-
DIMENSION
32
-
GRAM
32 0
4
-2 4
1 -2 4
1 -2 2 4
-2 0 0 -1 4
-2 2 -1 -2 0 4
-2 2 -1 0 1 1 4
-2 2 -2 -2 1 1 2 4
-2 2 0 0 0 2 1 1 4
0 1 1 -1 1 1 0 -1 1 4
0 -1 -1 -1 0 0 0 1 -2 -2 4
1 -1 -1 1 0 -1 -1 -1 0 0 0 4
1 -2 0 1 1 -1 0 0 -1 -1 0 1 4
-2 2 0 0 0 1 2 1 1 0 0 -1 -2 4
-1 1 1 -1 1 1 1 0 0 2 0 -1 -1 2 4
-1 0 1 -1 2 1 1 1 0 1 0 -1 1 0 2 4
-1 -1 1 0 2 0 1 0 0 1 1 0 0 0 1 1 4
-1 2 0 -1 0 2 1 0 1 1 0 -1 -2 2 1 0 0 4
-2 0 0 0 2 1 0 0 1 0 -1 0 1 0 0 1 0 0 4
0 -1 2 1 0 0 0 0 1 0 -1 -1 1 0 0 1 0 -1 0 4
0 1 -1 -2 0 0 -1 1 0 0 0 -1 -1 0 0 -1 -1 0 0 0 4
0 1 -1 -1 -1 0 -1 1 0 -1 0 0 -1 1 0 -1 -2 0 0 -1 2 4
-1 0 1 -1 1 1 0 0 0 1 0 -2 -1 0 1 1 1 1 0 0 0 0 4
1 0 0 1 -2 0 0 -1 0 0 0 0 -1 0 -1 -2 0 1 -2 -1 -1 0 1 4
0 0 0 1 -1 -1 -1 -1 0 0 -1 1 -1 0 0 -1 -1 -1 0 0 0 1 0 0 4
1 -2 1 1 -1 -1 -1 0 -1 -2 1 -1 1 -1 -1 0 0 -1 -1 1 0 0 1 1 0 4
-2 0 0 1 2 0 2 0 0 0 0 0 1 1 1 1 1 0 2 0 -1 -1 0 -1 0 -1 4
-1 2 -1 -2 0 2 0 1 1 1 -1 -1 -1 0 0 1 -1 1 1 -1 0 1 1 0 0 -1 -1 4
-1 0 0 1 1 0 1 0 0 0 0 0 0 1 0 0 0 1 1 0 0 -1 0 0 0 0 2 -1 4
1 1 0 -1 0 0 0 0 0 1 0 0 -1 0 0 0 0 1 -1 0 1 0 -1 0 -1 -1 -1 0 0 4
-1 0 0 -1 0 1 -1 0 0 0 1 -1 -1 0 0 0 0 1 1 -1 0 0 1 0 0 0 0 1 0 -1 4
-1 0 0 -1 2 0 1 1 -1 0 1 -1 0 1 1 1 1 0 0 1 1 0 1 -1 -1 0 1 -1 1 1 -1 4
-
DET
1
-
MINIMAL_NORM
4
-
MINVECS
-
KISSING_NUMBER
146880
-
DENSITY
-
HERMITE_NUMBER
4
-
GROUP_ORDER
2^21 * 3 * 5 * 31
-
GROUP_GENERATORS
3
32 32
-4 -13 3 -3 -1 1 -2 4 5 -1 -2 -3 5 5 -1 1 -5 2 -4 -9 4 -7 -1 2 5 -3 2 1 -3 2 -2 2
4 10 -4 6 0 0 -1 0 -5 0 0 3 -4 -4 2 -1 5 0 3 7 -2 4 2 -2 -4 2 0 0 1 -1 2 -1
-2 -1 3 -3 0 0 2 -1 0 -1 -1 1 1 0 0 -1 -1 -1 -1 -1 0 0 0 0 0 0 -1 0 1 0 0 0
-3 -5 2 -3 -1 -1 0 0 3 0 0 -2 3 2 -1 1 -2 0 -2 -4 1 -2 -1 1 2 -2 -1 0 0 0 -1 1
3 7 1 -2 2 0 3 -4 -2 0 1 2 -3 -2 -1 0 2 -2 2 4 -2 4 -1 0 -2 2 -1 -1 2 -2 1 -1
3 11 -2 4 1 0 1 -2 -5 0 1 3 -4 -5 2 -2 4 -1 3 7 -3 5 1 -2 -4 3 -1 0 2 -1 2 -1
3 8 -3 3 0 -1 1 -2 -3 0 0 2 -4 -5 2 -1 4 0 2 6 -3 5 0 -2 -4 2 -1 0 2 -2 2 0
3 6 -3 3 1 0 1 -1 -3 0 0 2 -4 -4 2 -1 3 0 2 6 -2 4 0 -1 -3 2 0 0 1 -1 2 -1
3 9 -2 4 1 0 0 -1 -4 -1 0 2 -3 -4 2 -1 4 -1 2 6 -2 4 1 -1 -3 1 -1 0 2 -1 2 -1
2 7 1 1 0 0 1 -1 -3 0 0 3 -2 -2 0 -1 2 -1 2 3 -1 2 1 -1 -2 2 0 0 1 -1 1 0
0 -1 0 -1 0 0 1 -1 1 0 1 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 -4 2 -2 0 0 -1 1 3 0 1 -2 2 3 -2 2 -2 0 -1 -4 2 -3 -1 2 3 -2 1 0 -1 0 -1 1
-1 -10 2 -4 2 1 -1 1 6 0 0 -5 3 4 -2 2 -4 1 -3 -7 2 -4 -4 3 5 -2 1 0 -2 0 -2 2
2 12 -3 3 -1 -2 2 -3 -6 1 1 4 -5 -6 2 -2 5 -1 4 9 -4 7 2 -3 -6 3 -2 -1 3 -2 2 -2
2 9 -1 1 0 -1 2 -3 -4 1 1 3 -4 -4 0 -1 3 -1 3 6 -3 5 1 -2 -4 3 -1 -1 2 -2 1 -1
2 7 0 0 2 0 3 -3 -3 1 1 2 -4 -4 0 -1 2 -1 2 5 -3 5 -1 -1 -3 3 -1 -1 2 -2 1 -1
1 4 1 -2 0 -1 3 -3 -1 -1 0 2 -2 -3 1 -1 1 -1 1 3 -2 3 0 -1 -2 1 -1 0 2 -1 1 0
2 10 -1 3 0 0 1 -2 -5 -1 0 4 -3 -4 2 -2 4 -1 2 6 -2 4 2 -2 -4 2 -1 0 2 -1 2 -1
2 4 1 0 3 1 1 -2 -1 1 2 0 -1 1 -2 1 0 -2 1 1 0 1 -1 1 0 1 -1 -1 0 -1 0 -1
-1 -2 0 -1 0 0 1 0 0 0 -1 -1 0 -1 1 -1 0 0 0 0 -1 1 -1 0 0 0 -1 0 1 0 0 0
0 -1 -1 1 0 1 -1 1 -1 0 -1 0 0 1 0 0 0 0 0 0 1 -1 1 0 0 0 1 0 -1 1 0 -1
0 -1 -1 1 0 0 -1 1 0 0 0 0 0 1 0 0 0 0 0 0 1 -1 1 0 0 0 1 0 -1 1 0 -1
0 5 -1 0 0 -1 2 -3 -2 0 0 2 -2 -3 1 -2 2 -1 1 4 -3 4 1 -2 -3 2 -1 0 2 0 1 -1
-2 -2 -1 1 -2 -1 -1 1 1 -1 -1 0 1 -1 2 -1 0 1 -1 -1 0 -1 1 -1 0 -1 0 1 0 1 0 1
-2 -3 0 0 -2 -1 -1 1 1 2 1 -1 2 3 -2 1 -1 0 0 -2 1 -2 1 0 1 -1 0 0 -1 1 -1 0
-5 -6 0 -2 -1 -1 1 -1 2 1 0 -2 2 0 0 0 -2 1 -2 -2 0 0 -1 0 1 -1 -1 0 0 1 -1 0
2 5 1 -1 1 0 1 -3 0 1 2 0 -1 0 -2 1 1 -2 1 1 -1 2 -1 0 -1 1 -1 -1 1 -2 0 0
2 6 -2 4 1 0 0 0 -3 1 1 2 -2 -2 1 -1 2 0 2 4 -1 2 1 -1 -2 2 0 0 0 0 1 -1
0 8 0 0 0 -1 2 -4 -3 2 2 2 -2 -2 -1 0 2 -2 2 4 -2 4 0 -1 -3 2 -2 -1 2 -2 1 -1
1 2 0 2 0 1 0 1 -2 -1 -1 2 -1 -1 1 0 1 0 1 1 1 0 1 0 -1 0 0 0 0 0 1 0
1 4 1 1 1 1 0 0 -2 0 1 2 -1 0 0 -1 0 -1 1 2 0 0 1 0 -1 1 0 0 0 0 0 -1
1 4 -1 -1 0 -1 3 -3 -2 0 0 2 -3 -3 1 -1 2 -1 2 4 -2 4 0 -1 -3 2 -1 0 2 -1 1 -1
32 32
-4 -10 1 -2 -1 0 -1 3 3 -1 -2 -2 3 2 1 0 -3 2 -3 -5 2 -4 -1 1 3 -2 1 1 -2 2 -1 1
5 6 0 2 1 1 -1 1 -2 -1 0 2 -2 0 0 0 2 -1 2 2 0 0 1 0 -1 1 1 0 0 -1 1 0
-2 -1 2 -2 0 0 2 -1 0 0 0 1 0 -1 0 -1 -1 0 -1 0 0 0 -1 0 0 1 0 1 0 0 0 0
-2 -7 1 -2 1 0 1 0 3 0 0 -2 1 1 0 0 -3 1 -2 -3 1 -2 -2 1 2 -1 0 1 -1 0 -1 1
2 13 -1 2 1 -1 3 -5 -5 1 2 4 -4 -5 0 -1 4 -2 3 8 -3 7 1 -2 -5 3 -2 -1 3 -2 2 -2
3 6 -1 3 0 1 -1 0 -2 1 1 1 -1 0 -1 1 2 -1 2 2 0 1 1 0 -1 1 0 -1 0 -1 1 0
4 8 0 1 1 0 2 -2 -3 -1 0 3 -4 -4 1 -1 3 -1 2 5 -2 4 0 -1 -3 2 0 0 2 -2 2 0
2 7 0 1 -1 -1 1 -2 -2 0 1 2 -1 -2 0 0 3 -1 2 3 -1 3 1 -1 -2 1 -1 0 2 -1 1 0
3 5 0 3 1 1 0 0 -2 0 1 2 -1 0 0 0 1 -1 1 2 1 0 1 0 -1 1 0 1 0 -1 1 0
3 8 0 2 1 1 0 0 -4 -2 -1 4 -3 -3 2 -2 3 -1 1 5 -1 2 1 -1 -3 2 1 1 1 -1 2 -1
-2 1 -1 -2 -2 -2 1 -2 0 1 0 -1 0 -2 0 0 1 0 1 1 -3 3 0 -1 -1 0 -2 -2 2 0 0 0
-1 -5 -1 0 1 0 -2 1 3 0 0 -3 2 2 0 1 -2 1 -2 -3 1 -2 -1 1 2 -2 0 0 -1 0 -1 1
-3 0 -1 2 -1 -1 1 -1 -1 2 1 0 0 -2 0 0 0 1 0 1 0 2 0 -1 -1 0 -1 0 0 0 0 0
3 1 2 -2 1 1 0 0 1 -1 0 0 0 1 -1 0 0 -1 0 -1 0 -1 -1 1 1 0 1 0 0 -1 0 1
2 6 0 0 0 0 0 -1 -2 0 0 2 -2 -2 0 -1 2 -1 1 3 -2 2 0 -1 -2 2 1 0 1 -1 1 0
1 6 0 1 0 0 1 -2 -2 1 1 2 -2 -2 -1 0 2 -1 1 3 -1 3 0 -1 -2 2 0 0 1 -1 1 0
1 8 -1 1 0 -1 2 -3 -3 -1 0 3 -3 -5 2 -2 3 -1 1 6 -3 5 1 -2 -4 2 -1 0 3 -1 2 -1
3 5 1 0 1 1 1 0 -2 0 0 2 -2 -1 0 -1 1 -1 2 2 -1 1 0 0 -1 2 0 -1 0 -1 1 0
1 4 -1 2 1 0 1 -2 -1 3 3 0 -1 0 -2 1 0 -1 2 2 0 2 0 0 -1 1 -1 -1 0 -1 0 -1
-1 0 1 1 0 1 1 0 -1 0 0 1 0 -1 0 0 0 0 -1 0 1 0 0 0 0 0 0 1 0 0 0 0
1 1 -2 2 0 0 -1 1 -1 0 0 0 0 0 1 0 1 0 1 1 0 0 1 0 0 0 0 0 0 1 0 -1
0 -6 0 -1 0 0 -3 2 4 1 1 -4 4 5 -2 2 -2 1 -1 -5 2 -4 -1 2 4 -2 1 0 -2 1 -2 1
1 5 0 0 0 0 1 -2 -2 0 0 2 -1 -1 0 -1 2 -1 1 3 -1 2 1 -1 -2 2 0 0 1 0 1 -1
0 -5 1 -1 0 1 -1 2 2 -2 -2 -1 2 2 1 -1 -1 1 -2 -3 1 -3 0 0 2 -1 1 1 -1 1 0 1
0 -8 0 -1 1 1 -3 3 4 -1 -1 -3 2 4 0 1 -3 1 -2 -5 2 -5 -1 2 3 -2 2 1 -2 1 -2 1
-4 -7 1 -1 -2 0 0 1 2 0 -1 -1 3 2 0 0 -2 1 -2 -4 2 -3 0 0 2 -1 0 1 -1 2 -1 1
2 7 -1 1 2 0 2 -3 -3 1 1 1 -4 -4 0 -1 2 -1 2 5 -3 5 -1 -1 -3 2 -1 -1 2 -2 1 -1
2 2 2 0 0 1 -1 1 0 0 1 1 1 3 -2 1 0 -1 1 -1 2 -2 1 1 1 0 1 0 -1 0 0 0
2 3 0 0 2 1 2 -2 -1 -1 0 1 -2 -2 1 -1 1 -1 1 2 -1 2 -1 0 -1 1 -1 0 1 -1 1 0
2 2 0 1 1 1 0 1 -1 -2 -1 1 -1 -1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0
-1 3 0 -1 -2 -1 1 -1 -2 1 0 1 -1 -1 0 -1 1 -1 2 2 -2 2 1 -1 -2 1 -1 -1 1 0 0 -1
2 7 -1 1 1 0 1 -3 -2 0 1 1 -2 -3 0 0 3 -1 1 4 -2 4 0 -1 -2 1 -1 -1 2 -1 1 -1
32 32
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 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 -1 0 0 0 0 0 0 0 1 0 0 0 1 -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 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 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 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 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 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 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 1 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 0 0 0 0 0 -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 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 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 -1 -1 0 0 0 0 0 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 -1 -1 0 0 0 0 0 0 1 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 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 2 1 0 1 1 0 -1 0 0 1 0 0 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 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
-2 -2 2 -2 -1 0 1 0 0 0 -1 1 1 1 -1 0 -1 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 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 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0
0 -2 1 -2 2 1 1 -1 1 0 0 -1 0 1 -1 0 -1 -1 -1 -1 0 0 -1 1 1 0 0 0 0 0 0 0
1 1 1 0 1 1 0 -1 1 0 1 -1 1 1 -1 0 0 -1 -1 -1 0 0 0 0 1 0 0 0 0 0 0 0
-2 0 3 -2 -1 0 1 0 -1 -1 -1 2 0 0 0 0 -1 -1 0 -1 1 -1 0 1 0 0 0 0 0 0 0 0
1 -1 0 0 1 1 -1 0 1 -1 0 -1 1 2 -1 1 0 -1 -1 -1 1 -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 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 0 0 0 0 0 1 0 0 0 0
2 1 1 2 2 2 -1 1 0 0 1 0 0 2 -1 1 -1 -1 0 -1 2 -2 0 1 1 0 1 0 -1 0 0 0
0 -5 1 0 0 1 -2 2 3 0 0 -2 3 4 -2 2 -2 1 -2 -5 3 -4 0 1 3 -2 1 0 -2 1 -1 1
0 0 -1 0 -1 -1 -1 0 1 1 1 -1 0 0 0 0 0 0 1 0 -1 0 0 0 0 0 0 0 0 0 0 1
0 0 1 0 1 1 1 0 -1 -1 -1 1 0 0 0 0 0 0 -1 0 1 0 0 0 0 0 0 0 0 0 1 0
-
BACHER_POLYNOMIALS
1 32
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
x^4872*1240 (32768x)
1 32
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 0 0 0
x^5208*24 x^5160*24 x^5016*8 x^5004*192 x^4872*800 x^4716*192 (39680x)
1 32
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0
x^5544*120 x^5208*480 x^5160*480 x^5016*160 (992x)
-
PROPERTIES
Unimodular=1
-
REFERENCES
H. Koch, B. B. Venkov,
\"Uber ganzzahlige unimodulare euklidische Gitter.
Journal f\"ur die reine und angewandte Mathematik 398 (1989), 144-168
H. Koch, B.B. Venkov, \"Uber gerade unimodulare
Gitter der Dimension 32, III
Mathem. Nachr. 152 (1991), 191-213
-
NOTES
-
LAST_LINE
Haftungsausschluss/Disclaimer
Gabriele Nebe