The Lattice HS20
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 13:16:36 CEST 2014
INDEX FILE |
ABBREVIATIONS
Contents of this file
NAME
DIMENSION
GRAM
DIVISORS (elementary divisors)
MINIMAL_NORM
KISSING_NUMBER
GROUP_ORDER
GROUP_GENERATORS
PROPERTIES
REFERENCES
LAST_LINE
-
NAME
HS20
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DIMENSION
20
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GRAM
20 0
4
0 4
0 0 4
-2 2 2 4
1 0 -1 -1 4
0 1 0 1 0 4
1 0 1 0 0 0 4
0 0 1 1 -2 2 2 4
1 1 2 1 -1 1 1 1 4
-1 1 0 0 -1 -1 -1 -1 0 4
-2 0 1 2 -1 1 -1 1 0 0 4
-2 1 -1 1 0 0 -2 -1 -2 2 2 4
0 -1 0 -1 2 -1 1 -1 -1 0 -1 0 4
1 0 -1 -1 1 2 0 0 0 -1 0 0 0 4
0 1 0 0 -1 0 2 1 1 0 -1 -1 0 0 4
1 1 0 0 -1 2 1 2 1 -1 0 -1 -2 2 2 4
1 0 0 0 2 1 0 -1 0 0 -1 0 0 1 -1 0 4
0 1 1 1 -1 2 -1 1 0 0 1 1 -1 0 0 1 0 4
0 -1 1 0 0 1 2 2 1 -1 0 -1 1 0 0 0 0 0 4
-1 0 1 1 -1 1 0 2 0 -1 1 0 0 -1 0 0 -2 2 2 4
-
DIVISORS (elementary divisors)
2^10
-
MINIMAL_NORM
4
-
KISSING_NUMBER
3960
-
GROUP_ORDER
2^14 * 3^2 * 5
-
GROUP_GENERATORS
#2
20
1 -1 -1 1 -1 1 1 -2 1 0 0 1 1 -1 -1 2 0 -1 0 1
1 0 0 1 0 0 -1 0 -1 1 1 -1 0 0 1 0 0 0 1 -1
0 -1 0 1 1 0 0 0 0 0 -1 1 -1 0 0 0 -1 1 1 -1
0 0 0 1 1 -1 -1 1 -1 1 0 -1 -1 1 1 -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 1 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 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 -1 0 1 -1 2 1 -3 0 0 0 1 0 -1 0 1 0 -1 0 1
0 1 1 -1 0 0 -2 1 -1 0 1 -1 0 0 1 -1 0 0 1 -1
0 0 0 1 2 -1 -1 1 -1 1 0 -1 -1 1 1 -1 -1 1 1 -1
0 1 0 0 1 -2 -2 2 -1 1 1 -2 0 1 1 -1 0 1 1 -1
0 0 0 -1 0 -1 0 1 1 -1 0 1 0 0 0 0 1 0 -1 1
1 0 -1 1 0 0 1 -1 0 1 1 -1 0 0 0 0 0 0 0 1
1 -1 0 1 -1 1 1 -1 0 0 0 1 0 0 0 0 0 -1 -1 1
1 -1 -1 2 0 1 1 -1 0 1 0 0 0 0 0 0 -1 0 0 0
0 1 0 0 0 -1 -1 1 0 0 0 -1 0 1 0 -1 0 1 1 -1
-1 0 0 0 1 -1 0 1 1 0 -1 1 0 0 -1 1 -1 1 0 -1
0 0 -1 0 -1 0 1 -1 1 -1 0 1 1 -1 -1 1 1 0 -1 1
-1 0 0 -1 0 0 1 0 1 -1 -1 2 0 -1 -1 1 0 0 -1 0
20
0 0 0 0 -1 1 1 -2 0 0 0 0 1 -1 -1 2 1 -1 -1 2
0 0 0 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 -1 0 0
0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 -1 0
0 0 0 0 1 0 0 1 0 0 0 0 -1 0 0 -1 -1 0 0 -1
-1 1 1 -2 0 0 0 1 1 -1 -1 2 0 0 0 0 1 -1 -1 1
0 1 0 0 0 -1 -1 1 0 -1 0 0 0 0 0 0 0 0 1 -1
0 0 1 0 1 0 -1 1 0 0 -1 0 0 0 0 0 -1 0 0 -1
1 0 0 1 0 0 -1 0 -1 0 0 -1 0 0 0 0 -1 0 1 -1
0 0 0 0 0 0 1 -1 0 0 0 0 0 -1 -1 1 0 0 0 0
0 0 0 0 -1 1 1 -1 0 0 0 0 0 0 0 -1 0 0 0 0
0 0 0 0 0 -1 0 1 0 0 0 0 -1 1 0 -1 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0 0 -1 1 1 -2 0 0 0 0
-1 1 2 -2 1 0 -1 2 0 -1 -1 1 -1 1 1 -2 0 0 0 -1
-1 1 0 0 1 -2 -1 2 0 0 0 -1 0 1 0 0 0 1 1 -1
1 -1 0 2 1 0 -1 0 -1 1 0 -1 0 0 1 0 -1 0 1 -1
1 0 -1 2 0 -1 -1 0 -1 1 1 -2 1 0 0 1 0 0 1 0
0 1 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
1 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
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PROPERTIES
MODULAR = 2
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REFERENCES
Found by Boris Hemkemeier and Rudolf Scharlau by a random search in the
neighborhood graph.
That there are only 3 extremal 2-modular lattices of dimension 20 is shown in
C. Bachoc, B. Venkov:
Modular forms, lattices and spherical designs in
``Reseaux euclidiens, ``designs'' spheriques et groupes, J. Martinet, (editor), L'Enseignement Mathematique, Monographie n. 37 (2001)'',
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LAST_LINE
Haftungsausschluss/Disclaimer
Gabriele Nebe