The Lattice (4+sqrt(5)) x 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 13:17:57 CEST 2014
INDEX FILE |
ABBREVIATIONS
Contents of this file
NAME
DIMENSION
DET
GRAM
DIVISORS
MINIMAL_NORM
KISSING_NUMBER
GROUP_ORDER
GROUP_NAME
GROUP_GENERATORS
PROPERTIES
NOTES
LAST_LINE
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NAME
(4+sqrt(5)) x Leech
-
DIMENSION
24
-
DET
11^12
-
GRAM
24
12 -1 3 0 6 -2 -1 -2 2 -2 4 3 3 -6 4 5 0 0 -3 5 0 6 1 2
-1 12 -1 6 -2 -1 0 6 5 -3 1 -1 0 -3 -3 0 -6 0 -6 0 -3 -3 3 3
3 -1 12 -1 3 3 0 -3 -3 -2 -3 0 6 -5 -2 1 -2 -3 4 6 -3 3 -6 5
0 6 -1 12 -2 0 0 3 3 -4 5 2 2 -1 -3 0 -6 2 -4 0 -3 -3 3 0
6 -2 3 -2 12 -3 -1 -2 0 -3 -2 0 4 -6 0 5 2 -2 -3 3 -1 6 1 3
-2 -1 3 0 -3 12 3 0 -3 3 -2 -2 -3 1 -2 -4 0 0 0 4 6 -6 -3 -1
-1 0 0 0 -1 3 12 -3 3 -3 -3 -6 -2 1 -6 -1 -1 -6 -2 3 5 -6 -6 6
-2 6 -3 3 -2 0 -3 12 2 3 2 -1 -3 -1 3 -5 -3 0 -4 -2 -2 -3 6 -3
2 5 -3 3 0 -3 3 2 12 -4 5 -1 -2 0 -1 0 -4 -3 -5 0 0 -3 3 4
-2 -3 -2 -4 -3 3 -3 3 -4 12 0 1 -3 5 6 -5 6 3 3 0 3 -2 0 -6
4 1 -3 5 -2 -2 -3 2 5 0 12 6 0 2 3 1 -2 3 -3 -3 -2 1 6 -3
3 -1 0 2 0 -2 -6 -1 -1 1 6 12 2 0 3 3 -1 6 3 -3 -5 6 5 -6
3 0 6 2 4 -3 -2 -3 -2 -3 0 2 12 -3 -3 2 -3 2 3 3 -6 6 0 4
-6 -3 -5 -1 -6 1 1 -1 0 5 2 0 -3 12 2 -5 3 3 3 -2 4 -6 -1 -5
4 -3 -2 -3 0 -2 -6 3 -1 6 3 3 -3 2 12 -2 3 3 2 0 1 3 3 -6
5 0 1 0 5 -4 -1 -5 0 -5 1 3 2 -5 -2 12 1 1 -3 -2 0 6 1 2
0 -6 -2 -6 2 0 -1 -3 -4 6 -2 -1 -3 3 3 1 12 1 1 -1 5 1 -3 -4
0 0 -3 2 -2 0 -6 0 -3 3 3 6 2 3 3 1 1 12 2 0 1 3 6 -6
-3 -6 4 -4 -3 0 -2 -4 -5 3 -3 3 3 3 2 -3 1 2 12 0 -3 3 -4 -2
5 0 6 0 3 4 3 -2 0 0 -3 -3 3 -2 0 -2 -1 0 0 12 3 -1 -5 5
0 -3 -3 -3 -1 6 5 -2 0 3 -2 -5 -6 4 1 0 5 1 -3 3 12 -6 -2 -1
6 -3 3 -3 6 -6 -6 -3 -3 -2 1 6 6 -6 3 6 1 3 3 -1 -6 12 3 0
1 3 -6 3 1 -3 -6 6 3 0 6 5 0 -1 3 1 -3 6 -4 -5 -2 3 12 -5
2 3 5 0 3 -1 6 -3 4 -6 -3 -6 4 -5 -6 2 -4 -6 -2 5 -1 0 -5 12
-
DIVISORS
11^12
-
MINIMAL_NORM
12
-
KISSING_NUMBER
37800
-
GROUP_ORDER
72576000 = 2^10 * 3^4 * 5^3 * 7
-
GROUP_NAME
2.J2 Y SL(2,5)
-
GROUP_GENERATORS
#3
24
-1 -1 0 -1 0 0 -1 0 1 0 0 1 1 -1 0 -1 -1 1 -2 -1 1 0 -2 0
2 0 1 1 -1 0 0 -1 -1 1 -1 -2 -2 2 -1 1 1 -1 1 1 -2 0 4 1
-1 -1 0 -1 -1 0 0 0 0 -1 0 1 1 -1 1 1 1 0 -1 1 -1 -2 0 0
-1 0 0 1 0 -1 -1 0 0 1 1 0 0 -1 -1 -1 -1 -1 0 0 3 2 -1 0
0 -1 1 -1 -1 1 1 -1 0 -1 -1 0 0 0 2 2 2 0 0 1 -4 -4 4 1
-2 1 -1 0 1 -2 -2 0 0 0 2 1 1 -2 -1 -2 -2 -1 0 0 5 2 -4 -2
0 0 -1 0 0 1 0 0 -1 -1 0 0 0 0 1 1 2 -1 1 1 -2 -2 2 1
1 1 0 2 1 -1 -2 1 1 2 0 -1 -2 2 -4 -2 -3 -1 1 -1 5 7 -3 -1
1 0 0 1 0 1 0 -1 0 1 -1 -1 -1 1 0 0 0 0 0 -1 -1 1 1 1
0 1 0 0 2 -1 -1 1 1 1 1 0 0 0 -2 -2 -3 1 0 -2 4 3 -5 -2
0 0 0 1 1 0 -1 0 1 1 0 0 0 0 -1 -2 -2 1 -1 -2 3 3 -3 0
0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 1 -1 -1 1 1 -1 0
1 -2 1 -1 -1 1 1 0 0 -1 -1 0 0 0 2 2 2 2 -1 0 -5 -5 2 2
0 0 0 0 1 0 1 0 0 0 0 0 0 -1 1 0 0 2 0 -1 -1 -2 -1 0
0 0 0 0 1 -1 -1 1 1 1 0 1 0 0 -2 -2 -3 1 -1 -2 4 3 -4 -1
-1 0 0 -1 -1 1 1 -1 -1 -1 0 0 1 0 2 2 2 -1 0 2 -3 -3 4 1
-2 0 0 -1 1 0 1 0 1 0 1 0 1 -1 1 0 -1 1 -1 0 0 0 -2 -2
0 0 1 0 0 -1 0 0 0 0 0 0 0 0 0 -1 -1 1 -1 -1 1 0 -1 0
0 -1 0 -1 0 0 1 1 0 -1 0 1 1 -1 1 0 0 2 -1 -1 -1 -2 -2 0
-1 -1 0 -1 0 -1 -1 0 1 0 0 1 1 -2 0 -1 -1 1 -2 -1 1 -1 -3 -1
-2 1 -1 0 1 -1 -1 0 0 0 1 1 1 -1 0 -1 -1 -1 0 0 3 1 -2 -1
1 -1 1 -1 -1 1 1 0 0 -1 -1 0 0 1 1 1 1 1 -1 0 -3 -2 2 1
2 1 1 2 0 0 -1 0 0 1 -1 -1 -2 3 -2 -1 -1 -1 1 -1 2 4 1 1
1 -1 0 -1 -1 1 1 -1 -1 -1 -1 0 0 0 2 2 3 0 0 1 -5 -5 4 2
24
0 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 0 1 0 1 0 0 0 1 1
1 0 1 0 0 0 0 0 0 1 0 -1 -1 1 -1 0 0 0 0 0 -1 0 1 0
0 0 0 -1 0 0 0 0 0 -1 0 1 1 -1 1 0 1 1 0 -1 0 -3 0 1
-1 0 0 0 1 -1 -1 1 1 1 1 0 0 -1 -2 -2 -2 1 -1 -1 3 3 -4 -1
0 0 0 0 -1 0 1 0 -1 -1 0 0 0 0 1 2 3 -1 2 2 -2 -3 4 1
-2 1 -2 -1 1 -1 -2 1 1 0 2 1 2 -2 -1 -2 -3 0 -1 -1 5 3 -6 -2
0 0 -1 0 1 0 -1 2 1 0 0 1 1 0 -1 -2 -2 1 -1 -2 3 3 -5 -1
0 -1 0 0 0 0 -1 0 1 1 0 -1 0 0 -1 0 -1 0 -1 0 0 1 -1 -1
-1 -1 0 -1 1 0 1 0 0 0 1 0 0 -1 1 0 0 2 -1 0 -2 -2 -1 0
0 0 -1 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 -1 0 0 1 1 -1 0
-2 -1 0 0 1 -1 0 0 1 1 1 0 0 -2 0 -1 -2 2 -2 -1 1 1 -4 -1
0 0 1 1 -1 0 1 -1 -1 0 -1 0 -1 1 1 1 2 -1 0 1 -2 -1 4 2
1 0 1 0 -1 1 1 0 -1 -1 -1 0 -1 1 1 2 3 0 1 1 -3 -3 4 2
-2 0 -1 0 1 -1 -1 1 1 1 1 1 1 -1 -1 -2 -4 0 -2 -1 5 5 -6 -2
0 0 0 1 0 -1 -1 -1 0 1 0 0 -1 0 -1 -1 -1 -1 0 0 2 2 0 0
0 0 1 0 -1 0 2 -1 -1 -1 0 0 0 -1 2 1 3 0 1 1 -3 -4 4 1
-1 0 -1 0 0 0 0 0 1 0 0 0 1 -1 1 0 -1 0 0 0 1 1 -2 -1
0 0 1 0 -1 0 1 -1 -1 0 0 -1 -1 0 1 1 1 0 0 1 -2 -1 3 1
2 0 1 0 -1 1 1 -1 -1 -1 -2 0 -1 2 1 1 2 0 0 0 -3 -2 4 2
0 1 -1 0 1 0 -1 1 0 0 1 0 0 0 -1 -1 -1 0 1 -1 3 2 -2 0
-2 0 -2 -1 1 -1 -1 1 1 0 2 1 2 -3 0 -2 -3 1 -1 -1 4 2 -6 -2
2 0 2 1 -2 1 2 -2 -2 -1 -2 -1 -2 2 2 3 5 -1 2 2 -6 -5 9 3
0 -1 1 0 -1 0 1 -1 -1 0 0 -1 -1 0 1 2 2 0 0 2 -4 -3 4 1
1 0 0 -1 0 1 1 0 -1 -1 0 0 0 0 1 1 2 1 1 0 -3 -4 2 1
24
1 0 0 1 0 0 0 0 0 1 0 -1 -1 1 -1 -1 -1 0 0 -1 1 3 -1 0
0 0 1 1 0 0 0 0 0 0 0 -1 -1 1 0 0 0 0 0 0 0 1 1 0
2 0 1 1 0 0 0 0 0 1 -1 -1 -1 2 -2 -1 0 0 0 -1 0 2 1 0
-1 -1 1 0 0 -1 0 0 1 1 0 0 0 -1 0 -1 -2 1 -2 -1 1 1 -2 -1
0 0 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 1 0 -1
2 0 1 1 -1 1 0 0 -1 0 -2 0 -1 2 0 1 2 -1 1 0 -2 -1 4 2
2 -1 0 1 0 1 0 1 0 0 -2 0 -1 2 0 1 1 0 1 0 -2 -1 2 2
0 1 0 1 0 0 -1 1 1 0 0 0 0 1 -1 -1 -1 -1 0 -1 3 4 -2 -1
0 1 0 1 0 0 0 0 -1 0 0 0 -1 2 0 0 0 -2 1 1 1 2 2 1
-1 0 -1 -1 0 0 -1 1 0 -1 1 1 1 -1 0 0 0 0 0 0 1 0 -2 0
0 0 1 0 -1 0 1 -1 -1 0 0 0 -1 1 1 1 1 -1 0 1 -2 -1 3 1
0 0 1 -1 0 -1 1 -1 -1 0 1 0 -1 -1 0 0 0 1 0 0 -1 -2 1 0
1 -1 1 -1 0 0 1 0 0 0 0 -1 0 0 0 0 0 2 -1 -1 -2 -2 0 0
0 0 0 -1 0 0 0 0 -1 -1 0 1 0 0 1 1 1 0 0 1 -2 -3 1 1
0 1 -1 0 0 0 -1 0 0 0 1 0 0 0 -1 -1 -1 -1 0 0 2 3 -2 -1
-1 -1 0 0 0 0 1 -1 0 0 0 0 0 -1 1 0 0 1 -1 0 -1 -1 0 0
-2 0 -1 -1 0 0 0 0 0 -1 1 1 1 -2 1 0 1 0 0 1 0 -1 -1 0
-1 0 0 -1 0 0 0 0 0 0 1 0 0 -1 0 0 -1 1 -1 0 0 0 -2 -1
1 0 0 -1 0 0 0 0 0 0 0 0 0 0 -1 0 0 1 0 0 -1 -1 0 0
2 0 0 1 0 0 -1 1 0 1 -1 -1 -1 2 -2 -1 -1 0 0 -1 1 3 -1 0
0 0 -1 1 0 1 -1 1 0 0 -1 1 0 1 0 0 0 -1 0 0 1 2 -1 1
0 0 0 -1 0 0 1 -1 0 0 1 -1 0 -1 0 0 0 1 0 0 -1 -1 0 -1
-1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 -1 -1 0 0 2 2 -1 -1
1 -1 0 0 0 1 1 0 0 0 -1 -1 0 1 1 1 1 1 0 0 -3 -2 2 1
-
PROPERTIES
INTEGRAL = 1
MODULAR=11
-
NOTES
The space of invariant forms of 2.J2 Y SL(2,5)
contains the Leech lattice. In this space one can construct a
sequence of modular lattices of level (n^2+5n+5) and minimum 2(2n+4)
for n=0,1,2,... of which the density tends to the density of the
Leech lattice.
For n=0 one finds the well known extremal 5-modular lattice
and for n=1, one gets this lattice.
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LAST_LINE
Haftungsausschluss/Disclaimer
Gabriele Nebe