The Lattice ext12m5
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 13:27:34 CEST 2014
INDEX FILE |
ABBREVIATIONS
Contents of this file
NAME
DIMENSION
DIVISORS
MINIMAL_NORM
KISSING_NUMBER
GROUP_ORDER
GROUP_NAME
GROUP_GENERATORS
GRAM
PROPERTIES
THETA
REMARKS
LAST_LINE
-
NAME
ext12m5
-
DIMENSION
12
-
DIVISORS
1 1 1 1 1 1 5 5 5 5 5 5
-
MINIMAL_NORM
4
-
KISSING_NUMBER
126
-
GROUP_ORDER
2^8 * 3^4 * 5
-
GROUP_NAME
W(E6)
-
GROUP_GENERATORS
4
12 12
-1 -2 -3 -2 -1 -2 0 0 0 0 0 0
1 2 2 2 1 1 0 0 0 0 0 0
0 0 1 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 -1 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 -1 1 0 0 0 -1
0 0 0 0 0 0 -2 2 0 0 0 -1
0 0 0 0 0 0 -3 2 1 -1 0 -1
0 0 0 0 0 0 -2 2 0 -1 0 0
0 0 0 0 0 0 -1 1 0 0 -1 0
0 0 0 0 0 0 -2 1 0 0 0 0
12 12
0 -1 -1 -1 0 0 0 0 0 0 0 0
0 1 2 2 1 1 0 0 0 0 0 0
0 0 -1 -1 -1 0 0 0 0 0 0 0
-1 -1 -1 -1 0 -1 0 0 0 0 0 0
1 1 2 1 0 1 0 0 0 0 0 0
1 1 1 1 1 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 -1 1 0 -1 1 1
0 0 0 0 0 0 -1 2 0 -2 1 1
0 0 0 0 0 0 -1 1 1 -2 1 0
0 0 0 0 0 0 0 0 1 -1 0 0
0 0 0 0 0 0 0 1 0 -1 1 0
12 12
-1 0 0 0 0 0 0 0 0 0 0 0
0 -1 -2 -1 0 -1 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0 0 0
0 0 -1 -1 -1 -1 0 0 0 0 0 0
1 2 2 2 1 1 0 0 0 0 0 0
0 0 0 0 0 0 -1 1 0 -1 1 0
0 0 0 0 0 0 0 1 0 -2 2 0
0 0 0 0 0 0 0 1 1 -3 2 0
0 0 0 0 0 0 0 1 0 -2 1 1
0 0 0 0 0 0 0 1 0 -1 0 0
0 0 0 0 0 0 0 1 0 -1 1 0
12 12
-1 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 -1 -1 -1 -1 0 0 0 0 0 0
0 0 0 1 1 0 0 0 0 0 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 0
0 0 0 0 0 0 -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 -1 0 1 -1 -1
0 0 0 0 0 0 0 -1 0 1 0 -1
0 0 0 0 0 0 0 -1 1 0 0 -1
0 0 0 0 0 0 0 -1 0 1 -1 0
-
GRAM
12 12
4 -2 0 0 0 0 1 0 0 0 0 0
-2 4 -2 0 0 0 0 1 0 0 0 0
0 -2 4 -2 0 -2 0 0 1 0 0 0
0 0 -2 4 -2 0 0 0 0 1 0 0
0 0 0 -2 4 0 0 0 0 0 1 0
0 0 -2 0 0 4 0 0 0 0 0 1
1 0 0 0 0 0 4 5 6 4 2 3
0 1 0 0 0 0 5 10 12 8 4 6
0 0 1 0 0 0 6 12 18 12 6 9
0 0 0 1 0 0 4 8 12 10 5 6
0 0 0 0 1 0 2 4 6 5 4 3
0 0 0 0 0 1 3 6 9 6 3 6
-
PROPERTIES
INTEGRAL =1
MODULAR = 5
-
THETA
1 0 126 1008 4410 12096 32760 67536
-
REMARKS
Found by Gabriele Nebe using the convenient construction described in Bachoc/Nebe (Crelle 1998)
If F is the GramMatrix of E6 then the GramMatrix of this lattice is the block matrix
2F I6
I6 3F^-1
Its automorphism group is diag(g,g^{-tr}) (the obvious subgroup) with g in W(E6).
-
LAST_LINE
Haftungsausschluss/Disclaimer
Gabriele Nebe