The Lattice S26
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 13:20:26 CEST 2014
INDEX FILE |
ABBREVIATIONS
Contents of this file
NAME
DIMENSION
DETERMINANT
MINIMAL_NORM
KISSING_NUMBER
GROUP_ORDER
PROPERTIES
REFERENCES
NOTES
GRAM
GROUP_GENERATORS
LAST_LINE
-
NAME
S26
-
DIMENSION
26
-
DETERMINANT
1
-
MINIMAL_NORM
3
-
KISSING_NUMBER
3120
-
GROUP_ORDER
18720000
-
PROPERTIES
-
REFERENCES
Borcherds' thesis (Ref [Bor1] of SPLAG).
G. Nebe: Finite subgroups of GL_n(Q) for 25 <= n <= 31,
Comm. Alg. 24(7), 2341-2397 (1996).
For more information see the Intro to 3rd edition of SPLAG.
-
NOTES
Discovered by J H Conway in the 1970's, proved to be unique
by R. E. Borcherds in his thesis.
Called S_26 in NJAS's notebooks, and S4(5).2 in the Nebe paper.
Group is Sp_4(5).2
-
GRAM
26 0
3
1 3
1 1 3
1 1 1 3
1 1 1 0 3
1 1 1 1 1 3
1 1 1 1 1 1 3
1 1 1 1 1 1 1 3
1 1 1 1 1 1 0 1 3
1 1 1 1 1 1 1 1 1 3
1 1 1 1 0 0 0 0 1 1 3
1 1 1 0 1 1 0 0 1 1 1 3
1 0 1 0 1 1 1 0 0 0 1 1 3
1 1 1 0 1 1 1 1 0 1 0 1 1 3
1 1 1 0 1 1 1 0 1 1 1 1 1 0 3
1 1 1 1 0 0 1 0 0 0 0 0 0 1 0 3
1 1 1 1 1 1 0 1 1 1 1 0 0 1 0 0 3
1 1 1 0 1 1 1 0 0 1 1 1 1 1 1 1 0 3
1 1 1 1 1 1 1 0 1 1 0 1 0 1 1 1 1 0 3
1 1 1 1 0 1 1 1 1 0 1 1 1 0 1 0 0 0 0 3
1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1 1 1 1 0 3
0 1 1 1 1 1 1 1 1 1 0 1 0 1 0 0 1 0 1 1 0 3
1 1 1 0 1 0 1 1 0 1 1 1 1 1 1 0 0 1 0 1 0 0 3
1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 0 1 1 1 1 3
1 0 1 1 0 0 1 0 1 0 1 0 1 0 1 1 0 0 1 1 0 0 1 0 3
1 1 0 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 1 1 1 1 1 1 1 3
-
GROUP_GENERATORS
2
26 26
0 0 0 -1 0 0 0 1 -1 0 0 0 -1 0 1 0 0 0 0 0 1 0 -1 1 1 0
0 -1 -1 -1 0 1 0 0 1 0 1 -1 1 -1 0 2 0 -1 1 0 0 1 2 0 -2 -1
0 -1 -1 -1 0 1 0 1 0 0 2 -1 0 0 0 1 -1 -1 1 0 1 1 1 0 -1 -1
0 -1 -1 -1 1 2 0 0 0 0 2 -1 -1 0 0 1 -1 -1 0 0 1 1 1 0 0 -1
0 0 0 0 0 0 -1 0 0 1 0 -1 1 0 0 1 0 0 1 1 -1 0 0 0 -1 0
0 -1 -1 -1 0 1 1 0 0 0 1 0 -1 0 0 1 0 -1 0 0 1 1 1 0 0 -1
0 -1 -1 -1 0 2 0 0 1 0 2 -2 1 -1 -1 2 -1 -1 2 0 0 1 2 0 -2 -1
0 -1 -1 0 0 1 1 0 1 0 1 0 0 0 0 1 0 -1 0 0 0 0 1 0 -1 -1
0 0 -1 -1 1 1 0 0 0 1 1 0 0 0 -1 1 0 -1 0 1 0 0 0 0 0 -1
0 -1 0 0 0 1 1 0 0 -1 1 0 -1 0 1 1 0 -1 -1 -1 1 1 1 0 -1 0
0 -1 -1 -2 1 2 0 1 0 0 3 -1 0 -1 -1 2 -1 -1 1 0 0 1 1 0 -1 -1
1 -1 2 0 -1 -1 0 0 -2 -2 -1 0 -3 1 4 -1 0 0 -2 -1 2 1 -1 1 0 3
0 -1 -1 -2 0 2 0 1 0 0 3 -1 0 -1 -1 2 -1 -1 2 0 0 1 1 0 -1 -1
0 -1 -1 -1 0 2 1 0 1 -1 3 -1 0 0 0 2 -1 -2 1 -1 0 1 2 0 -2 -1
-1 0 -1 -1 0 0 0 1 0 1 0 0 1 -1 -1 1 1 0 1 1 0 0 0 0 0 -1
1 0 2 1 -1 -2 -1 0 -1 -2 -2 -1 -2 1 4 -2 -1 1 -1 -1 2 1 -1 1 0 3
-1 0 -4 -2 2 3 1 1 1 2 4 0 2 -1 -4 3 0 -2 2 1 -1 0 2 -1 0 -5
1 -1 3 1 -1 -2 -1 0 -1 -2 -2 -1 -2 1 4 -1 -1 1 -1 -1 1 1 -1 1 -1 4
0 0 -1 -1 0 1 0 0 0 0 1 -1 0 0 0 1 0 -1 0 0 1 1 1 0 0 -1
0 -1 -2 -2 0 2 1 1 0 0 2 0 0 -1 -1 1 0 -1 1 0 1 1 1 0 0 -2
1 -1 0 -1 0 1 0 0 0 -1 2 -1 -1 0 1 1 -1 -1 0 -1 1 2 1 0 -1 0
0 -1 -2 -1 1 3 0 0 0 0 3 -1 0 0 -1 2 -1 -2 1 0 0 1 2 0 -1 -2
0 -1 0 -1 -1 1 1 1 0 -1 1 0 0 -1 1 1 0 -1 0 -1 1 1 1 0 -1 0
0 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 -1 1
0 0 -1 -2 0 2 0 1 0 0 2 -1 0 -1 -1 1 -1 -1 1 0 1 1 1 0 0 -1
1 -1 -1 -2 0 3 1 0 0 -1 3 -1 -1 -1 0 2 -1 -2 0 -1 1 2 2 0 -1 -1
26 26
0 -1 -2 -2 1 3 1 0 1 0 4 -1 0 0 -2 2 -1 -2 1 0 0 1 2 0 -1 -3
0 -1 -1 -1 0 1 1 1 0 -1 2 0 -1 0 0 1 0 -1 0 -1 1 1 1 0 0 -1
0 -1 -2 -2 1 3 0 0 1 1 4 -2 1 -1 -2 3 -1 -2 2 0 -1 1 2 0 -2 -2
0 -1 1 0 0 0 0 -1 1 0 0 -1 0 0 0 1 0 0 0 0 -1 0 1 0 -2 1
0 -1 -1 -1 0 1 2 1 -1 -1 2 1 -2 0 1 0 0 -1 -1 -1 2 1 0 0 1 -1
0 -1 -2 -1 0 2 2 0 2 0 3 0 1 -1 -2 2 0 -2 1 -1 0 1 3 -1 -2 -3
0 -1 -1 -1 0 1 1 0 0 0 1 0 -1 0 0 1 0 -1 0 0 1 1 1 0 0 -1
1 -3 -1 -2 0 4 1 0 1 -2 5 -2 -1 -1 0 3 -2 -3 1 -2 1 3 4 0 -3 -1
0 -1 -1 -2 1 3 1 0 1 0 4 -1 0 -1 -2 2 -1 -2 1 -1 0 1 2 0 -1 -2
0 -1 -1 -1 -1 1 2 0 1 -1 1 0 0 -1 0 1 1 -1 0 -1 1 1 2 0 -1 -1
1 0 1 -1 0 1 0 0 -1 -1 1 -1 -2 0 1 0 -1 -1 -1 -1 1 1 0 1 0 1
1 0 2 1 -2 -2 0 0 -1 -2 -2 0 -2 1 4 -2 0 1 -2 -2 2 1 -1 1 0 3
1 0 1 0 0 0 0 0 -2 -1 0 0 -3 1 3 -1 -1 -1 -2 -1 2 1 -1 1 1 1
0 -1 -1 -1 0 2 1 0 1 -1 3 -1 0 0 0 2 -1 -2 1 -1 0 1 2 0 -2 -1
0 0 -1 0 0 0 1 0 0 0 0 1 -1 0 0 0 1 0 -1 0 1 0 0 0 1 -1
-1 0 -1 -1 1 0 -1 0 1 2 1 -1 2 0 -2 2 0 0 2 2 -2 -1 0 0 -1 -1
0 -1 -2 -2 1 4 2 0 2 0 5 -1 1 -1 -3 3 -1 -3 1 -1 -1 1 3 -1 -2 -3
0 0 -2 -2 0 1 1 1 0 0 2 0 0 -1 -1 1 0 -1 1 0 1 1 1 0 0 -2
-1 0 -1 0 0 0 1 0 1 1 0 1 1 0 -1 1 1 0 0 0 -1 -1 0 0 0 -1
1 -1 1 0 0 1 0 -1 0 -1 1 -1 -2 1 1 0 -1 -1 -1 -1 1 1 1 0 -1 1
-1 0 -2 -2 1 2 1 1 0 1 3 0 1 -1 -2 2 0 -2 1 0 0 0 1 0 0 -3
1 -1 2 1 -1 -1 1 -1 0 -2 -1 0 -2 1 3 -1 0 0 -2 -2 1 1 0 0 -1 3
1 -1 1 0 -1 0 0 0 -1 -2 0 -1 -3 1 3 -1 -1 0 -1 -1 2 1 0 1 0 2
1 -1 1 -1 -1 1 1 0 0 -2 1 -1 -2 0 2 0 -1 -1 -1 -2 2 2 1 0 -1 1
0 0 1 0 1 0 -1 -1 0 1 0 -1 -1 1 0 0 -1 0 0 1 -1 -1 -1 1 0 1
1 -1 2 0 -1 0 1 0 -1 -2 0 0 -3 1 3 -1 -1 -1 -2 -2 2 1 0 1 0 2
-
LAST_LINE
Haftungsausschluss/Disclaimer
Gabriele Nebe