The Lattice P_48n
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 13:12:30 CEST 2014
INDEX FILE |
ABBREVIATIONS
Contents of this file
NAME
DIMENSION
DET
GRAM
DIVISORS
MINIMAL_NORM
KISSING_NUMBER
HERMITE_NUMBER
GROUP_ORDER
GROUP_NAME
GROUP_GENERATORS
PROPERTIES
REFERENCES
NOTES
LAST_LINE
-
NAME
P_48n
-
DIMENSION
48
-
DET
1
-
GRAM
48 0
6
3 6
3 1 6
-2 1 -3 6
1 2 2 -2 6
-1 1 0 -1 3 6
-1 -1 1 -2 -1 1 6
0 1 1 -1 3 0 1 6
-2 0 -2 3 0 0 -3 0 6
1 2 2 0 3 0 -1 3 2 6
-2 0 -2 3 0 -1 0 2 2 0 6
1 -1 0 0 1 1 0 -1 -1 -1 1 6
2 0 2 -2 2 -1 -1 0 -2 1 -2 2 6
-1 -1 1 -2 1 2 3 2 0 1 0 1 -1 6
0 2 -1 3 -1 -1 -2 0 3 1 2 -2 -3 -2 6
-2 -2 -1 -1 -2 -1 2 1 0 -1 1 -1 -2 1 1 6
-2 -3 0 0 -1 0 -1 -2 2 0 -1 0 -1 0 0 0 6
1 2 0 -1 3 1 -2 2 0 2 0 0 1 -1 1 1 -2 6
0 -2 1 0 1 -1 0 1 -1 0 1 3 3 1 -2 -1 -1 -1 6
1 -1 1 -3 -1 1 3 -1 -3 -2 -3 -1 0 1 -2 0 -1 -1 0 6
3 3 1 0 2 1 0 0 -1 0 1 2 1 -1 0 -2 -2 0 1 0 6
0 1 2 -2 0 -1 0 0 -1 1 -3 -4 0 -1 1 2 1 1 -2 1 -2 8
-1 1 1 -1 3 3 0 0 1 1 0 1 1 1 -1 0 1 1 0 -1 2 1 6
1 2 -1 0 1 2 -2 -2 1 -1 0 1 -1 -1 1 -1 0 2 -2 0 2 -1 2 6
-3 0 -2 3 1 2 -2 -1 4 0 1 1 -1 1 1 -1 2 -1 1 -2 0 -1 3 1 8
-2 1 -2 1 1 1 0 1 0 -1 1 -2 -1 0 0 -1 -2 0 0 1 0 0 0 1 2 6
0 0 -2 -1 -1 1 2 -1 -2 -1 -1 1 0 0 -1 2 -1 1 -1 1 0 0 0 1 -2 -1 6
1 -1 1 0 1 -1 -3 -1 1 1 -1 1 2 -2 1 -2 2 0 2 -1 0 0 0 0 1 -1 -1 6
-2 0 1 2 0 2 1 1 2 2 0 -2 -3 2 1 -1 2 -2 -1 0 -2 0 0 -1 2 1 -2 -1 8
-3 -3 -1 -1 -1 1 1 -1 0 -1 -1 0 -1 1 0 3 3 0 -1 0 -3 2 1 -1 1 -1 1 0 0 6
-1 -2 1 -2 1 2 2 1 -1 -1 0 2 1 3 -3 0 1 -2 2 1 0 -1 2 -1 1 0 -1 -1 1 1 6
2 1 0 -2 2 2 -1 -1 -1 0 -1 1 1 -1 0 -1 -1 2 -1 1 1 0 0 2 -2 0 1 1 -2 0 0 6
-3 -3 0 -2 -1 1 2 -1 1 -1 0 0 0 2 -1 2 3 -2 0 1 -2 1 2 0 1 -1 1 0 0 3 3 0 8
0 0 -1 2 -1 1 1 -2 0 -1 1 1 -1 0 0 -2 0 -2 0 1 2 -3 0 0 1 0 0 -1 1 -1 0 0 -1 6
-1 1 1 -1 3 2 -1 1 0 2 0 -1 0 0 0 0 1 2 -1 -2 0 2 2 1 0 1 0 0 1 1 0 1 0 -2 6
-2 -2 -3 2 -1 -2 -2 0 1 -1 2 0 0 -2 1 0 0 0 2 -1 -1 -1 -2 0 0 2 0 2 -1 0 -1 0 0 -1 0 6
-3 0 -3 1 2 1 -2 2 3 0 3 -1 -1 -1 1 1 0 2 -1 -3 -1 0 2 1 2 2 0 0 -1 1 0 0 1 -2 2 2 8
-3 0 -3 3 -2 1 0 -2 1 -2 1 -1 -2 -2 1 0 1 -1 -2 -1 -1 0 0 1 1 1 1 -1 1 1 -1 0 1 1 0 1 2 6
2 -1 -1 0 -3 -3 1 -1 -3 -3 1 1 0 -2 0 1 -2 -1 1 2 1 -1 -3 0 -4 0 2 0 -3 -1 -1 1 -1 1 -3 2 -2 0 8
1 -1 0 1 -2 -2 -1 0 0 0 -1 -1 -1 -1 0 0 0 0 0 1 -1 0 -3 -1 -2 -1 -1 0 1 -1 -2 -1 -3 0 -2 1 -2 0 2 6
0 -2 0 -1 1 0 1 2 -2 -1 0 1 0 1 -2 0 -1 0 2 1 -1 -1 -2 -2 -1 1 0 0 0 0 1 0 -2 0 -1 1 0 -1 2 2 6
-2 -1 -1 1 -1 -1 2 2 -1 0 1 -2 -1 1 -1 0 -1 -1 0 1 -2 0 -2 -3 -1 2 -1 -2 2 0 0 -2 -1 1 -1 1 0 1 1 2 2 6
-2 -3 -3 0 0 2 -1 -1 2 -1 -1 0 -1 0 0 1 1 1 0 1 -2 -1 0 0 2 0 1 1 0 2 0 1 0 1 -1 1 2 1 -1 1 2 0 8
-3 -1 -1 2 1 0 1 2 0 0 4 2 0 1 0 0 0 -1 3 -2 0 -2 0 -1 2 2 -1 0 1 0 2 -1 1 0 1 2 1 1 0 -2 1 1 -2 8
1 -1 0 1 -3 -4 -2 -3 -1 -1 -1 0 1 -3 1 0 1 -1 0 -1 -1 1 -3 0 -2 -1 0 2 -2 0 -2 0 -1 -1 0 2 -2 0 2 1 -1 -1 -2 0 8
2 -1 1 -2 1 -1 -2 1 1 1 -1 0 1 1 -1 0 1 0 1 0 0 0 0 0 0 -1 -1 1 0 -1 1 0 0 -1 0 0 0 -3 0 1 1 -1 1 -2 0 6
-3 -1 -1 1 0 2 2 -1 0 -1 1 1 0 0 0 1 0 0 0 0 0 -1 2 0 1 0 1 -1 0 2 1 0 2 1 0 0 1 2 -1 -2 -1 0 1 1 -1 -3 6
-1 -1 -2 1 -2 -2 1 -1 0 0 1 0 -1 0 1 2 0 0 0 0 -1 1 -1 -1 0 0 2 0 -1 1 -2 0 0 1 -1 1 -1 0 2 0 0 1 1 0 1 -1 0 6
-
DIVISORS
1
-
MINIMAL_NORM
6
-
KISSING_NUMBER
52416000
-
HERMITE_NUMBER
6
-
GROUP_ORDER
2^7 *3^2*5*7*13
-
GROUP_NAME
(SL_2(13) otimes SL_2(5)).2^2
-
GROUP_GENERATORS
5
48 48
-3 1 -1 1 3 -1 3 -1 4 -2 -1 1 -2 -2 0 -1 0 1 3 -2 0 1 -1 0 -3 2 1 1 -1 1 1 0 0 0 0 -3 -2 -1 0 0 -1 1 0 -1 0 1 -1 -1
-3 1 -2 -2 -1 0 2 2 4 -2 1 1 0 -1 -2 -2 -2 1 1 -1 1 3 -2 1 0 -1 0 2 1 1 0 0 0 1 1 -2 -1 -1 1 0 0 0 0 -1 1 0 0 -1
2 1 -3 -1 3 0 3 0 3 -2 2 0 0 -1 -1 0 0 0 2 -1 -2 1 0 0 -1 1 0 1 0 1 0 -1 0 1 1 -1 -2 0 0 1 -1 0 0 -1 1 0 0 -1
1 -3 1 2 -2 0 0 2 0 -1 1 0 1 1 0 -2 -1 1 -2 1 1 1 0 0 2 -1 1 0 0 0 0 1 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0
-2 2 -1 -5 -5 3 0 4 1 0 2 0 4 1 -2 0 -1 -1 -1 0 1 2 -2 2 3 -3 -2 1 2 0 -2 0 -1 2 1 1 1 0 2 0 1 -1 0 -1 2 -1 1 0
1 2 -2 -2 -1 2 0 1 -1 1 1 -1 3 2 -3 2 -2 -1 -2 1 0 1 -2 2 2 -4 -4 2 2 0 -1 -2 0 2 1 2 2 1 2 0 1 -2 0 0 1 -1 1 1
1 2 -2 -1 1 1 1 -1 0 1 1 0 0 -1 -1 2 0 -1 1 0 -2 0 0 1 -1 0 -2 1 0 0 0 -1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
-3 0 1 -4 -4 0 1 3 3 -1 1 2 0 -2 1 -3 0 0 1 -1 0 1 1 0 1 1 2 -1 0 0 -1 2 -1 0 0 -1 -1 -1 0 0 0 1 0 -1 1 0 0 -1
-1 0 1 -1 -1 0 -1 0 0 0 0 1 0 0 1 -1 1 1 0 0 1 0 0 -1 0 1 1 -1 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0
-3 2 0 -4 -3 1 1 2 3 -1 1 2 0 -2 0 -1 1 0 2 -1 0 1 0 0 0 1 1 0 0 0 -1 1 -1 0 0 -1 -1 -1 0 0 0 1 0 -1 1 0 0 -1
2 -2 0 2 -1 -1 1 2 0 -1 -1 0 0 0 0 -1 -1 1 -1 0 0 0 1 0 1 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0
1 2 -1 2 2 1 1 0 -1 0 -1 -1 2 1 -1 3 0 -1 -1 1 -1 -1 -1 1 0 -2 -3 1 0 0 0 -1 0 1 1 1 1 1 1 0 0 -1 0 0 0 0 0 1
1 -2 2 2 0 1 0 2 0 -1 0 -1 1 1 0 0 1 -1 -1 0 0 0 0 0 1 0 1 -1 -1 0 -1 1 -1 0 0 1 0 0 0 0 -1 0 0 0 0 0 0 0
-1 3 -1 -2 1 0 0 -2 0 1 0 1 -1 -1 0 1 1 0 1 0 -1 -1 1 -1 -2 1 -1 0 0 0 0 -1 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 -1 0 0 1 0 2 1 -1 2 0 2 0 0 0 0 -1 0 1 -1 1 0 1 1 -1 0 0 0 0 0 0 0 0 1 0 -1 0 0 0 1 0 0 -1 1 -1 0 0
1 0 1 1 0 -2 -2 -1 -1 1 -2 1 -2 -1 2 1 2 -1 1 1 -1 -2 3 -1 -2 2 1 -2 -1 -1 0 1 0 -1 -1 0 0 0 -2 -1 1 1 0 1 -1 0 0 0
-3 5 -1 -5 1 0 -1 -2 -2 2 -1 1 0 -1 1 2 0 -1 1 0 -1 -2 -1 0 0 -1 -2 0 1 0 1 -2 0 1 0 1 1 1 1 1 0 -1 0 0 0 0 0 1
1 0 0 -1 -4 1 -1 3 0 0 0 0 2 1 -1 1 0 -1 -1 1 1 1 0 1 2 -2 -1 0 1 -1 -2 1 0 1 1 1 1 0 0 -1 2 0 0 0 1 -1 1 0
3 -4 2 4 -1 1 1 2 -1 -1 1 -2 2 2 0 -1 1 0 -2 0 0 0 1 0 2 0 1 -1 -1 0 -1 2 -1 -1 0 1 0 0 0 0 -1 0 0 0 0 0 0 0
2 -1 -1 1 1 2 1 -2 -1 1 2 -2 1 1 -1 1 0 0 0 -1 0 1 -1 1 0 0 -1 1 0 0 0 -1 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0
-2 -1 -1 -1 -3 2 2 3 2 -1 2 -1 3 1 -2 -2 -2 1 -1 -1 2 3 -3 2 2 -2 -1 2 1 1 0 0 -1 1 1 -1 0 0 2 0 0 -1 0 -1 1 0 0 0
-3 2 0 -4 -1 -1 -1 0 1 0 0 1 -2 -2 1 -1 1 -1 3 -1 -1 0 1 0 -1 1 2 -1 0 0 0 1 -1 0 -1 -1 -1 -1 -1 0 0 1 0 0 0 0 0 -1
-1 2 -1 -4 -4 2 -1 3 -1 1 1 -1 4 2 -2 1 -1 -1 -2 1 1 1 -2 2 3 -4 -3 1 2 0 -1 -1 -1 2 1 2 2 1 2 0 1 -2 0 0 1 -1 1 1
0 2 -3 -2 -2 2 0 0 -1 1 1 -2 4 3 -3 2 -2 1 -2 0 3 2 -4 2 2 -4 -4 3 3 0 0 -2 1 2 1 1 2 1 2 0 2 -2 0 0 1 -1 1 1
-1 0 0 -3 -4 3 -3 2 -3 2 3 -1 5 3 -1 0 -1 -1 -4 2 2 1 -2 1 4 -4 -2 0 2 0 -1 -1 -1 0 0 2 2 1 2 0 1 -2 0 0 1 -1 1 1
2 -4 0 -1 -5 3 -1 2 -1 1 4 -2 4 3 -2 -2 -2 1 -4 0 3 3 -2 1 4 -2 0 1 2 0 -1 0 0 0 0 1 1 0 1 0 1 -1 0 0 1 -1 1 0
-2 3 0 1 0 0 -1 -1 -2 2 -3 0 0 0 -1 4 0 -2 0 1 0 -1 -1 2 -1 -2 -3 1 0 -1 0 0 0 1 0 1 2 0 0 -1 1 0 0 1 -1 0 0 1
0 3 0 -1 2 2 -1 1 -1 0 2 -1 4 1 0 3 1 -3 -1 2 -2 -1 -1 1 2 -2 -1 0 0 0 0 -1 -1 1 1 2 0 1 1 1 0 -1 0 -1 1 -1 0 1
3 -3 -3 -1 -1 2 4 0 3 -1 4 0 2 0 -1 -2 -2 2 0 -1 0 3 -1 1 1 0 0 2 1 1 0 -1 0 0 1 -1 -1 0 0 1 0 0 -1 -1 1 0 0 -1
-4 6 2 -2 0 -2 -3 -2 -4 3 -4 1 -2 -1 1 3 2 -2 1 1 -1 -4 1 0 -1 -1 -2 -2 0 -1 0 0 0 0 -1 2 2 0 0 -1 1 0 1 1 -1 0 0 1
1 1 -1 0 2 1 2 0 -1 0 -1 -1 1 0 0 1 0 -1 0 -1 -2 -1 0 1 0 -1 -2 0 0 0 0 -1 0 1 0 1 1 0 1 1 -1 -1 0 0 0 0 0 1
2 1 -1 2 3 0 1 -1 1 -1 -2 0 -1 0 -1 2 0 0 1 -1 -1 0 0 0 -2 1 -1 1 0 0 0 -1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
2 2 -1 0 5 -1 0 -3 -3 1 -2 -1 -1 0 0 3 0 -1 0 -1 -2 -2 0 0 -1 0 -2 0 0 0 1 -2 1 1 -1 2 1 1 0 1 -1 -1 0 1 -1 0 0 1
2 -3 0 2 -1 2 0 1 -2 1 1 -1 2 2 -1 0 -2 0 -3 1 1 1 -1 1 2 -2 -2 1 0 0 0 -1 0 0 1 1 1 1 1 0 0 -1 0 0 0 0 0 1
1 0 -1 -2 -3 1 0 2 0 0 1 -1 2 1 -2 0 -1 0 -1 0 1 1 -1 1 2 -2 -1 1 2 0 -1 0 0 1 1 1 1 0 1 0 1 -1 0 0 1 -1 1 0
3 -7 2 3 -3 2 2 2 0 -1 2 -3 3 2 0 -3 -1 2 -3 -1 2 2 -1 1 4 -1 2 0 0 0 0 2 -1 -1 0 1 0 0 0 0 0 0 -1 0 0 0 0 0
-3 1 3 1 -2 -3 -2 1 -3 1 -5 1 -2 0 1 0 1 0 0 0 1 -2 1 0 0 0 0 -2 0 -1 -1 2 1 0 -1 1 2 -1 0 -1 0 0 1 1 -1 0 0 0
-1 2 2 1 0 -3 -2 -1 -3 1 -3 1 -2 1 1 1 1 0 0 1 0 -2 1 -1 0 0 0 -1 0 -1 0 1 1 0 0 1 2 0 0 0 0 0 1 1 -1 0 0 0
1 -3 2 5 2 -1 2 -1 1 -1 -1 -1 -1 0 1 -1 0 1 0 -1 0 0 0 0 -1 2 2 0 -2 0 2 1 0 -1 0 -1 -1 0 -1 0 -1 1 0 0 -1 1 -1 0
-3 -3 4 1 -2 -3 1 -2 3 -1 -1 2 -5 -2 3 -5 2 4 3 -2 2 0 2 -3 -2 5 6 -2 -2 0 1 3 -1 -3 -2 -3 -2 -1 -2 -1 -1 3 0 0 -1 2 -1 -2
-5 2 5 -2 -5 -1 -2 1 0 1 -3 2 -2 -1 2 -2 2 0 1 0 1 -2 2 -1 0 1 2 -3 -1 -1 -1 3 -1 -1 -1 0 1 -1 0 -1 0 2 1 1 0 1 0 0
1 -5 1 -1 -3 1 0 1 1 0 4 0 1 0 1 -3 -1 1 -1 0 1 2 0 0 2 1 3 0 0 0 0 1 -1 -1 0 -1 -1 0 -1 0 0 1 -1 -1 0 0 0 -1
-5 4 5 0 -5 1 -5 0 -5 4 -3 1 1 1 2 3 4 -3 0 2 1 -3 1 1 0 -1 -1 -3 -1 -2 -2 2 -1 -1 -2 2 3 -1 0 -2 2 1 1 1 -1 0 0 1
2 1 0 -2 -1 1 -1 2 -1 0 1 -1 2 1 -1 1 -1 -1 -2 1 -1 -1 0 0 3 -2 -1 0 1 0 0 0 0 1 1 2 1 1 1 1 0 -1 0 0 1 -1 1 1
2 -2 0 2 2 -2 -2 -2 0 -1 -1 0 -2 0 1 0 0 1 0 0 1 -1 1 -2 -2 2 2 0 0 0 1 0 1 -1 0 -1 -1 0 -2 0 0 1 0 1 -1 0 0 0
-5 -4 5 1 -3 -3 0 0 2 -1 -4 2 -4 -1 4 -6 1 3 1 -2 3 -1 3 -3 -2 4 5 -3 -2 0 0 3 -1 -3 -2 -2 -1 -1 -1 -1 -1 2 0 1 -1 2 -1 -1
4 0 -2 0 -2 3 0 4 -2 0 2 -1 4 2 -2 2 -1 -2 -3 2 -2 1 0 2 3 -4 -4 1 1 -1 -2 -1 0 2 2 3 1 1 1 0 1 -2 0 0 1 -1 1 1
3 1 -3 -2 -1 3 -3 0 -1 2 3 0 3 0 -1 4 -1 -3 -1 2 -1 1 -1 2 0 -1 -2 2 1 0 0 -2 0 1 1 0 0 1 -1 0 2 0 -1 0 0 -1 1 1
48 48
-10 7 3 -8 -6 0 -3 0 1 2 -1 3 -2 -3 1 -1 2 0 3 -1 2 -1 -1 0 -1 1 1 -1 0 0 0 1 -1 0 -2 -1 0 -1 0 -1 1 2 1 0 0 1 0 0
-9 4 3 -5 -7 1 0 2 0 1 -2 2 0 -2 1 -2 1 0 1 -2 2 0 -1 1 1 -1 0 -1 0 -1 -1 2 -1 0 -1 0 1 -2 1 -1 1 1 1 0 0 1 0 0
-4 4 1 -5 -5 0 -3 2 -1 2 -2 1 -1 -1 -1 0 -1 -1 0 0 1 -1 0 1 1 -2 -2 0 1 0 -1 0 0 1 0 1 1 0 1 -1 1 0 1 1 0 0 1 1
-1 -3 3 0 -3 0 -1 2 1 -1 1 1 1 1 1 -3 0 1 -2 0 2 1 0 -1 1 1 3 -1 0 0 0 1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
2 -2 0 -1 -4 4 1 3 0 0 3 -3 5 2 -2 0 -2 -1 -3 0 1 2 -2 3 4 -4 -2 1 1 0 -1 0 -1 1 1 2 1 0 1 0 1 -1 0 0 1 -1 1 1
4 -3 -1 2 3 3 1 1 -1 -1 4 -4 5 2 0 1 -1 -2 -3 2 -2 0 0 1 3 -3 -1 0 0 0 0 -1 -1 0 1 2 0 1 1 1 0 -2 -1 -1 1 -1 0 1
-1 4 -2 -1 3 -1 -1 -1 -1 1 -3 1 -2 -2 0 3 0 -2 2 0 -2 -2 1 0 -2 0 -2 0 0 0 0 -1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
-3 2 1 -3 -5 0 -1 2 1 1 -2 2 0 -1 -1 0 -1 -1 0 -1 2 1 -1 2 0 -1 -1 0 1 0 -1 1 0 1 0 0 1 -1 0 -1 1 1 1 1 0 0 1 0
3 -4 1 4 -1 0 2 1 1 -2 1 -1 2 2 0 -2 0 2 -2 0 1 1 0 0 1 0 1 0 0 0 0 1 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0
-2 0 2 -2 -6 0 -1 2 1 0 -1 1 0 0 -1 -2 -1 1 -1 -1 3 1 -1 1 1 -1 0 0 1 0 -1 1 0 0 0 0 1 -1 0 -1 1 1 1 1 0 0 1 0
0 0 0 0 0 0 1 0 1 0 -1 1 0 0 0 0 0 0 0 -1 0 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 0 0
2 -1 0 -2 2 2 -2 0 1 0 4 -1 2 1 0 0 -1 -1 -2 2 -1 0 0 -1 1 0 1 0 0 1 1 -2 -1 0 0 1 -1 2 0 1 0 -1 -1 -1 1 -1 0 1
5 -7 0 -1 -2 2 -1 3 2 -1 4 -2 2 2 -2 -3 -4 1 -4 1 2 3 -1 0 4 -2 1 1 1 1 0 -1 -1 0 1 1 -1 2 0 0 0 -1 -1 0 1 -1 1 1
0 3 -2 0 3 -1 0 -1 0 0 -2 1 -1 -1 0 2 0 -1 1 0 -2 -1 1 0 -2 0 -2 0 0 0 0 -1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
-6 2 5 2 -4 -2 2 0 -1 0 -5 2 -4 -2 2 -3 4 2 3 -3 1 -2 2 -1 -2 3 2 -3 -2 -1 -1 4 0 -2 -2 -1 1 -3 0 -1 -1 2 2 1 -1 2 -1 -1
1 -3 1 4 3 -5 3 -2 3 -2 -4 2 -7 -3 2 -4 0 3 3 -2 -1 -1 4 -3 -4 5 4 -1 -2 1 1 2 1 -2 -1 -3 -2 -1 -2 0 -2 2 0 1 -1 2 -1 -2
5 5 -5 3 6 1 0 -3 -5 2 1 -2 3 2 -2 8 1 -3 0 3 -4 -2 -1 2 0 -3 -6 2 1 -1 0 -3 2 2 2 2 2 1 1 1 1 -2 0 -1 0 -2 0 1
2 -7 3 2 -3 0 3 1 3 -2 1 -1 -1 0 1 -6 -1 3 -1 -2 2 2 1 -1 1 2 4 -1 -1 1 0 2 -1 -2 -1 -1 -1 -1 -1 0 -1 1 0 1 0 1 0 -1
3 -6 1 -3 -3 3 -3 3 2 0 5 -1 3 2 -1 -3 -3 0 -4 1 2 3 -1 0 3 -1 2 0 1 1 0 -1 -2 0 0 1 -1 2 0 0 0 -1 -1 0 1 -1 1 1
-1 1 0 -1 1 0 -2 -1 -2 1 0 -1 0 0 1 1 1 -1 1 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0
-6 4 2 -6 -4 3 -1 2 1 1 2 0 3 -1 0 0 0 -2 0 0 0 0 -2 2 2 -2 0 0 0 0 0 0 -2 1 0 1 0 0 1 0 1 0 0 -1 1 0 0 1
-3 3 0 0 -2 -2 1 -1 -2 1 -4 1 -4 -2 0 0 1 1 3 -2 0 -1 1 0 -1 0 -1 0 0 -1 -1 2 1 0 -1 -1 1 -2 0 -1 0 1 1 1 -1 1 0 -1
5 -2 -2 2 3 2 3 1 0 -1 2 -3 3 1 -1 1 -2 -1 -2 1 -3 0 0 1 3 -3 -2 1 0 0 0 -1 0 1 2 2 0 1 1 1 0 -2 -1 -1 1 -1 0 1
-1 2 -1 2 2 2 3 -2 -3 1 1 -2 3 1 1 2 2 -1 0 0 -2 -1 -1 1 1 -1 -2 0 -1 -1 0 0 0 0 0 1 1 -1 1 1 0 -1 0 -1 0 0 -1 0
5 -6 0 3 0 3 2 3 0 -2 4 -3 5 3 0 -2 -2 0 -4 1 -1 2 0 1 4 -3 0 0 0 0 0 0 -1 0 2 2 0 1 1 1 0 -2 -1 -1 1 -1 0 1
-1 -1 -1 1 -1 1 2 2 -2 0 -1 -1 2 1 0 0 -1 -1 -1 -1 0 1 -1 2 2 -3 -2 0 0 -1 -1 1 0 1 1 1 1 -1 1 0 0 -1 0 0 0 0 0 0
3 -2 -2 2 5 1 2 -3 0 0 2 -1 0 -1 1 1 1 0 1 0 -2 0 0 -1 -1 2 1 0 -1 0 1 -1 0 -1 -1 -1 -1 0 -1 1 -1 0 -1 -1 0 0 -1 -1
1 1 1 1 -1 2 -1 0 -4 2 1 -2 2 2 -1 2 2 -1 -1 1 0 -1 -1 1 2 -2 -2 -1 0 -1 -1 0 0 0 0 2 2 0 1 0 0 -1 1 0 0 -1 0 1
-1 3 1 -1 -2 -1 -4 2 -3 1 -2 1 1 1 0 2 0 -2 -2 2 0 -2 1 0 1 -2 -2 -1 1 -1 -1 0 1 1 1 2 2 0 1 0 1 -1 1 1 0 -1 1 1
6 -1 -2 8 8 -3 2 -4 -3 0 -3 -1 -4 0 1 3 2 0 2 1 -4 -3 4 -2 -3 2 -1 -1 -2 -1 0 0 2 -1 0 0 0 0 -1 0 -1 0 0 0 -1 0 -1 -1
3 4 -5 -2 5 1 -2 0 -2 1 1 -1 3 1 -2 6 -2 -4 -1 3 -3 -1 -1 2 1 -4 -5 2 2 0 0 -3 1 3 2 2 1 2 1 1 1 -2 -1 -1 1 -2 1 2
0 0 0 0 1 0 0 -2 -1 0 1 -2 0 1 0 0 1 1 0 0 1 -1 0 -1 0 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0
10 -2 -7 6 9 1 2 -3 -5 1 1 -4 3 3 -2 6 -1 -2 -2 2 -4 -1 0 1 1 -3 -6 2 1 -1 0 -3 2 1 2 2 1 2 1 1 0 -3 -1 -1 0 -2 0 1
3 -1 0 -1 3 2 -1 0 1 -1 4 -2 3 1 0 1 0 -1 -1 1 -1 0 -1 0 1 0 1 0 0 1 1 -2 -1 0 0 1 -1 2 0 1 0 -1 -1 -1 1 -1 0 1
0 1 -2 1 0 1 3 1 -1 0 -1 -1 1 0 -1 1 -1 -1 0 0 -2 0 0 2 1 -3 -3 1 0 -1 -1 0 0 1 1 1 1 -1 1 0 0 -1 0 0 0 0 0 0
4 -3 -1 4 1 2 0 0 -3 1 2 -2 3 3 -1 2 1 -1 -2 1 0 1 -1 1 1 -1 -1 0 0 -1 -1 0 0 0 0 1 1 0 0 0 0 -1 0 0 0 -1 0 0
7 -2 -5 5 5 2 5 -1 0 -1 1 -2 3 1 -1 4 0 -1 0 0 -3 1 -1 2 0 -1 -3 2 0 -1 0 -1 1 1 2 0 0 0 0 1 0 -1 -1 -1 0 -1 0 0
3 -2 -3 2 4 1 1 -2 -1 0 3 -2 3 2 -1 2 -1 0 -2 1 0 1 -2 0 1 -1 -1 2 1 0 1 -2 0 0 1 0 0 1 1 1 0 -2 -1 -1 0 -1 0 0
-8 7 2 -4 0 -3 -3 -4 -1 3 -3 3 -5 -3 2 1 4 0 5 -1 1 -3 0 -2 -4 4 2 -2 -1 0 1 1 0 -1 -3 -2 0 -1 -1 -1 0 2 1 0 -1 1 -1 -1
-4 1 4 -5 -6 -2 -6 1 3 0 2 3 -1 0 1 -3 1 1 0 1 4 1 0 -2 -1 3 5 -1 1 1 0 1 -1 -1 -1 -2 -1 0 -1 -1 1 2 1 1 0 0 1 -1
0 1 -1 -5 -3 2 -3 1 1 1 4 0 3 1 -1 1 -1 -1 -1 1 2 2 -2 1 1 -1 0 1 2 1 0 -1 -1 1 0 0 0 1 0 0 1 0 0 0 1 -1 1 0
-2 2 1 1 1 -4 -2 -1 0 0 -4 2 -4 -1 0 1 1 0 2 -1 1 -1 1 -1 -3 2 1 -1 0 0 0 1 1 0 -1 -1 0 -1 -1 -1 0 1 1 1 -1 0 0 -1
11 -14 2 6 2 1 2 1 5 -5 7 -4 2 3 1 -6 -1 4 -3 1 1 3 2 -3 2 3 7 -1 -1 2 1 1 -2 -3 0 -1 -3 2 -2 1 -2 0 -2 0 1 0 0 -1
-1 1 1 1 0 1 -2 1 -4 2 -1 0 2 2 0 2 0 -3 -2 1 0 -1 0 1 1 -2 -2 -1 0 -1 -1 0 0 0 0 2 2 0 1 0 0 -1 1 0 0 -1 0 1
-4 4 0 1 1 -4 0 -3 -2 1 -6 3 -6 -2 1 1 2 1 4 -2 0 -2 1 -1 -4 3 0 0 -1 -1 0 1 2 0 -1 -2 1 -2 -1 0 -1 2 1 1 -2 1 -1 -1
2 1 -4 -5 -1 2 0 0 3 0 5 0 3 0 -2 1 -2 0 0 0 0 3 -3 2 1 -1 -1 3 2 1 1 -2 0 2 1 -1 -1 1 0 1 1 0 -1 -1 1 -1 1 0
12 -11 -3 8 4 3 5 2 1 -3 4 -5 4 3 -2 -1 -3 1 -4 0 -1 3 0 1 3 -1 0 2 0 1 0 -1 0 0 2 1 -1 1 0 1 -1 -2 -2 -1 1 -1 0 0
-1 -2 3 1 1 -4 2 -2 3 -2 -3 2 -7 -3 2 -5 1 4 3 -3 0 -1 3 -4 -3 5 5 -2 -2 1 1 2 0 -2 -2 -2 -2 -1 -2 0 -2 2 0 1 -1 2 -1 -2
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4 -6 -1 2 1 0 3 0 3 -2 3 -1 0 1 -1 -4 -3 3 -2 -1 1 3 -1 -1 1 1 2 2 0 2 1 -1 0 0 1 -1 -2 1 0 1 -1 -1 -1 0 0 1 0 -1
3 -8 1 -3 -6 1 1 2 5 -2 4 -1 0 1 -1 -8 -4 5 -3 -2 4 4 0 -2 3 1 4 1 1 2 0 0 -1 -1 0 -1 -2 1 0 0 0 0 -1 1 1 1 1 -1
1 -3 1 3 -1 2 1 1 0 0 3 -1 2 1 0 -1 1 0 -1 1 0 1 0 0 1 0 1 0 -1 0 -1 1 -1 -1 0 0 -1 0 0 -1 0 0 0 -1 0 0 -1 -1
-2 5 1 -1 2 -2 -2 -1 -3 1 -4 0 -3 -1 1 2 2 -1 2 0 -2 -4 2 -1 -1 0 -1 -2 -1 -1 0 0 1 0 -1 1 1 0 0 0 0 0 1 1 0 0 0 1
4 -1 0 1 -2 0 -1 1 2 -1 -1 1 -1 0 -1 0 0 0 0 1 0 0 2 -1 -1 1 0 0 0 0 -2 0 1 0 1 0 0 0 -1 -1 1 1 0 1 0 0 1 0
1 -2 2 -1 -5 0 -2 1 1 0 0 0 0 1 -1 -1 -1 1 -1 0 3 1 0 0 1 0 1 0 1 0 -1 1 0 0 0 0 1 0 -1 -1 1 1 0 1 0 0 1 0
-6 5 -2 -4 -3 1 0 0 -1 2 0 1 2 -1 -1 2 -1 -1 1 -1 1 1 -3 3 0 -2 -3 2 2 0 0 0 0 2 0 -1 1 -1 1 0 1 0 0 -1 0 0 0 0
1 2 -3 1 0 1 2 1 0 0 -1 0 1 -1 -1 2 0 -2 1 0 -2 0 1 2 -1 -1 -3 1 0 0 -2 0 1 1 1 0 0 -1 0 0 1 0 0 0 0 0 0 0
-1 2 1 1 1 -3 -2 -1 -3 1 -4 1 -4 -1 1 1 2 -1 2 0 -1 -3 3 -1 -2 1 0 -2 -1 -1 -1 1 1 -1 -1 0 1 -1 -1 -1 0 1 1 1 -1 0 0 0
0 4 -1 1 1 -2 -2 -2 -3 2 -4 2 -3 -1 1 3 2 -2 2 1 -2 -3 3 -1 -3 1 -2 -1 -1 -1 -2 0 2 0 0 0 1 -1 -1 -1 1 1 1 1 -1 0 0 0
0 5 -4 -1 4 -1 1 0 -1 0 -3 1 -2 -3 -1 3 -1 -2 2 0 -4 -2 1 1 -2 -1 -4 1 0 0 0 -2 2 2 1 0 0 0 0 1 0 -1 0 0 0 0 0 1
-1 7 -1 2 4 -4 -1 -2 -3 1 -7 2 -4 -1 -1 5 1 -1 2 1 -2 -4 1 -1 -3 0 -4 0 0 -1 0 -1 3 2 1 1 2 0 0 0 0 -1 1 1 -1 0 0 1
-3 2 4 0 0 -3 -2 -1 1 0 -4 3 -4 -1 1 -1 2 1 1 0 1 -2 1 -3 -2 3 2 -2 -1 0 0 1 0 -1 -1 0 0 0 0 -1 -1 1 1 1 -1 1 0 0
-5 6 0 1 -1 -1 0 -1 -3 2 -4 1 0 -1 0 4 2 -2 2 0 0 -2 0 2 -2 -1 -3 0 0 -1 -1 1 1 1 0 0 2 -2 0 -1 1 1 1 0 -1 0 -1 0
5 -5 -2 -2 -1 1 0 2 1 -1 4 -2 1 1 -1 -3 -3 1 -2 0 0 2 0 0 3 -1 1 1 1 1 0 -1 0 0 1 0 -1 1 0 1 0 -1 -1 0 1 -1 1 0
1 2 -7 -4 0 5 2 1 -2 2 5 -2 7 1 -3 4 -3 -3 -2 0 -1 3 -5 5 3 -5 -6 4 3 0 0 -3 0 3 2 1 1 0 2 2 1 -3 -1 -2 1 -2 1 1
-1 -1 4 6 0 -2 -2 0 -4 1 -4 0 -2 1 2 1 3 -1 0 2 0 -3 3 -1 -1 0 1 -3 -2 -2 -1 3 0 -2 -1 1 2 -1 -1 -2 0 1 1 1 -1 0 -1 0
6 -4 -3 -4 -4 3 0 2 4 -1 5 0 3 1 -2 -2 -3 1 -2 0 1 4 -1 0 2 0 0 2 2 1 -1 -2 0 1 2 0 -1 1 0 1 1 -1 -1 0 1 -1 2 0
0 8 -1 4 8 -2 1 -3 -3 1 -6 1 -3 -2 0 7 4 -3 4 1 -5 -5 1 0 -4 1 -4 -1 -2 -1 0 -1 2 1 0 1 1 0 0 0 -1 0 1 0 -1 0 -1 1
-2 -1 0 -3 -2 2 0 -1 3 0 4 0 2 0 0 -1 0 1 1 -1 2 3 -3 1 0 1 2 1 1 1 1 0 -2 0 -1 -2 -1 0 0 0 0 1 -1 -1 0 0 0 -1
5 -5 -2 1 1 0 3 0 3 -2 1 -1 -1 0 -3 -3 -5 3 -2 -2 1 3 -2 0 1 0 0 3 1 2 1 -2 1 1 1 -1 -1 1 0 1 -1 -1 -1 1 0 1 1 0
5 -9 -2 -1 -3 7 3 2 4 -1 10 -3 7 3 -1 -4 -2 2 -4 0 2 6 -3 1 4 -1 2 2 1 1 0 -1 -3 -1 0 0 -2 1 1 0 0 -1 -2 -2 2 -1 0 -1
2 -4 1 0 -4 1 0 1 1 0 0 0 0 1 -2 -2 -2 2 -2 -1 3 2 -1 0 1 0 0 1 1 0 -1 0 0 0 0 0 1 0 0 -1 0 0 0 1 0 0 1 0
6 -8 0 0 -2 0 0 -1 3 -1 3 -1 -1 1 -1 -4 -3 4 -2 -1 3 3 0 -2 1 2 3 1 1 1 1 -1 0 -1 0 -1 -1 1 -1 0 0 0 -1 1 0 0 1 -1
0 1 4 0 -4 -1 -4 1 -3 1 -4 -1 -1 2 0 1 3 0 0 1 2 -3 2 -1 1 -1 0 -3 0 -2 -2 2 0 -1 -1 2 3 0 0 -2 1 1 1 2 0 -1 1 1
3 -1 -1 -4 -1 2 -1 0 3 -1 3 -1 2 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 0 0 -1 -1 0 0 0 -1 1 0 0 1 1 -1 0 1 -1 1 0
-3 5 -3 -4 -1 0 -1 -2 -2 3 -1 1 1 0 -1 3 -2 -1 0 0 1 0 -3 2 0 -2 -4 2 2 0 1 -2 1 2 1 0 2 0 1 1 1 -1 0 0 0 0 1 1
2 0 3 6 4 -4 -1 -1 -3 0 -6 1 -5 0 3 1 4 -1 2 2 -3 -5 5 -3 -2 2 1 -4 -3 -2 -1 2 1 -2 0 1 1 0 -1 -1 -1 1 1 1 -1 0 -1 0
-1 2 4 4 0 -1 -4 -1 -5 2 -3 -1 -2 0 3 3 5 -3 2 2 -2 -5 5 -1 -2 1 1 -4 -3 -2 -2 3 0 -2 -2 1 1 -1 -2 -2 1 2 1 1 -1 0 -1 0
-1 3 0 -3 -3 1 -3 2 -2 1 1 0 3 2 -1 2 0 -2 -2 2 1 0 -1 1 2 -3 -2 0 2 -1 -1 0 0 1 1 2 2 0 1 0 1 -1 1 0 1 -2 1 1
-2 -1 2 4 1 -2 2 0 1 -1 -3 1 -2 -1 0 0 1 1 1 -1 1 0 0 0 -2 2 1 0 -1 0 0 2 0 0 -1 -1 0 -1 -1 -1 -1 1 0 0 -1 1 -1 -1
3 -1 -2 -3 1 -1 -1 -1 3 -1 1 2 -2 -1 0 -1 -2 1 0 0 0 1 0 -2 -1 2 1 1 1 1 1 -2 1 1 1 -1 -1 1 -1 1 0 0 -1 0 0 0 1 0
-7 3 3 -1 -2 -2 -3 -1 -3 3 -3 2 -2 -1 1 1 2 -1 1 1 2 -2 0 0 -1 0 0 -2 0 -1 0 2 -1 -1 -2 0 2 -1 0 -2 0 1 1 0 -1 0 -1 0
-6 6 2 -1 0 -2 -2 -2 -2 1 -4 1 -2 -1 0 2 1 0 2 -1 1 -2 -2 1 -2 0 -1 0 0 0 1 0 1 1 -1 -1 2 -1 0 0 0 1 1 1 -1 1 0 1
8 -5 -2 1 -1 3 0 2 2 -1 4 -1 3 1 -1 0 -2 -1 -3 2 -1 2 1 0 2 -1 0 1 0 0 -1 -1 0 0 2 1 -1 1 -1 0 1 0 -1 0 1 -1 1 0
4 0 -1 3 9 -3 1 -2 1 -2 -2 1 -4 -2 2 1 1 -1 2 1 -5 -3 4 -3 -3 3 1 -2 -2 0 1 -1 1 -1 0 0 -2 1 -1 1 -2 0 0 0 0 0 -1 0
2 -1 -1 -1 -1 0 1 3 3 -2 0 1 -1 -1 -1 -2 -1 0 0 0 -1 1 2 0 0 0 0 0 0 0 -1 0 0 0 1 0 -1 0 0 0 0 0 0 0 1 0 1 0
-4 1 3 -2 -4 -1 -1 0 -1 1 -3 1 -3 -1 1 -2 1 2 1 -2 2 -1 1 -1 0 1 1 -2 0 -1 0 2 0 -1 -2 0 1 -1 0 -1 0 1 1 1 0 1 0 0
1 6 -8 -3 7 3 3 -2 -1 1 3 -1 4 -1 -2 6 -2 -3 1 0 -4 1 -5 4 0 -3 -6 4 2 1 2 -5 1 4 2 0 0 1 2 3 0 -3 -1 -2 1 -1 0 1
1 1 -2 2 4 2 2 -1 -1 0 2 -1 2 0 1 2 1 -2 1 0 -3 0 0 1 -1 0 -1 0 -1 0 0 -1 0 0 0 0 -1 0 0 1 0 -1 0 -1 0 0 -1 0
-1 9 -5 -1 3 3 1 0 -2 1 1 0 5 0 -1 8 2 -5 2 2 -4 -1 -2 4 -1 -3 -6 2 1 -1 -1 -2 1 3 2 1 1 0 1 1 2 -1 0 -2 1 -1 0 1
-3 4 1 0 1 0 -1 0 -1 0 -2 1 0 -1 2 2 3 -2 2 0 -2 -2 2 0 -2 1 0 -2 -1 -1 -1 1 0 0 -1 0 0 -1 0 0 0 1 1 0 0 0 -1 0
0 1 -1 0 0 -1 0 -1 0 0 -2 1 -1 0 -1 2 0 0 2 -2 1 1 -2 2 -2 1 -1 1 1 0 0 0 1 1 0 -1 1 -1 -1 0 0 1 0 0 -1 0 1 0
-3 9 1 2 3 -3 -2 -2 -4 2 -8 2 -4 -3 2 6 5 -3 4 2 -4 -7 4 -1 -4 1 -3 -3 -2 -2 -1 1 2 0 -1 1 2 -1 -1 -1 1 1 1 1 -1 0 -1 1
3 -1 0 4 7 -1 0 -2 -2 0 1 -1 0 1 2 2 2 -1 -1 2 -3 -2 1 -2 0 1 1 -1 -2 -1 2 -1 0 -1 0 1 -1 1 0 1 -1 -1 0 -1 0 -1 -1 0
-1 1 0 4 4 0 2 -1 -1 0 -2 0 -1 -1 0 2 1 -2 1 0 -2 -1 0 1 -2 0 -1 0 -2 0 0 0 0 0 0 0 0 -1 0 0 -1 0 0 0 -1 1 -1 0
-8 5 2 -7 -5 -2 -3 1 1 1 -3 3 -3 -2 -1 -1 -1 1 2 -2 3 0 -2 0 0 0 0 0 2 0 0 1 0 1 -1 -1 1 -1 1 -1 0 1 1 0 0 0 1 0
2 7 -8 -4 5 2 -1 -3 -4 3 0 0 3 0 -1 8 -1 -4 1 1 -4 -1 -3 3 -1 -3 -7 3 2 -1 1 -5 2 4 2 1 2 1 1 2 1 -2 -1 -1 0 -2 1 2
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5 -1 -2 1 3 -1 1 -2 3 -2 0 -1 -2 -1 -1 0 -1 1 1 0 -1 0 1 -1 -1 1 1 1 0 1 1 -1 1 0 1 -1 -1 0 -1 1 0 1 -1 1 0 0 1 0
4 -1 -1 4 5 -1 1 -2 -1 0 -2 -1 -2 0 -1 2 0 -1 0 0 -2 -1 1 0 -1 0 -1 0 -1 0 0 -1 1 0 0 1 0 0 0 0 -1 0 0 1 -1 0 0 0
8 -4 -4 -5 -5 6 0 5 5 -2 7 -3 6 2 -4 -1 -4 0 -3 1 1 5 -2 2 5 -4 -1 3 3 1 -1 -2 -1 2 3 1 -1 2 1 1 2 -1 -2 0 3 -2 3 1
-6 1 8 11 2 -6 0 -3 -7 2 -11 1 -7 0 3 1 6 1 2 -1 1 -6 4 -2 -4 3 1 -5 -4 -2 -1 5 1 -3 -4 1 3 -3 -1 -3 -2 2 3 2 -3 2 -3 -1
2 1 -4 -5 -2 3 0 2 1 0 2 0 3 0 -2 1 -3 -2 -1 0 -1 2 -1 2 2 -3 -3 2 2 0 -1 -2 0 2 2 1 0 1 1 1 1 -1 -1 0 1 -1 1 1
-2 4 -3 -6 -3 2 -3 2 -2 2 1 1 3 1 -1 2 -2 -3 -2 2 -1 0 0 1 2 -4 -4 1 2 -1 -2 -2 0 2 2 2 1 1 2 0 2 -2 0 0 1 -1 1 1
2 -4 -1 -4 -2 3 0 4 5 -2 8 1 3 0 0 -4 -2 0 -2 1 -1 4 0 -1 2 1 3 1 0 1 0 -1 -2 0 1 -1 -4 2 0 1 0 -1 -1 -2 2 0 1 -1
1 -2 -4 -2 -1 2 3 1 3 -1 3 1 2 -1 -1 -1 -3 0 0 -2 0 4 -2 2 0 0 -1 3 1 1 0 -1 0 1 1 -2 -2 0 0 1 0 0 -1 -1 0 0 0 -1
-11 9 6 1 -1 -3 -3 -1 -6 3 -8 1 -2 0 2 3 5 -2 2 1 1 -6 1 0 -1 -2 -2 -4 -1 -2 -1 3 0 -1 -2 2 4 -2 2 -2 0 1 3 1 -1 0 -2 1
2 1 -2 -3 -1 2 -2 2 0 0 2 -1 3 1 -2 2 -1 -2 -1 1 0 1 -1 2 2 -3 -2 1 2 0 -1 -1 0 1 1 1 0 1 1 0 1 0 0 0 1 -2 1 1
-8 6 6 5 2 -5 0 -2 -4 1 -9 4 -7 -3 4 1 6 -1 4 -1 -2 -6 5 -3 -5 4 1 -5 -4 -2 -1 4 1 -2 -3 0 1 -3 -1 -1 -2 2 3 1 -2 2 -3 -1
-4 3 4 -2 -4 -2 -2 2 1 -1 -3 3 -3 -2 1 -2 1 0 1 0 0 -2 3 -2 -1 2 2 -2 -1 0 -1 2 0 0 0 0 0 -1 -1 0 0 1 1 1 0 1 0 0
9 -7 -3 -2 -2 2 1 2 6 -3 5 -1 2 1 -3 -3 -4 3 -2 -1 2 5 -2 0 2 1 2 3 2 2 1 -2 0 1 2 -1 -2 2 -1 1 0 0 -2 0 1 -1 2 0
-2 -3 0 -7 -6 3 0 7 6 -3 5 2 3 -1 0 -6 -4 0 -2 0 0 4 0 0 3 -1 2 1 1 1 -1 0 -2 1 2 -1 -3 1 1 1 0 -1 -1 -1 2 0 1 0
-9 9 4 5 1 -2 0 -4 -7 4 -8 0 -3 -1 1 5 6 -2 4 -1 0 -5 0 2 -3 -1 -3 -3 -2 -2 -1 3 1 -1 -3 1 4 -4 1 -2 0 2 3 1 -2 1 -2 0
-6 9 -3 -9 -3 4 -2 1 -1 3 4 1 5 -1 -1 4 0 -4 1 1 -1 1 -5 4 2 -4 -4 2 2 0 0 -2 -1 3 1 0 1 0 2 1 2 -1 0 -2 1 -1 1 1
0 1 -1 -3 -3 4 -2 3 -2 1 4 -3 7 3 -2 2 -1 -2 -3 2 2 1 -3 3 4 -5 -3 1 3 0 -1 -1 -1 1 1 2 2 1 2 0 2 -2 0 -1 2 -2 1 2
1 5 -3 -4 1 0 -2 -2 -2 2 -1 0 0 -1 -1 4 -1 -3 1 0 -2 -1 -1 2 0 -2 -3 1 1 0 0 -2 1 2 1 1 1 0 0 1 1 0 0 1 -1 -1 1 1
-4 1 1 -2 -6 0 2 2 3 -1 -2 3 -1 -2 0 -2 0 2 2 -3 2 2 -1 1 -1 2 1 1 0 0 0 2 0 1 0 -2 0 -2 -1 0 0 2 0 0 0 1 0 -1
2 -2 -2 -3 0 1 1 0 3 -1 4 0 1 0 1 -2 -1 1 0 0 -1 2 0 -1 1 1 2 1 0 0 1 -1 -1 0 1 -1 -2 1 0 1 0 0 -1 -1 1 0 1 -1
1 3 -2 0 5 -1 0 -3 0 0 -1 1 -2 -1 -1 3 0 -1 1 0 -2 -1 0 -1 -2 1 -1 1 0 0 1 -2 1 1 0 0 0 0 0 1 -1 0 0 0 0 0 0 0
5 3 -8 -6 -1 9 0 2 -2 3 8 -5 11 3 -5 7 -3 -5 -3 2 -1 4 -8 8 6 -9 -8 5 4 0 0 -5 -1 4 3 3 2 2 3 1 3 -3 -2 -2 2 -3 3 3
3 2 -3 -8 -3 7 -4 5 0 1 7 -2 9 3 -3 3 -3 -5 -5 4 -1 2 -3 3 6 -7 -4 2 3 0 -1 -4 -2 3 3 4 1 3 3 1 2 -3 -1 -1 3 -3 2 3
-5 9 3 -3 2 -4 -5 -3 -3 2 -6 2 -5 -2 2 3 3 -3 3 1 -2 -6 3 -2 -3 1 0 -3 -1 -1 0 0 1 0 -1 1 2 -1 0 0 0 1 2 2 -1 0 0 1
-8 2 4 2 -4 2 1 3 -4 2 -3 0 3 1 1 1 3 -2 0 0 1 -1 -1 3 2 -3 -2 -2 -1 -2 -2 3 -2 0 -1 2 3 -2 2 -2 0 0 1 -1 0 0 -2 0
-2 -3 2 4 2 -3 2 -2 0 0 -4 3 -5 -2 4 -2 2 1 3 -2 -1 -1 3 -2 -4 5 3 -2 -3 -1 0 2 0 -2 -2 -2 -1 -1 -2 -1 -2 2 0 0 -2 2 -2 -2
0 7 -3 -4 4 0 -3 -2 -2 2 2 1 1 -1 0 5 1 -4 1 2 -4 -2 -1 1 -1 0 -2 1 0 0 1 -3 1 2 1 0 0 1 0 2 1 -1 0 -1 0 -1 1 1
0 4 -1 1 -1 2 0 -2 -3 2 -1 -2 3 1 -1 5 2 -1 1 0 1 -1 -2 3 0 -2 -3 1 1 -1 0 0 1 1 0 1 3 -1 0 0 2 1 0 0 0 -1 0 1
-2 -3 3 1 -6 0 -1 5 -1 0 -2 0 1 2 1 -3 0 0 -2 1 2 0 3 0 2 -2 0 -2 0 -1 -3 3 -1 -1 0 1 1 0 1 -2 1 0 1 1 1 0 0 0
-2 -2 -1 -7 -8 7 -1 5 1 1 9 -2 9 2 -2 -2 -3 -1 -4 1 3 5 -5 4 6 -5 -1 2 3 1 -1 -1 -3 1 1 1 0 1 2 0 2 -2 -1 -2 2 -1 1 1
-1 -2 -2 -6 -6 3 0 5 5 -2 4 2 3 0 -1 -4 -4 1 -2 0 2 4 -1 0 2 -1 0 2 2 1 -1 -1 -1 1 2 -1 -2 1 1 0 1 -1 -1 -1 2 0 1 0
-6 12 -2 -5 2 -3 -2 -4 -2 2 -6 3 -5 -4 1 4 2 -2 5 0 -3 -5 2 -1 -4 0 -3 -1 0 -1 0 -1 2 1 0 0 1 -1 0 0 1 1 1 1 -1 1 0 1
-6 9 -1 -9 -5 4 -3 3 0 2 3 1 5 0 -1 3 2 -3 1 1 0 0 -3 2 2 -3 -3 0 2 -1 -1 -1 -1 2 1 1 1 0 3 0 2 -1 1 -2 2 -2 1 1
-1 -3 5 8 5 -5 1 -1 -1 -2 -6 1 -7 -1 3 -3 3 2 1 0 -2 -4 6 -5 -3 4 4 -4 -4 -1 0 3 0 -3 -2 0 -1 -1 -1 -1 -3 1 1 1 -1 2 -2 -1
3 2 -6 -7 -1 3 -3 1 -1 2 3 0 4 1 -3 4 -4 -3 -2 1 0 2 -3 3 2 -4 -5 3 4 0 -1 -4 1 3 2 1 1 2 1 1 2 -2 -1 0 1 -2 2 2
-8 6 4 3 1 -6 0 -5 -3 2 -9 4 -7 -3 3 2 4 1 5 -3 1 -4 1 -1 -6 5 1 -2 -2 -1 1 3 2 -1 -3 -2 2 -3 -2 -1 -1 3 2 1 -3 2 -2 -1
-4 3 2 1 1 0 -1 -2 -5 3 -2 0 1 1 2 2 2 -2 -1 1 0 -3 0 0 0 -1 -2 -2 -1 -1 0 0 0 -1 -1 2 2 0 1 0 0 -1 1 0 -1 0 -2 1
-5 4 3 3 1 -4 -2 -3 -5 3 -6 1 -4 0 1 2 3 0 1 0 1 -4 1 -1 -3 1 -1 -2 -1 -1 0 1 1 -1 -2 1 2 -1 0 -2 0 0 2 1 -2 1 -1 0
-1 4 3 5 4 -4 0 -5 -1 0 -5 2 -6 -3 4 2 6 0 5 0 -3 -5 4 -3 -6 6 3 -3 -4 -1 1 2 1 -2 -2 -2 0 -2 -3 0 -1 3 1 1 -2 2 -1 -1
3 -1 -1 4 3 -2 1 -3 -2 0 -2 -2 -1 1 0 2 0 1 0 0 0 -1 0 0 -1 0 -1 1 0 0 1 0 1 0 0 0 1 0 -1 0 0 0 0 1 -1 0 0 0
-1 -2 0 3 -1 -1 3 0 1 -1 -2 1 -2 -2 2 -2 0 1 2 -1 -1 0 2 0 -2 2 1 0 -2 0 0 2 0 -1 0 -2 -1 -1 -2 0 0 1 -1 0 -1 2 -1 -1
7 -14 2 8 1 -1 3 1 2 -3 2 -1 -1 2 2 -6 -1 4 -3 -1 1 3 3 -3 0 4 5 -1 -2 0 0 2 -1 -3 -1 -1 -3 1 -2 -1 -2 0 -1 0 -1 1 -1 -2
-9 8 1 -2 -2 -2 -3 -2 -6 4 -4 1 0 0 2 3 2 -2 1 2 0 -4 0 1 0 -3 -3 -1 0 -2 0 1 0 0 0 1 3 -1 1 -1 2 0 1 0 -1 0 -1 1
-7 4 2 0 -4 -2 2 1 0 0 -5 5 -4 -3 1 -1 3 1 4 -4 1 0 1 0 -4 4 0 -1 -1 -1 -2 3 1 0 -2 -2 0 -3 -1 -1 -1 2 2 0 -1 2 -1 -2
1 -4 3 3 0 -3 0 -3 1 -1 -2 -1 -5 0 0 -3 1 5 2 -3 4 0 0 -2 -2 4 4 -1 0 1 1 2 1 -2 -3 -2 0 -1 -2 -1 -1 2 0 1 -1 1 0 -1
0 0 -2 -4 -3 1 0 2 2 -1 0 0 2 0 0 -1 -2 0 0 0 1 1 -1 1 2 -2 -1 1 2 0 0 0 0 1 2 -1 1 0 1 1 1 0 -1 0 1 -1 1 1
-2 -2 4 -2 -5 2 -3 3 -1 1 3 0 2 2 0 -3 0 0 -3 1 2 1 0 -1 3 -1 2 -2 0 0 -1 1 -2 -1 -1 2 0 1 1 -1 0 -1 1 0 1 0 0 0
-4 7 2 2 4 -1 -2 -1 -5 2 -3 1 -1 -2 3 5 5 -5 3 2 -5 -5 2 1 -2 0 -1 -3 -3 -2 0 1 0 0 -1 1 1 -1 -1 0 0 1 1 -1 -1 0 -1 1
48 48
-4 0 1 0 -4 0 3 2 2 -1 -2 1 -1 -1 -1 -3 0 2 1 -3 2 2 -1 1 0 0 0 0 0 0 -1 2 0 0 0 -1 0 -2 1 -1 0 1 1 0 0 1 0 -1
-1 -2 1 0 -3 0 1 1 1 0 0 1 0 0 0 -2 0 1 0 -2 2 2 -1 1 0 1 1 0 0 0 0 1 0 0 0 -1 0 -1 0 0 0 1 1 0 0 0 0 -1
-2 5 -3 1 1 2 1 -1 -3 2 0 0 2 0 -1 4 2 -2 1 0 -1 0 -2 2 -1 -1 -4 1 0 -1 -1 -1 1 1 0 0 1 -1 1 0 1 -1 1 -1 0 0 0 0
1 0 1 -3 -1 1 -2 2 0 0 2 0 2 0 1 0 0 -2 -1 2 -1 -1 1 0 2 -1 1 -1 0 0 0 0 -1 0 1 1 0 1 0 1 0 0 0 0 1 -1 0 1
0 -3 0 -2 -4 1 -1 0 0 1 2 0 0 1 -1 -2 -1 2 -1 -2 3 3 -2 0 1 1 1 1 1 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 1 -1
1 -3 0 1 0 0 0 0 0 0 1 0 0 1 0 -1 0 1 0 -1 1 2 -1 0 0 1 1 0 0 0 0 0 0 0 -1 -1 0 0 0 0 -1 0 0 0 0 0 0 -1
0 -2 1 4 4 -2 1 -2 1 -1 0 0 -1 0 1 -1 0 1 0 0 0 0 0 -1 -2 2 2 0 -1 1 1 0 0 -1 -1 -1 -1 0 -1 0 -1 0 0 0 -1 1 -1 -1
1 -8 3 2 -2 -2 0 -1 1 -1 1 -1 -2 1 0 -6 -3 4 -3 -2 4 2 0 -2 1 2 4 0 0 1 1 1 0 -2 -1 -1 -1 0 -1 0 -1 0 0 1 -1 1 0 -1
3 1 -4 -4 -1 3 -1 1 -2 1 3 -1 4 1 -1 3 -1 -2 -1 1 -1 1 -2 2 3 -3 -3 2 2 -1 0 -2 0 2 2 1 1 1 1 1 1 -1 -1 -1 1 -2 1 1
0 -1 -2 -4 -4 3 0 2 0 0 4 -1 4 1 -1 -1 -2 0 -2 -1 2 3 -3 2 3 -2 -1 2 2 0 0 -1 0 1 1 0 0 0 1 1 1 -1 0 -1 1 -1 1 0
1 -3 1 -3 2 -2 -2 -1 3 -1 2 1 -2 -1 1 -3 -2 1 -1 1 0 0 1 -3 0 2 4 0 0 1 2 -1 -1 -1 0 -1 -2 2 -1 1 -1 0 -1 0 0 0 0 0
-3 3 0 -6 -3 1 -1 2 2 0 1 1 0 -1 -1 -1 0 0 1 0 0 1 -1 0 1 -1 0 0 1 0 -1 0 -1 1 0 0 0 0 1 0 0 0 0 0 1 0 1 0
0 3 -2 -4 -4 3 -1 2 0 1 2 -1 3 1 -3 2 -1 -1 -1 0 1 2 -3 3 2 -3 -3 2 2 0 -1 -1 0 2 1 1 1 0 1 0 2 -1 0 0 1 -1 2 1
4 -3 -4 2 4 1 4 -2 1 -1 3 -2 1 0 -2 0 -2 2 0 -2 0 3 -3 1 0 0 -1 3 1 1 1 -2 1 1 0 -1 -1 0 0 1 -1 -1 -1 -1 0 0 0 -1
-3 3 2 -1 0 -2 -2 0 -1 0 -2 2 -2 -1 2 0 2 -1 1 1 -1 -3 3 -2 -1 1 1 -2 -1 -1 0 1 0 -1 0 0 0 0 0 0 0 1 1 0 0 0 -1 0
0 1 0 3 4 -3 -1 -2 -2 0 -2 1 -2 0 2 1 1 -1 0 2 -2 -3 3 -2 -2 1 0 -1 -1 -1 0 0 1 -1 0 0 0 0 -1 0 0 0 0 0 -1 0 -1 0
-2 7 -2 -2 1 2 -2 -1 -5 3 -1 0 2 0 1 6 3 -4 2 2 -3 -3 0 1 0 -2 -4 -1 0 -2 -1 -1 0 1 0 2 2 0 1 0 1 -1 0 -1 0 -1 0 1
-4 1 3 -2 -5 -1 -3 1 -1 1 -2 1 -2 0 1 -2 1 0 0 0 2 -1 2 -1 0 0 1 -2 0 -1 -1 2 0 -1 -1 0 1 -1 0 -1 1 1 1 1 0 0 0 0
3 -2 -2 -2 0 1 1 1 2 -1 3 -1 1 0 -2 -1 -3 1 -2 0 0 2 -1 0 2 -1 0 2 1 1 0 -1 0 1 1 0 -1 1 0 1 0 -1 -1 0 1 0 1 0
0 -3 2 5 1 -2 3 0 2 -2 -3 0 -3 -1 1 -3 0 2 1 -2 0 0 2 -1 -2 2 2 -1 -2 1 0 2 0 -1 -1 -1 -1 -1 -1 -1 -1 1 0 1 -1 2 -1 -1
0 -2 1 -2 -3 0 0 1 2 0 1 1 0 0 0 -2 0 1 0 -1 1 2 -1 0 1 1 2 0 0 0 0 1 -1 0 0 -1 0 0 0 0 0 1 0 0 0 0 1 -1
0 4 -1 5 3 -1 0 -2 -5 2 -4 1 -1 0 2 5 3 -3 1 1 -3 -4 3 0 -3 1 -3 -1 -2 -2 -1 0 2 0 0 1 1 -1 -1 0 1 0 1 0 -1 0 -1 0
4 0 -4 -4 -3 4 -2 1 -1 2 5 0 5 2 -1 3 0 -2 -1 1 0 3 -3 2 2 -1 -2 2 2 -1 -1 -2 0 2 1 0 1 1 0 1 2 -1 -1 -1 1 -2 2 0
-5 1 2 -4 -5 -1 0 1 3 0 -2 3 -3 -2 1 -3 1 2 3 -2 2 1 0 -1 -1 2 2 -1 0 0 0 2 -1 0 -1 -2 0 -1 0 -1 0 2 0 0 0 1 0 -1
5 0 -5 -3 -1 4 -1 0 -3 2 5 -1 5 2 -1 4 0 -2 -1 1 -1 2 -2 2 2 -2 -3 2 2 -1 -1 -2 0 2 1 1 1 1 0 1 1 -1 -1 -1 1 -2 1 0
3 -7 1 1 0 -2 0 -2 3 -1 1 0 -3 0 0 -4 -3 4 -1 -2 3 2 0 -2 -1 3 4 1 0 2 2 0 0 -1 -1 -2 -2 1 -2 0 -1 1 -1 1 -1 1 0 -1
-3 -4 4 1 -2 -3 1 2 5 -3 -1 2 -3 -1 2 -6 0 3 1 -1 2 1 2 -2 -1 3 5 -2 -1 1 0 3 -1 -2 -1 -2 -2 -1 -1 -1 -1 2 0 0 0 1 -1 -2
-2 4 -1 -5 -4 1 -1 2 1 0 0 1 0 -1 -1 0 0 0 1 0 0 0 0 0 1 -1 -1 0 1 0 -1 0 0 1 1 0 0 0 1 0 1 0 0 0 1 0 1 0
1 -3 0 3 1 2 1 1 -2 0 2 -1 3 1 1 0 0 -1 -1 0 0 1 -1 1 1 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 1 -1 -1 0 -1 0 0 -1 0
2 4 -2 0 3 1 -3 -2 -4 2 0 0 1 1 1 5 2 -3 0 3 -3 -3 2 -1 -1 -1 -3 -1 0 -2 -1 -2 1 0 0 2 1 1 0 0 1 -1 0 0 0 -1 0 1
4 -4 -2 0 2 1 1 0 1 -1 3 -1 1 1 -2 -1 -3 1 -2 -1 0 3 -2 0 1 0 0 2 1 1 0 -2 0 1 0 0 -1 1 0 1 -1 -2 -1 0 0 0 1 0
-4 2 0 -3 0 -1 1 1 3 -1 -2 2 -3 -2 -1 -2 -2 1 1 -2 0 1 -1 0 -1 0 0 1 0 1 1 -1 0 1 0 -1 -1 0 1 0 -1 0 0 0 0 1 0 0
-1 3 -2 -2 3 0 0 -1 1 0 1 1 0 -1 0 2 0 -1 1 1 -2 0 -1 0 -1 0 -1 1 0 0 1 -2 -1 1 0 0 -1 1 0 0 0 -1 -1 -1 0 0 0 0
3 -4 0 0 0 3 0 2 2 -1 5 -2 4 1 0 -1 0 -1 -1 1 0 2 -1 1 2 -1 2 0 0 1 0 0 -2 -1 0 0 -1 1 0 0 0 0 -1 -1 1 -1 0 0
1 1 -2 0 4 0 -2 -3 -3 2 1 0 0 1 1 3 1 -1 0 1 -1 -1 0 -1 -1 1 -1 0 0 -1 1 -2 1 0 -1 0 0 1 0 1 0 -1 0 -1 0 -1 0 0
0 -3 3 -2 -1 -2 -1 1 4 -2 -1 1 -4 -2 1 -4 -2 2 0 0 0 -1 3 -3 0 2 4 -1 -1 1 1 1 -1 -1 0 -1 -2 1 -1 0 -1 1 -1 1 0 1 0 0
5 -6 1 -1 -1 -1 -4 -1 1 0 2 0 -1 2 0 -2 -2 2 -3 1 2 1 1 -3 1 2 3 0 1 0 1 -1 0 -1 0 0 -1 2 -1 0 0 0 -1 1 0 -1 1 0
-2 2 2 1 1 0 -1 0 -1 1 -1 0 1 0 1 2 1 -2 0 2 -1 -2 0 1 0 -1 0 -1 -1 0 1 0 -1 0 0 1 1 0 0 0 0 0 0 0 0 0 -1 1
-7 0 5 -2 -2 -5 1 1 7 -3 -4 4 -7 -5 2 -7 -1 3 3 -2 1 -1 3 -3 -3 4 6 -2 -2 2 1 3 -1 -2 -1 -3 -3 -1 -2 -1 -1 3 0 1 -1 3 -1 -1
-3 4 2 3 0 -1 1 0 -3 1 -6 0 -2 -2 1 2 2 -2 2 0 -2 -4 2 1 -1 -1 -2 -2 -2 -1 -1 2 1 0 0 1 2 -2 0 -1 0 1 1 1 -1 1 -1 1
-2 -4 3 3 0 -2 1 -1 1 -1 -2 0 -4 -1 0 -4 -1 3 0 -2 2 0 1 -2 -1 2 3 -1 -1 1 0 2 0 -2 -2 -1 -1 -1 -1 -1 -2 1 0 1 -1 2 -1 -1
3 -5 3 6 2 -1 1 -1 0 -1 0 -2 -1 0 1 -2 0 1 -1 0 0 -1 2 -1 -1 2 3 -1 -2 1 1 1 0 -2 -1 0 -1 0 -2 0 -1 0 0 1 -1 1 -1 0
7 -7 -2 3 0 2 1 2 0 -2 3 -3 3 3 -1 -1 -2 1 -3 1 0 2 1 0 3 -2 0 1 1 0 -1 0 0 0 1 1 0 1 0 0 0 -1 -2 0 1 -1 0 0
-4 3 1 -4 3 -4 -3 -3 0 1 -1 2 -4 -2 1 0 -1 0 1 0 0 -2 0 -2 -2 2 2 0 0 1 2 -1 0 0 -1 -1 -1 1 -1 1 -1 0 0 0 -1 1 0 0
-3 6 1 -2 1 -1 -1 0 0 0 -2 1 -1 -1 0 2 2 -1 1 1 -1 -3 1 -1 -1 0 -1 -1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 1
2 -5 -1 2 -1 1 4 1 2 -2 1 -1 0 0 -1 -3 -1 3 0 -3 1 3 -1 0 1 1 1 1 0 0 0 1 0 0 0 -1 -1 -1 0 0 -1 0 -1 -1 0 0 0 -1
2 1 1 -2 1 1 -3 1 0 0 3 0 2 1 1 1 1 -2 -1 3 -2 -1 1 -1 1 0 1 -1 0 0 0 -1 -1 0 0 1 0 2 0 1 0 -1 0 0 1 -1 0 1
-1 1 1 0 1 -2 0 0 1 -1 -1 1 -2 -1 2 -1 1 0 1 1 -1 -2 3 -2 -1 1 2 -1 -1 0 0 1 0 -1 0 0 -1 0 -1 0 0 1 0 0 0 0 -1 -1
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PROPERTIES
Uniodular=1
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REFERENCES
G. Nebe, Some cyclo-quaternionic lattices, J. Alg. 199 (1998), 472-498.
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NOTES
Using the classification of finite simple groups one may prove that this is indeed the full
automorphism group
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LAST_LINE
Haftungsausschluss/Disclaimer
Gabriele Nebe