The Lattice Shimada_86
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 13:26:49 CEST 2014
INDEX FILE |
ABBREVIATIONS
Contents of this file
NAME
DIM
GRAM
GRAM_MATRIX (in Maple)
DET
PROPERTIES
MINIMAL_NORM
KISSING_NUMBER
GROUP_NAME
REMARKS
REFERENCES
LAST_LINE
-
NAME
Shimada_86
-
DIM
86
-
GRAM
86 86
16
8 8
8 4 8
8 4 4 8
8 4 4 4 8
8 4 4 4 4 8
8 4 4 4 4 4 8
8 4 4 4 4 4 4 8
0 2 2 2 2 2 2 2 8
8 4 4 4 4 4 4 4 0 8
-8 4 4 4 4 4 4 4 0 4 8
-4 2 4 4 4 4 2 2 4 4 4 8
-8 4 4 4 4 4 4 4 0 4 4 2 8
4 4 2 4 4 2 4 2 4 4 2 4 4 8
4 4 4 4 2 2 2 4 4 2 2 4 4 4 8
8 4 4 4 4 4 4 4 0 4 4 2 4 2 2 8
0 2 2 2 2 0 0 0 4 2 0 4 0 4 4 2 8
4 4 4 2 4 4 2 2 4 2 2 4 4 4 4 4 4 8
0 2 2 2 2 0 0 0 4 0 0 4 0 4 4 0 4 4 8
4 4 2 4 4 4 2 2 4 2 2 4 2 4 4 2 4 4 4 8
0 2 2 2 2 0 0 0 4 0 2 4 2 4 4 0 4 4 4 4 8
8 4 4 4 4 4 4 4 0 4 4 2 4 2 2 4 0 2 0 2 0 8
8 4 4 4 4 4 4 4 0 4 4 2 4 2 2 4 0 2 0 2 0 4 8
8 4 4 4 4 4 4 4 0 4 4 2 4 2 2 4 0 2 0 2 0 4 4 8
0 2 2 2 2 0 0 0 4 0 0 3 0 3 3 0 4 3 4 3 4 2 2 2 8
8 4 4 4 4 4 4 4 0 4 4 2 4 2 2 4 0 2 0 2 0 4 4 4 0 8
-8 4 4 4 4 4 4 4 0 4 4 2 4 2 2 4 0 2 0 2 0 4 4 4 0 4 8
-4 2 4 4 4 2 2 2 3 2 2 4 2 3 3 2 3 3 3 3 3 4 2 2 4 4 4 8
-8 4 4 4 4 4 4 4 0 4 4 2 4 2 2 4 0 2 0 2 0 4 4 4 0 4 4 2 8
4 4 2 4 4 2 2 2 3 2 2 3 2 4 3 2 3 3 3 4 3 2 4 2 4 4 2 4 4 8
4 4 4 4 2 2 2 2 3 2 2 3 2 3 4 2 3 3 3 3 3 2 2 4 4 2 2 4 4 4 8
8 4 4 4 4 4 4 4 0 4 4 2 4 2 2 4 0 2 0 2 0 4 4 4 0 4 4 2 4 2 2 8
0 2 2 2 2 0 0 0 4 0 0 3 0 3 3 0 4 3 4 3 4 0 0 0 4 2 0 4 0 4 4 2 8
4 4 4 2 4 2 2 2 3 2 2 3 2 3 3 2 3 4 3 3 3 4 2 2 4 2 2 4 4 4 4 4 4 8
0 2 2 2 2 0 0 0 4 0 0 3 0 3 3 0 4 3 4 3 4 0 0 0 4 0 0 4 0 4 4 0 4 4 8
4 4 2 4 4 2 2 2 3 2 2 3 2 4 3 2 3 3 3 4 3 4 2 2 4 2 2 4 2 4 4 2 4 4 4 8
0 0 0 0 0 1 1 1 2 1 1 2 1 2 2 1 2 2 2 2 2 1 1 1 2 1 1 2 1 2 2 1 2 2 2 2 8
8 4 4 4 4 4 4 4 0 4 4 2 4 2 2 4 0 2 0 2 0 4 4 4 0 4 4 2 4 2 2 4 0 2 0 2 0 8
-8 4 4 4 4 4 4 4 0 4 4 2 4 2 2 4 0 2 0 2 0 4 4 4 0 4 4 2 4 2 2 4 0 2 0 2 0 4 8
-4 2 2 2 2 2 4 4 3 2 2 1 2 2 2 2 0 1 0 1 0 4 2 2 1 2 2 2 2 1 1 2 0 2 0 2 2 4 4 8
-8 4 4 4 4 4 4 4 0 4 4 2 4 2 2 4 0 2 0 2 0 4 4 4 0 4 4 2 4 2 2 4 0 2 0 2 0 4 4 2 8
4 2 2 2 2 4 2 4 3 2 2 2 2 1 2 2 0 2 0 2 0 2 4 2 1 2 2 1 2 2 1 2 0 1 0 1 2 4 2 4 4 8
4 2 2 2 2 4 4 4 3 2 2 2 2 2 2 2 0 2 0 2 0 2 2 4 1 2 2 1 2 1 2 2 0 1 0 1 2 2 2 4 4 4 8
8 4 4 4 4 4 4 4 0 4 4 2 4 2 2 4 0 2 0 2 0 4 4 4 0 4 4 2 4 2 2 4 0 2 0 2 0 4 4 2 4 2 2 8
0 0 0 0 0 2 2 2 4 0 0 1 0 1 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 0 4 0 4 4 2 8
4 2 2 2 2 4 4 2 3 2 2 2 2 2 1 2 0 2 0 2 0 4 2 2 1 2 2 2 2 1 1 2 0 2 0 2 2 2 2 4 4 4 4 4 4 8
0 0 0 0 0 2 2 2 4 0 0 1 0 1 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 4 0 4 4 0 4 4 8
4 2 2 2 2 4 2 4 3 2 2 2 2 1 2 2 0 2 0 2 0 4 2 2 1 2 2 2 2 1 1 2 0 2 0 2 2 2 2 4 2 4 4 2 4 4 4 8
0 1 1 1 1 0 0 0 2 1 1 3 1 3 3 1 4 3 4 3 4 1 1 1 4 0 0 2 0 2 2 0 2 2 2 2 4 1 1 2 1 2 2 1 2 2 2 2 8
8 4 4 4 4 4 4 4 0 4 4 2 4 2 2 4 0 2 0 2 0 4 4 4 0 4 4 2 4 2 2 4 0 2 0 2 0 4 4 2 4 2 2 4 0 2 0 2 0 8
4 2 2 2 2 2 2 2 0 2 2 1 2 1 2 4 3 2 0 2 0 2 2 2 0 4 2 2 2 2 1 2 1 1 0 1 2 4 2 2 2 2 1 2 1 1 0 1 2 4 8
4 2 2 2 2 2 2 2 0 4 2 2 2 2 2 2 3 1 0 2 0 2 2 2 0 2 2 1 2 1 2 2 1 1 0 2 2 2 2 1 2 1 1 2 1 1 0 1 2 2 4 8
4 2 2 2 2 2 2 2 0 4 2 2 2 2 1 4 3 2 0 1 0 2 2 2 0 4 2 2 2 2 1 2 1 1 0 1 2 2 2 1 2 1 2 2 1 1 0 2 2 2 4 4 8
4 2 2 2 2 2 2 2 0 2 2 1 2 1 2 4 3 2 0 2 0 2 2 2 0 2 2 1 2 1 1 4 1 2 0 1 2 2 2 1 2 1 1 4 1 2 0 1 2 2 4 4 4 8
0 0 0 0 0 0 0 0 0 2 0 1 0 1 1 2 4 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 2 0 4 4 4 4 8
4 2 2 2 2 2 2 2 0 4 2 2 2 2 2 2 3 1 0 2 0 2 2 2 0 4 2 2 2 2 1 2 1 1 0 1 2 2 2 1 2 1 1 4 1 2 0 1 2 2 4 4 4 4 4 8
0 1 1 1 1 1 1 1 4 0 1 3 1 3 3 0 2 3 4 3 4 0 0 0 2 1 0 2 0 2 2 1 4 2 2 2 4 1 0 2 0 2 2 1 4 2 2 2 4 1 2 2 2 2 2 2 8
4 2 2 2 2 2 2 2 0 2 2 1 2 2 1 2 0 2 3 2 0 2 2 2 0 2 4 2 2 1 1 2 0 1 0 1 2 2 2 2 2 1 2 2 0 1 1 1 2 4 2 1 1 1 0 1 2 8
4 2 2 2 2 2 2 2 0 2 2 2 2 2 2 2 0 1 3 2 0 2 2 2 0 2 4 2 2 1 1 2 0 1 0 1 2 2 2 1 2 1 1 2 0 1 1 2 2 2 1 2 2 2 0 1 2 4 8
4 2 2 2 2 2 2 2 0 2 2 2 2 1 2 2 0 2 3 2 0 2 2 2 0 2 4 2 2 1 1 2 0 1 0 1 2 2 2 1 2 1 1 2 0 2 1 1 2 2 1 1 1 1 0 2 2 4 4 8
0 1 1 1 1 1 1 1 4 1 1 3 1 3 3 1 4 3 2 3 4 0 0 0 2 0 1 2 1 2 2 0 2 2 2 2 4 0 0 2 0 2 2 0 2 2 4 2 4 1 2 2 2 2 2 2 4 2 2 2 8
0 4 0 0 0 0 0 0 2 0 0 0 0 2 2 0 2 2 2 2 2 0 0 0 2 0 0 0 0 2 2 0 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 8
0 0 0 0 0 2 0 0 1 2 2 4 0 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 0 1 0 2 1 0 2 1 1 1 2 0 1 1 1 0 1 1 2 1 1 1 2 2 8
0 2 0 0 0 2 0 0 2 2 0 3 0 2 2 0 2 2 2 2 1 2 0 0 2 0 0 1 0 1 1 0 1 2 1 2 2 0 0 1 0 1 1 0 1 2 1 2 2 2 1 1 1 0 1 1 2 1 1 1 2 4 4 8
0 0 0 0 0 2 0 0 1 2 2 4 0 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 1 0 1 2 0 1 1 2 1 2 0 1 2 1 0 2 1 2 1 1 1 2 0 4 4 8
0 2 0 0 0 2 0 0 2 0 2 3 0 1 2 0 1 2 2 2 2 2 0 0 2 0 0 1 0 1 1 0 1 2 1 2 2 0 0 1 0 1 1 0 1 2 1 2 2 0 0 0 0 0 0 0 2 1 1 1 2 2 4 4 4 8
2 1 2 2 2 1 2 2 3 1 1 2 2 3 3 2 3 3 3 3 3 2 1 1 2 2 2 4 1 2 2 1 2 2 2 2 4 2 2 4 1 2 2 1 2 2 2 2 4 2 4 2 2 2 2 2 4 4 2 2 4 1 2 2 2 2 8
0 0 0 0 0 0 2 0 1 2 0 1 2 4 1 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 1 0 0 1 0 1 1 1 1 2 0 0 1 1 0 1 1 2 1 1 0 2 0 1 1 1 1 2 8
0 0 0 0 0 0 2 0 1 2 0 1 2 4 1 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 1 0 0 1 0 1 1 2 1 2 0 0 1 1 0 2 2 2 1 2 0 2 0 1 1 2 1 2 4 8
0 0 2 0 0 0 2 0 2 2 0 2 0 3 1 0 2 1 2 0 1 2 0 0 2 0 0 2 0 0 1 0 1 2 1 1 2 0 0 2 0 0 1 0 1 2 1 1 2 0 0 1 1 0 1 1 2 1 1 0 2 0 1 2 1 1 2 4 4 8
2 2 1 2 2 2 1 2 3 1 2 3 1 2 3 2 3 3 3 4 3 2 1 1 2 1 1 2 1 2 2 1 2 2 2 4 4 1 1 2 1 2 2 1 2 2 2 4 4 1 2 2 2 2 2 2 4 2 2 2 4 1 2 2 2 4 4 2 2 2 8
0 0 0 0 0 0 0 2 1 0 0 1 2 1 4 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 1 2 2 2 0 1 1 1 1 2 0 1 1 0 1 1 1 2 0 1 1 2 0 1 1 1 1 2 2 1 1 2 8
2 2 2 2 1 2 2 1 3 2 2 3 1 3 2 2 3 3 3 3 3 2 1 1 2 1 1 2 1 2 2 1 2 2 2 2 4 1 1 2 2 2 2 2 2 4 2 2 4 1 2 2 2 2 2 2 4 2 2 2 4 1 2 2 2 2 4 2 2 4 4 2 8
0 0 0 0 0 2 0 0 1 0 0 1 0 0 1 0 1 1 1 4 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 1 1 0 1 1 1 1 2 0 1 2 1 2 1 1 2 1 2 1 2 0 1 1 1 1 2 0 0 0 2 2 2 8
2 2 1 2 2 1 2 2 3 2 2 3 2 4 3 2 3 3 3 2 3 1 2 1 2 2 1 2 2 4 2 1 2 2 2 2 4 1 1 2 1 2 2 1 2 2 2 2 4 1 2 2 4 2 2 2 4 2 2 2 4 1 2 2 2 2 4 2 2 2 4 2 4 2 8
0 0 0 0 0 0 0 0 0 2 0 1 0 1 1 2 4 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 2 0 3 3 3 3 4 3 0 0 0 0 2 0 1 1 1 1 2 2 1 1 2 2 2 2 2 8
0 1 1 1 1 1 1 1 4 0 1 3 1 3 3 0 2 3 4 3 4 1 1 1 4 0 0 2 0 2 2 0 2 2 2 2 4 0 0 2 0 2 2 0 2 2 2 2 4 0 0 0 0 0 0 0 4 2 2 2 4 1 2 2 2 4 4 2 2 2 4 2 4 2 4 2 8
4 4 2 4 4 2 2 2 3 2 2 3 2 4 3 2 3 3 3 4 3 2 2 2 3 2 2 3 2 4 3 2 3 3 3 4 0 2 4 2 2 1 1 2 0 1 0 1 2 2 1 1 1 1 0 2 2 1 1 2 2 2 0 1 0 1 2 0 0 0 2 1 2 0 2 1 2 8
0 0 0 0 0 1 1 1 2 1 1 2 1 2 2 1 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 1 2 1 2 2 0 2 2 2 2 4 1 2 2 2 2 2 2 4 2 2 2 4 0 2 2 2 2 4 4 2 2 4 4 4 4 4 4 4 2 8
2 2 2 1 2 1 2 2 3 2 2 3 1 3 3 1 3 2 3 3 3 2 1 1 2 1 1 2 2 2 2 2 2 4 2 2 4 2 2 4 1 2 2 1 2 2 2 2 4 1 2 2 2 2 2 2 4 2 2 2 4 1 2 2 2 2 4 2 2 4 4 2 4 2 4 2 4 2 4 8
0 1 1 1 1 1 1 1 4 1 1 3 1 3 3 1 4 3 2 3 4 1 1 1 4 0 0 2 0 2 2 0 2 2 2 2 4 0 0 2 0 2 2 0 2 2 2 2 4 0 2 2 2 2 4 2 4 0 0 0 4 1 2 2 2 2 4 2 2 2 4 2 4 2 4 2 4 2 4 4 8
2 1 2 2 2 1 2 2 3 1 1 2 2 3 3 2 3 3 3 3 3 1 1 2 2 1 1 2 1 2 2 2 2 2 2 2 4 1 1 2 1 2 2 2 2 2 2 2 4 1 2 2 2 4 2 2 4 2 2 2 4 0 0 0 0 0 4 2 2 2 4 2 4 2 4 2 4 2 4 4 4 8
0 0 0 0 0 1 1 1 2 1 1 2 1 2 2 1 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 1 0 2 0 2 2 1 4 2 2 2 4 0 2 2 2 2 2 2 4 2 2 2 4 1 4 2 2 2 4 2 2 2 4 2 4 2 4 2 4 0 4 4 4 4 8
2 2 1 2 2 1 2 2 3 2 2 3 2 4 3 2 3 3 3 2 3 2 1 1 2 1 1 2 1 2 2 1 2 2 2 4 4 2 2 4 1 2 2 1 2 2 2 2 4 1 2 2 2 2 2 2 4 2 2 2 4 1 2 2 2 2 4 2 2 2 4 2 4 0 4 2 4 4 4 4 4 4 4 8
0 0 0 0 0 1 1 1 2 1 1 2 1 2 2 1 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 2 0 2 2 0 2 2 4 2 4 0 2 2 2 2 4 2 4 2 2 2 4 0 2 2 4 2 4 2 4 2 4 2 4 2 4 2 4 0 4 4 4 4 4 4 8
1 1 1 1 1 1 1 1 2 -1 -1 -1 1 1 1 1 1 2 1 2 1 1 1 1 2 -1 -1 -1 1 1 1 1 1 2 1 2 1 -1 -1 -1 1 1 1 1 1 2 1 2 1 1 0 1 0 2 1 0 2 0 0 0 2 1 -2 0 -2 0 1 1 1 1 3 1 3 2 2 1 3 1 2 1 3 4 2 1 2 10
-
GRAM_MATRIX (in Maple)
[[16, 8, 8, 8, 8, 8, 8, 8, 0, 8, 8, 4, 8, 4, 4, 8, 0, 4, 0, 4, 0, 8, 8
, 8, 0, 8, 8, 4, 8, 4, 4, 8, 0, 4, 0, 4, 0, 8, 8, 4, 8, 4, 4, 8, 0, 4, 0, 4, 0
, 8, 4, 4, 4, 4, 0, 4, 0, 4, 4, 4, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 2, 0, 2
, 0, 0, 4, 0, 2, 0, 2, 0, 2, 0, 1], [8, 8, 4, 4, 4, 4, 4, 4, 2, 4, 4, 2, 4, 4,
4, 4, 2, 4, 2, 4, 2, 4, 4, 4, 2, 4, 4, 2, 4, 4, 4, 4, 2, 4, 2, 4, 0, 4, 4, 2,
4, 2, 2, 4, 0, 2, 0, 2, 1, 4, 2, 2, 2, 2, 0, 2, 1, 2, 2, 2, 1, 4, 0, 2, 0, 2,
1, 0, 0, 0, 2, 0, 2, 0, 2, 0, 1, 4, 0, 2, 1, 1, 0, 2, 0, 1], [8, 4, 8, 4, 4, 4
, 4, 4, 2, 4, 4, 4, 4, 2, 4, 4, 2, 4, 2, 2, 2, 4, 4, 4, 2, 4, 4, 4, 4, 2, 4, 4
, 2, 4, 2, 2, 0, 4, 4, 2, 4, 2, 2, 4, 0, 2, 0, 2, 1, 4, 2, 2, 2, 2, 0, 2, 1, 2
, 2, 2, 1, 0, 0, 0, 0, 0, 2, 0, 0, 2, 1, 0, 2, 0, 1, 0, 1, 2, 0, 2, 1, 2, 0, 1
, 0, 1], [8, 4, 4, 8, 4, 4, 4, 4, 2, 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 4, 2, 4, 4,
4, 2, 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 4, 0, 4, 4, 2, 4, 2, 2, 4, 0, 2, 0, 2, 1,
4, 2, 2, 2, 2, 0, 2, 1, 2, 2, 2, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 2, 0, 2,
0, 1, 4, 0, 1, 1, 2, 0, 2, 0, 1], [8, 4, 4, 4, 8, 4, 4, 4, 2, 4, 4, 4, 4, 4, 2
, 4, 2, 4, 2, 4, 2, 4, 4, 4, 2, 4, 4, 4, 4, 4, 2, 4, 2, 4, 2, 4, 0, 4, 4, 2, 4
, 2, 2, 4, 0, 2, 0, 2, 1, 4, 2, 2, 2, 2, 0, 2, 1, 2, 2, 2, 1, 0, 0, 0, 0, 0, 2
, 0, 0, 0, 2, 0, 1, 0, 2, 0, 1, 4, 0, 2, 1, 2, 0, 2, 0, 1], [8, 4, 4, 4, 4, 8,
4, 4, 2, 4, 4, 4, 4, 2, 2, 4, 0, 4, 0, 4, 0, 4, 4, 4, 0, 4, 4, 2, 4, 2, 2, 4,
0, 2, 0, 2, 1, 4, 4, 2, 4, 4, 4, 4, 2, 4, 2, 4, 0, 4, 2, 2, 2, 2, 0, 2, 1, 2,
2, 2, 1, 0, 2, 2, 2, 2, 1, 0, 0, 0, 2, 0, 2, 2, 1, 0, 1, 2, 1, 1, 1, 1, 1, 1,
1, 1], [8, 4, 4, 4, 4, 4, 8, 4, 2, 4, 4, 2, 4, 4, 2, 4, 0, 2, 0, 2, 0, 4, 4, 4
, 0, 4, 4, 2, 4, 2, 2, 4, 0, 2, 0, 2, 1, 4, 4, 4, 4, 2, 4, 4, 2, 4, 2, 2, 0, 4
, 2, 2, 2, 2, 0, 2, 1, 2, 2, 2, 1, 0, 0, 0, 0, 0, 2, 2, 2, 2, 1, 0, 2, 0, 2, 0
, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1], [8, 4, 4, 4, 4, 4, 4, 8, 2, 4, 4, 2, 4, 2, 4,
4, 0, 2, 0, 2, 0, 4, 4, 4, 0, 4, 4, 2, 4, 2, 2, 4, 0, 2, 0, 2, 1, 4, 4, 4, 4,
4, 4, 4, 2, 2, 2, 4, 0, 4, 2, 2, 2, 2, 0, 2, 1, 2, 2, 2, 1, 0, 0, 0, 0, 0, 2,
0, 0, 0, 2, 2, 1, 0, 2, 0, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1], [0, 2, 2, 2, 2, 2, 2
, 2, 8, 0, 0, 4, 0, 4, 4, 0, 4, 4, 4, 4, 4, 0, 0, 0, 4, 0, 0, 3, 0, 3, 3, 0, 4
, 3, 4, 3, 2, 0, 0, 3, 0, 3, 3, 0, 4, 3, 4, 3, 2, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0
, 0, 4, 2, 1, 2, 1, 2, 3, 1, 1, 2, 3, 1, 3, 1, 3, 0, 4, 3, 2, 3, 4, 3, 2, 3, 2
, 2], [8, 4, 4, 4, 4, 4, 4, 4, 0, 8, 4, 4, 4, 4, 2, 4, 2, 2, 0, 2, 0, 4, 4, 4,
0, 4, 4, 2, 4, 2, 2, 4, 0, 2, 0, 2, 1, 4, 4, 2, 4, 2, 2, 4, 0, 2, 0, 2, 1, 4,
2, 4, 4, 2, 2, 4, 0, 2, 2, 2, 1, 0, 2, 2, 2, 0, 1, 2, 2, 2, 1, 0, 2, 0, 2, 2,
0, 2, 1, 2, 1, 1, 1, 2, 1, -1], [8, 4, 4, 4, 4, 4, 4, 4, 0, 4, 8, 4, 4, 2, 2,
4, 0, 2, 0, 2, 2, 4, 4, 4, 0, 4, 4, 2, 4, 2, 2, 4, 0, 2, 0, 2, 1, 4, 4, 2, 4,
2, 2, 4, 0, 2, 0, 2, 1, 4, 2, 2, 2, 2, 0, 2, 1, 2, 2, 2, 1, 0, 2, 0, 2, 2, 1,
0, 0, 0, 2, 0, 2, 0, 2, 0, 1, 2, 1, 2, 1, 1, 1, 2, 1, -1], [4, 2, 4, 4, 4, 4,
2, 2, 4, 4, 4, 8, 2, 4, 4, 2, 4, 4, 4, 4, 4, 2, 2, 2, 3, 2, 2, 4, 2, 3, 3, 2,
3, 3, 3, 3, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 1, 2, 3, 2, 1, 2, 2, 1, 1, 2, 3, 1,
2, 2, 3, 0, 4, 3, 4, 3, 2, 1, 1, 2, 3, 1, 3, 1, 3, 1, 3, 3, 2, 3, 3, 2, 2, 3,
2, -1], [8, 4, 4, 4, 4, 4, 4, 4, 0, 4, 4, 2, 8, 4, 4, 4, 0, 4, 0, 2, 2, 4, 4,
4, 0, 4, 4, 2, 4, 2, 2, 4, 0, 2, 0, 2, 1, 4, 4, 2, 4, 2, 2, 4, 0, 2, 0, 2, 1,
4, 2, 2, 2, 2, 0, 2, 1, 2, 2, 2, 1, 0, 0, 0, 0, 0, 2, 2, 2, 0, 1, 2, 1, 0, 2,
0, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1], [4, 4, 2, 4, 4, 2, 4, 2, 4, 4, 2, 4, 4, 8, 4
, 2, 4, 4, 4, 4, 4, 2, 2, 2, 3, 2, 2, 3, 2, 4, 3, 2, 3, 3, 3, 4, 2, 2, 2, 2, 2
, 1, 2, 2, 1, 2, 1, 1, 3, 2, 1, 2, 2, 1, 1, 2, 3, 2, 2, 1, 3, 2, 1, 2, 1, 1, 3
, 4, 4, 3, 2, 1, 3, 0, 4, 1, 3, 4, 2, 3, 3, 3, 2, 4, 2, 1], [4, 4, 4, 4, 2, 2,
2, 4, 4, 2, 2, 4, 4, 4, 8, 2, 4, 4, 4, 4, 4, 2, 2, 2, 3, 2, 2, 3, 2, 3, 4, 2,
3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 2, 3, 2, 2, 2, 1, 2, 1, 2, 3, 1,
2, 2, 3, 2, 1, 2, 1, 2, 3, 1, 1, 1, 3, 4, 2, 1, 3, 1, 3, 3, 2, 3, 3, 3, 2, 3,
2, 1], [8, 4, 4, 4, 4, 4, 4, 4, 0, 4, 4, 2, 4, 2, 2, 8, 2, 4, 0, 2, 0, 4, 4, 4
, 0, 4, 4, 2, 4, 2, 2, 4, 0, 2, 0, 2, 1, 4, 4, 2, 4, 2, 2, 4, 0, 2, 0, 2, 1, 4
, 4, 2, 4, 4, 2, 2, 0, 2, 2, 2, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 2, 0, 2, 2
, 0, 2, 1, 1, 1, 2, 1, 2, 1, 1], [0, 2, 2, 2, 2, 0, 0, 0, 4, 2, 0, 4, 0, 4, 4,
2, 8, 4, 4, 4, 4, 0, 0, 0, 4, 0, 0, 3, 0, 3, 3, 0, 4, 3, 4, 3, 2, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 4, 0, 3, 3, 3, 3, 4, 3, 2, 0, 0, 0, 4, 2, 1, 2, 1, 1, 3,
1, 1, 2, 3, 1, 3, 1, 3, 4, 2, 3, 2, 3, 4, 3, 2, 3, 2, 1], [4, 4, 4, 2, 4, 4, 2
, 2, 4, 2, 2, 4, 4, 4, 4, 4, 4, 8, 4, 4, 4, 2, 2, 2, 3, 2, 2, 3, 2, 3, 3, 2, 3
, 4, 3, 3, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 1, 2, 3, 2, 2, 1, 2, 2, 1, 1, 3, 2, 1
, 2, 3, 2, 1, 2, 1, 2, 3, 1, 1, 1, 3, 1, 3, 1, 3, 1, 3, 3, 2, 2, 3, 3, 2, 3, 2
, 2], [0, 2, 2, 2, 2, 0, 0, 0, 4, 0, 0, 4, 0, 4, 4, 0, 4, 4, 8, 4, 4, 0, 0, 0,
4, 0, 0, 3, 0, 3, 3, 0, 4, 3, 4, 3, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0,
0, 0, 0, 0, 0, 0, 4, 3, 3, 3, 2, 2, 1, 2, 1, 2, 3, 1, 1, 2, 3, 1, 3, 1, 3, 0,
4, 3, 2, 3, 2, 3, 2, 3, 2, 1], [4, 4, 2, 4, 4, 4, 2, 2, 4, 2, 2, 4, 2, 4, 4, 2
, 4, 4, 4, 8, 4, 2, 2, 2, 3, 2, 2, 3, 2, 4, 3, 2, 3, 3, 3, 4, 2, 2, 2, 1, 2, 2
, 2, 2, 1, 2, 1, 2, 3, 2, 2, 2, 1, 2, 1, 2, 3, 2, 2, 2, 3, 2, 1, 2, 1, 2, 3, 0
, 0, 0, 4, 1, 3, 4, 2, 1, 3, 4, 2, 3, 3, 3, 2, 2, 2, 2], [0, 2, 2, 2, 2, 0, 0,
0, 4, 0, 2, 4, 2, 4, 4, 0, 4, 4, 4, 4, 8, 0, 0, 0, 4, 0, 0, 3, 0, 3, 3, 0, 4,
3, 4, 3, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0,
0, 4, 2, 1, 1, 1, 2, 3, 1, 1, 1, 3, 1, 3, 1, 3, 0, 4, 3, 2, 3, 4, 3, 2, 3, 2,
1], [8, 4, 4, 4, 4, 4, 4, 4, 0, 4, 4, 2, 4, 2, 2, 4, 0, 2, 0, 2, 0, 8, 4, 4, 2
, 4, 4, 4, 4, 2, 2, 4, 0, 4, 0, 4, 1, 4, 4, 4, 4, 2, 2, 4, 0, 4, 0, 4, 1, 4, 2
, 2, 2, 2, 0, 2, 0, 2, 2, 2, 0, 0, 0, 2, 0, 2, 2, 0, 0, 2, 2, 0, 2, 0, 1, 0, 1
, 2, 0, 2, 1, 1, 0, 2, 0, 1], [8, 4, 4, 4, 4, 4, 4, 4, 0, 4, 4, 2, 4, 2, 2, 4,
0, 2, 0, 2, 0, 4, 8, 4, 2, 4, 4, 2, 4, 4, 2, 4, 0, 2, 0, 2, 1, 4, 4, 2, 4, 4,
2, 4, 0, 2, 0, 2, 1, 4, 2, 2, 2, 2, 0, 2, 0, 2, 2, 2, 0, 0, 0, 0, 0, 0, 1, 0,
0, 0, 1, 0, 1, 0, 2, 0, 1, 2, 0, 1, 1, 1, 0, 1, 0, 1], [8, 4, 4, 4, 4, 4, 4, 4
, 0, 4, 4, 2, 4, 2, 2, 4, 0, 2, 0, 2, 0, 4, 4, 8, 2, 4, 4, 2, 4, 2, 4, 4, 0, 2
, 0, 2, 1, 4, 4, 2, 4, 2, 4, 4, 0, 2, 0, 2, 1, 4, 2, 2, 2, 2, 0, 2, 0, 2, 2, 2
, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 2, 0, 1, 1, 2, 0, 1, 0, 1
], [0, 2, 2, 2, 2, 0, 0, 0, 4, 0, 0, 3, 0, 3, 3, 0, 4, 3, 4, 3, 4, 2, 2, 2, 8,
0, 0, 4, 0, 4, 4, 0, 4, 4, 4, 4, 2, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 4, 0, 0,
0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 2, 0, 2, 0, 2, 2, 0, 0, 2, 2, 0, 2, 0, 2, 0, 4,
3, 0, 2, 4, 2, 0, 2, 0, 2], [8, 4, 4, 4, 4, 4, 4, 4, 0, 4, 4, 2, 4, 2, 2, 4, 0
, 2, 0, 2, 0, 4, 4, 4, 0, 8, 4, 4, 4, 4, 2, 4, 2, 2, 0, 2, 1, 4, 4, 2, 4, 2, 2
, 4, 0, 2, 0, 2, 0, 4, 4, 2, 4, 2, 0, 4, 1, 2, 2, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0
, 0, 1, 0, 1, 0, 2, 0, 0, 2, 0, 1, 0, 1, 0, 1, 0, -1], [8, 4, 4, 4, 4, 4, 4, 4
, 0, 4, 4, 2, 4, 2, 2, 4, 0, 2, 0, 2, 0, 4, 4, 4, 0, 4, 8, 4, 4, 2, 2, 4, 0, 2
, 0, 2, 1, 4, 4, 2, 4, 2, 2, 4, 0, 2, 0, 2, 0, 4, 2, 2, 2, 2, 0, 2, 0, 4, 4, 4
, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 2, 0, 1, 0, 1, 0, 1, 0,
-1], [4, 2, 4, 4, 4, 2, 2, 2, 3, 2, 2, 4, 2, 3, 3, 2, 3, 3, 3, 3, 3, 4, 2, 2, 4
, 4, 4, 8, 2, 4, 4, 2, 4, 4, 4, 4, 2, 2, 2, 2, 2, 1, 1, 2, 0, 2, 0, 2, 2, 2, 2
, 1, 2, 1, 0, 2, 2, 2, 2, 2, 2, 0, 0, 1, 0, 1, 4, 0, 0, 2, 2, 0, 2, 0, 2, 0, 2
, 3, 0, 2, 2, 2, 0, 2, 0, -1], [8, 4, 4, 4, 4, 4, 4, 4, 0, 4, 4, 2, 4, 2, 2, 4
, 0, 2, 0, 2, 0, 4, 4, 4, 0, 4, 4, 2, 8, 4, 4, 4, 0, 4, 0, 2, 1, 4, 4, 2, 4, 2
, 2, 4, 0, 2, 0, 2, 0, 4, 2, 2, 2, 2, 0, 2, 0, 2, 2, 2, 1, 0, 0, 0, 0, 0, 1, 0
, 0, 0, 1, 0, 1, 0, 2, 0, 0, 2, 0, 2, 0, 1, 0, 1, 0, 1], [4, 4, 2, 4, 4, 2, 2,
2, 3, 2, 2, 3, 2, 4, 3, 2, 3, 3, 3, 4, 3, 2, 4, 2, 4, 4, 2, 4, 4, 8, 4, 2, 4,
4, 4, 4, 2, 2, 2, 1, 2, 2, 1, 2, 0, 1, 0, 1, 2, 2, 2, 1, 2, 1, 0, 2, 2, 1, 1,
1, 2, 2, 0, 1, 0, 1, 2, 0, 0, 0, 2, 0, 2, 0, 4, 0, 2, 4, 0, 2, 2, 2, 0, 2, 0,
1], [4, 4, 4, 4, 2, 2, 2, 2, 3, 2, 2, 3, 2, 3, 4, 2, 3, 3, 3, 3, 3, 2, 2, 4, 4
, 2, 2, 4, 4, 4, 8, 2, 4, 4, 4, 4, 2, 2, 2, 1, 2, 1, 2, 2, 0, 1, 0, 1, 2, 2, 1
, 2, 1, 1, 0, 1, 2, 1, 1, 1, 2, 2, 0, 1, 0, 1, 2, 0, 0, 1, 2, 0, 2, 0, 2, 0, 2
, 3, 0, 2, 2, 2, 0, 2, 0, 1], [8, 4, 4, 4, 4, 4, 4, 4, 0, 4, 4, 2, 4, 2, 2, 4,
0, 2, 0, 2, 0, 4, 4, 4, 0, 4, 4, 2, 4, 2, 2, 8, 2, 4, 0, 2, 1, 4, 4, 2, 4, 2,
2, 4, 0, 2, 0, 2, 0, 4, 2, 2, 2, 4, 0, 2, 1, 2, 2, 2, 0, 0, 0, 0, 0, 0, 1, 0,
0, 0, 1, 0, 1, 0, 1, 0, 0, 2, 0, 2, 0, 2, 0, 1, 0, 1], [0, 2, 2, 2, 2, 0, 0, 0
, 4, 0, 0, 3, 0, 3, 3, 0, 4, 3, 4, 3, 4, 0, 0, 0, 4, 2, 0, 4, 0, 4, 4, 2, 8, 4
, 4, 4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 1, 1, 1, 0, 1, 4, 0, 0, 0
, 2, 2, 0, 1, 0, 1, 2, 0, 0, 1, 2, 0, 2, 0, 2, 0, 2, 3, 0, 2, 2, 2, 0, 2, 0, 1
], [4, 4, 4, 2, 4, 2, 2, 2, 3, 2, 2, 3, 2, 3, 3, 2, 3, 4, 3, 3, 3, 4, 2, 2, 4,
2, 2, 4, 4, 4, 4, 4, 4, 8, 4, 4, 2, 2, 2, 2, 2, 1, 1, 2, 0, 2, 0, 2, 2, 2, 1,
1, 1, 2, 0, 1, 2, 1, 1, 1, 2, 2, 0, 2, 0, 2, 2, 0, 0, 2, 2, 0, 2, 0, 2, 0, 2,
3, 0, 4, 2, 2, 0, 2, 0, 2], [0, 2, 2, 2, 2, 0, 0, 0, 4, 0, 0, 3, 0, 3, 3, 0, 4
, 3, 4, 3, 4, 0, 0, 0, 4, 0, 0, 4, 0, 4, 4, 0, 4, 4, 8, 4, 2, 0, 0, 0, 0, 0, 0
, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 2, 0, 1, 0, 1, 2, 0, 0
, 1, 2, 0, 2, 0, 2, 0, 2, 3, 0, 2, 2, 2, 0, 2, 0, 1], [4, 4, 2, 4, 4, 2, 2, 2,
3, 2, 2, 3, 2, 4, 3, 2, 3, 3, 3, 4, 3, 4, 2, 2, 4, 2, 2, 4, 2, 4, 4, 2, 4, 4,
4, 8, 2, 2, 2, 2, 2, 1, 1, 2, 0, 2, 0, 2, 2, 2, 1, 2, 1, 1, 0, 1, 2, 1, 1, 1,
2, 2, 0, 2, 0, 2, 2, 0, 0, 1, 4, 0, 2, 0, 2, 0, 2, 4, 0, 2, 2, 2, 0, 4, 0, 2],
[0, 0, 0, 0, 0, 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 1, 1, 1, 2, 1,
1, 2, 1, 2, 2, 1, 2, 2, 2, 2, 8, 0, 0, 2, 0, 2, 2, 0, 2, 2, 2, 2, 4, 0, 2, 2,
2, 2, 2, 2, 4, 2, 2, 2, 4, 0, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4, 2, 4, 0,
4, 4, 4, 4, 4, 4, 4, 1], [8, 4, 4, 4, 4, 4, 4, 4, 0, 4, 4, 2, 4, 2, 2, 4, 0, 2
, 0, 2, 0, 4, 4, 4, 0, 4, 4, 2, 4, 2, 2, 4, 0, 2, 0, 2, 0, 8, 4, 4, 4, 4, 2, 4
, 2, 2, 0, 2, 1, 4, 4, 2, 2, 2, 0, 2, 1, 2, 2, 2, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0
, 1, 0, 1, 0, 1, 0, 0, 2, 0, 2, 0, 1, 1, 2, 0, -1], [8, 4, 4, 4, 4, 4, 4, 4, 0
, 4, 4, 2, 4, 2, 2, 4, 0, 2, 0, 2, 0, 4, 4, 4, 0, 4, 4, 2, 4, 2, 2, 4, 0, 2, 0
, 2, 0, 4, 8, 4, 4, 2, 2, 4, 0, 2, 0, 2, 1, 4, 2, 2, 2, 2, 0, 2, 0, 2, 2, 2, 0
, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 4, 1, 2, 0, 1, 0, 2, 0, -1],
[4, 2, 2, 2, 2, 2, 4, 4, 3, 2, 2, 1, 2, 2, 2, 2, 0, 1, 0, 1, 0, 4, 2, 2, 1, 2,
2, 2, 2, 1, 1, 2, 0, 2, 0, 2, 2, 4, 4, 8, 2, 4, 4, 2, 4, 4, 4, 4, 2, 2, 2, 1,
1, 1, 0, 1, 2, 2, 1, 1, 2, 0, 1, 1, 1, 1, 4, 1, 1, 2, 2, 1, 2, 0, 2, 0, 2, 2,
2, 4, 2, 2, 2, 4, 2, -1], [8, 4, 4, 4, 4, 4, 4, 4, 0, 4, 4, 2, 4, 2, 2, 4, 0,
2, 0, 2, 0, 4, 4, 4, 0, 4, 4, 2, 4, 2, 2, 4, 0, 2, 0, 2, 0, 4, 4, 2, 8, 4, 4,
4, 0, 4, 0, 2, 1, 4, 2, 2, 2, 2, 0, 2, 0, 2, 2, 2, 0, 0, 0, 0, 0, 0, 1, 0, 0,
0, 1, 2, 2, 0, 1, 0, 0, 2, 1, 1, 0, 1, 0, 1, 0, 1], [4, 2, 2, 2, 2, 4, 2, 4, 3
, 2, 2, 2, 2, 1, 2, 2, 0, 2, 0, 2, 0, 2, 4, 2, 1, 2, 2, 1, 2, 2, 1, 2, 0, 1, 0
, 1, 2, 4, 2, 4, 4, 8, 4, 2, 4, 4, 4, 4, 2, 2, 2, 1, 1, 1, 0, 1, 2, 1, 1, 1, 2
, 0, 2, 1, 1, 1, 2, 0, 0, 0, 2, 2, 2, 1, 2, 0, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1],
[4, 2, 2, 2, 2, 4, 4, 4, 3, 2, 2, 2, 2, 2, 2, 2, 0, 2, 0, 2, 0, 2, 2, 4, 1, 2,
2, 1, 2, 1, 2, 2, 0, 1, 0, 1, 2, 2, 2, 4, 4, 4, 8, 2, 4, 4, 4, 4, 2, 2, 1, 1,
2, 1, 0, 1, 2, 2, 1, 1, 2, 0, 1, 1, 2, 1, 2, 1, 1, 1, 2, 2, 2, 1, 2, 0, 2, 1,
2, 2, 2, 2, 2, 2, 2, 1], [8, 4, 4, 4, 4, 4, 4, 4, 0, 4, 4, 2, 4, 2, 2, 4, 0, 2
, 0, 2, 0, 4, 4, 4, 0, 4, 4, 2, 4, 2, 2, 4, 0, 2, 0, 2, 0, 4, 4, 2, 4, 2, 2, 8
, 2, 4, 0, 2, 1, 4, 2, 2, 2, 4, 0, 4, 1, 2, 2, 2, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0
, 1, 0, 2, 0, 1, 0, 0, 2, 0, 1, 0, 2, 1, 1, 0, 1], [0, 0, 0, 0, 0, 2, 2, 2, 4,
0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 2, 2, 0, 4, 0, 4, 4, 2, 8, 4, 4, 4, 2, 0, 1, 1, 1, 1, 0, 1, 4, 0, 0, 0, 2,
0, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 0, 2, 0, 2, 2, 2, 2, 4, 2, 2, 1], [4
, 2, 2, 2, 2, 4, 4, 2, 3, 2, 2, 2, 2, 2, 1, 2, 0, 2, 0, 2, 0, 4, 2, 2, 1, 2, 2
, 2, 2, 1, 1, 2, 0, 2, 0, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 8, 4, 4, 2, 2, 1, 1, 1
, 2, 0, 2, 2, 1, 1, 2, 2, 0, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 4, 1, 2, 0, 2, 1, 2
, 2, 2, 2, 2, 2, 2, 2], [0, 0, 0, 0, 0, 2, 2, 2, 4, 0, 0, 1, 0, 1, 1, 0, 0, 1,
0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 4, 0, 4, 4, 0,
4, 4, 8, 4, 2, 0, 0, 0, 0, 0, 0, 0, 2, 1, 1, 1, 4, 0, 1, 1, 2, 1, 2, 1, 2, 1,
2, 1, 2, 1, 2, 0, 2, 0, 2, 2, 2, 2, 2, 2, 4, 1], [4, 2, 2, 2, 2, 4, 2, 4, 3, 2
, 2, 2, 2, 1, 2, 2, 0, 2, 0, 2, 0, 4, 2, 2, 1, 2, 2, 2, 2, 1, 1, 2, 0, 2, 0, 2
, 2, 2, 2, 4, 2, 4, 4, 2, 4, 4, 4, 8, 2, 2, 1, 1, 2, 1, 0, 1, 2, 1, 2, 1, 2, 0
, 1, 2, 1, 2, 2, 1, 1, 1, 4, 1, 2, 1, 2, 0, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2], [0,
1, 1, 1, 1, 0, 0, 0, 2, 1, 1, 3, 1, 3, 3, 1, 4, 3, 4, 3, 4, 1, 1, 1, 4, 0, 0,
2, 0, 2, 2, 0, 2, 2, 2, 2, 4, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 2, 8, 0, 2, 2, 2,
2, 2, 2, 4, 2, 2, 2, 4, 1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4,
4, 4, 4, 4, 4, 4, 1], [8, 4, 4, 4, 4, 4, 4, 4, 0, 4, 4, 2, 4, 2, 2, 4, 0, 2, 0
, 2, 0, 4, 4, 4, 0, 4, 4, 2, 4, 2, 2, 4, 0, 2, 0, 2, 0, 4, 4, 2, 4, 2, 2, 4, 0
, 2, 0, 2, 0, 8, 4, 2, 2, 2, 0, 2, 1, 4, 2, 2, 1, 0, 0, 2, 0, 0, 2, 0, 0, 0, 1
, 0, 1, 0, 1, 0, 0, 2, 1, 1, 0, 1, 0, 1, 0, 1], [4, 2, 2, 2, 2, 2, 2, 2, 0, 2,
2, 1, 2, 1, 2, 4, 3, 2, 0, 2, 0, 2, 2, 2, 0, 4, 2, 2, 2, 2, 1, 2, 1, 1, 0, 1,
2, 4, 2, 2, 2, 2, 1, 2, 1, 1, 0, 1, 2, 4, 8, 4, 4, 4, 4, 4, 2, 2, 1, 1, 2, 0,
1, 1, 1, 0, 4, 0, 0, 0, 2, 1, 2, 1, 2, 3, 0, 1, 2, 2, 2, 2, 2, 2, 2, 0], [4, 2
, 2, 2, 2, 2, 2, 2, 0, 4, 2, 2, 2, 2, 2, 2, 3, 1, 0, 2, 0, 2, 2, 2, 0, 2, 2, 1
, 2, 1, 2, 2, 1, 1, 0, 2, 2, 2, 2, 1, 2, 1, 1, 2, 1, 1, 0, 1, 2, 2, 4, 8, 4, 4
, 4, 4, 2, 1, 2, 1, 2, 0, 1, 1, 2, 0, 2, 1, 1, 1, 2, 1, 2, 2, 2, 3, 0, 1, 2, 2
, 2, 2, 2, 2, 2, 1], [4, 2, 2, 2, 2, 2, 2, 2, 0, 4, 2, 2, 2, 2, 1, 4, 3, 2, 0,
1, 0, 2, 2, 2, 0, 4, 2, 2, 2, 2, 1, 2, 1, 1, 0, 1, 2, 2, 2, 1, 2, 1, 2, 2, 1,
1, 0, 2, 2, 2, 4, 4, 8, 4, 4, 4, 2, 1, 2, 1, 2, 0, 1, 1, 1, 0, 2, 1, 1, 1, 2,
0, 2, 1, 4, 3, 0, 1, 2, 2, 2, 2, 2, 2, 2, 0], [4, 2, 2, 2, 2, 2, 2, 2, 0, 2, 2
, 1, 2, 1, 2, 4, 3, 2, 0, 2, 0, 2, 2, 2, 0, 2, 2, 1, 2, 1, 1, 4, 1, 2, 0, 1, 2
, 2, 2, 1, 2, 1, 1, 4, 1, 2, 0, 1, 2, 2, 4, 4, 4, 8, 4, 4, 2, 1, 2, 1, 2, 0, 0
, 0, 0, 0, 2, 0, 0, 0, 2, 1, 2, 2, 2, 3, 0, 1, 2, 2, 2, 4, 2, 2, 2, 2], [0, 0,
0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 1, 1, 2, 4, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 4, 4, 4, 4,
8, 4, 2, 0, 0, 0, 2, 0, 1, 1, 2, 0, 2, 1, 2, 1, 2, 1, 2, 1, 2, 4, 0, 0, 2, 2,
4, 2, 2, 2, 4, 1], [4, 2, 2, 2, 2, 2, 2, 2, 0, 4, 2, 2, 2, 2, 2, 2, 3, 1, 0, 2
, 0, 2, 2, 2, 0, 4, 2, 2, 2, 2, 1, 2, 1, 1, 0, 1, 2, 2, 2, 1, 2, 1, 1, 4, 1, 2
, 0, 1, 2, 2, 4, 4, 4, 4, 4, 8, 2, 1, 1, 2, 2, 0, 1, 1, 1, 0, 2, 1, 2, 1, 2, 1
, 2, 1, 2, 3, 0, 2, 2, 2, 2, 2, 2, 2, 2, 0], [0, 1, 1, 1, 1, 1, 1, 1, 4, 0, 1,
3, 1, 3, 3, 0, 2, 3, 4, 3, 4, 0, 0, 0, 2, 1, 0, 2, 0, 2, 2, 1, 4, 2, 2, 2, 4,
1, 0, 2, 0, 2, 2, 1, 4, 2, 2, 2, 4, 1, 2, 2, 2, 2, 2, 2, 8, 2, 2, 2, 4, 1, 2,
2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4, 0, 4, 2, 4, 4, 4, 4, 4, 4, 4, 2], [4, 2, 2
, 2, 2, 2, 2, 2, 0, 2, 2, 1, 2, 2, 1, 2, 0, 2, 3, 2, 0, 2, 2, 2, 0, 2, 4, 2, 2
, 1, 1, 2, 0, 1, 0, 1, 2, 2, 2, 2, 2, 1, 2, 2, 0, 1, 1, 1, 2, 4, 2, 1, 1, 1, 0
, 1, 2, 8, 4, 4, 2, 0, 1, 1, 1, 1, 4, 1, 1, 1, 2, 0, 2, 1, 2, 0, 2, 1, 2, 2, 0
, 2, 2, 2, 2, 0], [4, 2, 2, 2, 2, 2, 2, 2, 0, 2, 2, 2, 2, 2, 2, 2, 0, 1, 3, 2,
0, 2, 2, 2, 0, 2, 4, 2, 2, 1, 1, 2, 0, 1, 0, 1, 2, 2, 2, 1, 2, 1, 1, 2, 0, 1,
1, 2, 2, 2, 1, 2, 2, 2, 0, 1, 2, 4, 8, 4, 2, 0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1,
2, 2, 2, 0, 2, 1, 2, 2, 0, 2, 2, 2, 2, 0], [4, 2, 2, 2, 2, 2, 2, 2, 0, 2, 2, 2
, 2, 1, 2, 2, 0, 2, 3, 2, 0, 2, 2, 2, 0, 2, 4, 2, 2, 1, 1, 2, 0, 1, 0, 1, 2, 2
, 2, 1, 2, 1, 1, 2, 0, 2, 1, 1, 2, 2, 1, 1, 1, 1, 0, 2, 2, 4, 4, 8, 2, 0, 1, 1
, 1, 1, 2, 0, 0, 0, 2, 1, 2, 1, 2, 0, 2, 2, 2, 2, 0, 2, 2, 2, 2, 0], [0, 1, 1,
1, 1, 1, 1, 1, 4, 1, 1, 3, 1, 3, 3, 1, 4, 3, 2, 3, 4, 0, 0, 0, 2, 0, 1, 2, 1,
2, 2, 0, 2, 2, 2, 2, 4, 0, 0, 2, 0, 2, 2, 0, 2, 2, 4, 2, 4, 1, 2, 2, 2, 2, 2,
2, 4, 2, 2, 2, 8, 1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 4, 4,
4, 4, 4, 4, 2], [0, 4, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 2, 0, 2, 2, 2, 2, 2
, 0, 0, 0, 2, 0, 0, 0, 0, 2, 2, 0, 2, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 8, 2, 4, 0, 2, 1, 0, 0, 0, 1, 0, 1
, 0, 1, 0, 1, 2, 0, 1, 1, 0, 1, 1, 0, 1], [0, 0, 0, 0, 0, 2, 0, 0, 1, 2, 2, 4,
0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2,
0, 1, 0, 2, 1, 0, 2, 1, 1, 1, 2, 0, 1, 1, 1, 0, 1, 1, 2, 1, 1, 1, 2, 2, 8, 4,
4, 4, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 0, 2, 2, 2, 0, 4, 2, 2, -2], [0, 2, 0,
0, 0, 2, 0, 0, 2, 2, 0, 3, 0, 2, 2, 0, 2, 2, 2, 2, 1, 2, 0, 0, 2, 0, 0, 1, 0,
1, 1, 0, 1, 2, 1, 2, 2, 0, 0, 1, 0, 1, 1, 0, 1, 2, 1, 2, 2, 2, 1, 1, 1, 0, 1,
1, 2, 1, 1, 1, 2, 4, 4, 8, 4, 4, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2,
0, 2, 2, 2, 0], [0, 0, 0, 0, 0, 2, 0, 0, 1, 2, 2, 4, 0, 1, 1, 0, 1, 1, 1, 1, 1
, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 1, 2, 0, 1, 1, 2
, 1, 2, 0, 1, 2, 1, 0, 2, 1, 2, 1, 1, 1, 2, 0, 4, 4, 8, 4, 2, 1, 2, 1, 2, 1, 2
, 1, 2, 1, 2, 0, 2, 2, 2, 0, 2, 2, 4, -2], [0, 2, 0, 0, 0, 2, 0, 0, 2, 0, 2, 3
, 0, 1, 2, 0, 1, 2, 2, 2, 2, 2, 0, 0, 2, 0, 0, 1, 0, 1, 1, 0, 1, 2, 1, 2, 2, 0
, 0, 1, 0, 1, 1, 0, 1, 2, 1, 2, 2, 0, 0, 0, 0, 0, 0, 0, 2, 1, 1, 1, 2, 2, 4, 4
, 4, 8, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 4, 1, 2, 2, 2, 0, 2, 2, 2, 0], [2, 1, 2,
2, 2, 1, 2, 2, 3, 1, 1, 2, 2, 3, 3, 2, 3, 3, 3, 3, 3, 2, 1, 1, 2, 2, 2, 4, 1,
2, 2, 1, 2, 2, 2, 2, 4, 2, 2, 4, 1, 2, 2, 1, 2, 2, 2, 2, 4, 2, 4, 2, 2, 2, 2,
2, 4, 4, 2, 2, 4, 1, 2, 2, 2, 2, 8, 2, 2, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 4, 4,
4, 4, 4, 4, 1], [0, 0, 0, 0, 0, 0, 2, 0, 1, 2, 0, 1, 2, 4, 1, 0, 1, 1, 1, 0, 1
, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1
, 1, 2, 0, 0, 1, 1, 0, 1, 1, 2, 1, 1, 0, 2, 0, 1, 1, 1, 1, 2, 8, 4, 4, 2, 2, 2
, 0, 2, 2, 2, 0, 4, 2, 2, 2, 2, 2, 2, 1], [0, 0, 0, 0, 0, 0, 2, 0, 1, 2, 0, 1,
2, 4, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0,
0, 1, 0, 0, 1, 0, 1, 1, 2, 1, 2, 0, 0, 1, 1, 0, 2, 2, 2, 1, 2, 0, 2, 0, 1, 1,
2, 1, 2, 4, 8, 4, 2, 1, 2, 0, 2, 1, 2, 0, 2, 2, 2, 2, 2, 2, 4, 1], [0, 0, 2, 0
, 0, 0, 2, 0, 2, 2, 0, 2, 0, 3, 1, 0, 2, 1, 2, 0, 1, 2, 0, 0, 2, 0, 0, 2, 0, 0
, 1, 0, 1, 2, 1, 1, 2, 0, 0, 2, 0, 0, 1, 0, 1, 2, 1, 1, 2, 0, 0, 1, 1, 0, 1, 1
, 2, 1, 1, 0, 2, 0, 1, 2, 1, 1, 2, 4, 4, 8, 2, 1, 4, 0, 2, 1, 2, 0, 2, 4, 2, 2
, 2, 2, 2, 1], [2, 2, 1, 2, 2, 2, 1, 2, 3, 1, 2, 3, 1, 2, 3, 2, 3, 3, 3, 4, 3,
2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 4, 4, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2,
4, 4, 1, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 4, 1, 2, 2, 2, 4, 4, 2, 2, 2, 8, 2, 4,
2, 4, 2, 4, 2, 4, 4, 4, 4, 4, 4, 4, 3], [0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 1, 2
, 1, 4, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0
, 1, 2, 2, 2, 0, 1, 1, 1, 1, 2, 0, 1, 1, 0, 1, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 1
, 1, 2, 2, 1, 1, 2, 8, 2, 2, 2, 2, 2, 1, 4, 2, 2, 2, 2, 2, 2, 1], [2, 2, 2, 2,
1, 2, 2, 1, 3, 2, 2, 3, 1, 3, 2, 2, 3, 3, 3, 3, 3, 2, 1, 1, 2, 1, 1, 2, 1, 2,
2, 1, 2, 2, 2, 2, 4, 1, 1, 2, 2, 2, 2, 2, 2, 4, 2, 2, 4, 1, 2, 2, 2, 2, 2, 2,
4, 2, 2, 2, 4, 1, 2, 2, 2, 2, 4, 2, 2, 4, 4, 2, 8, 2, 4, 2, 4, 2, 4, 4, 4, 4,
4, 4, 4, 3], [0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 4, 1, 0
, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1
, 2, 0, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 0, 1, 1, 1, 1, 2, 0, 0, 0, 2, 2, 2, 8
, 2, 2, 2, 0, 4, 2, 2, 2, 2, 0, 2, 2], [2, 2, 1, 2, 2, 1, 2, 2, 3, 2, 2, 3, 2,
4, 3, 2, 3, 3, 3, 2, 3, 1, 2, 1, 2, 2, 1, 2, 2, 4, 2, 1, 2, 2, 2, 2, 4, 1, 1,
2, 1, 2, 2, 1, 2, 2, 2, 2, 4, 1, 2, 2, 4, 2, 2, 2, 4, 2, 2, 2, 4, 1, 2, 2, 2,
2, 4, 2, 2, 2, 4, 2, 4, 2, 8, 2, 4, 2, 4, 4, 4, 4, 4, 4, 4, 2], [0, 0, 0, 0, 0
, 0, 0, 0, 0, 2, 0, 1, 0, 1, 1, 2, 4, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 3, 3, 3, 3, 4, 3, 0
, 0, 0, 0, 2, 0, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 8, 2, 1, 4, 2, 2, 2, 2
, 2, 2, 1], [0, 1, 1, 1, 1, 1, 1, 1, 4, 0, 1, 3, 1, 3, 3, 0, 2, 3, 4, 3, 4, 1,
1, 1, 4, 0, 0, 2, 0, 2, 2, 0, 2, 2, 2, 2, 4, 0, 0, 2, 0, 2, 2, 0, 2, 2, 2, 2,
4, 0, 0, 0, 0, 0, 0, 0, 4, 2, 2, 2, 4, 1, 2, 2, 2, 4, 4, 2, 2, 2, 4, 2, 4, 2,
4, 2, 8, 2, 4, 4, 4, 4, 4, 4, 4, 3], [4, 4, 2, 4, 4, 2, 2, 2, 3, 2, 2, 3, 2, 4
, 3, 2, 3, 3, 3, 4, 3, 2, 2, 2, 3, 2, 2, 3, 2, 4, 3, 2, 3, 3, 3, 4, 0, 2, 4, 2
, 2, 1, 1, 2, 0, 1, 0, 1, 2, 2, 1, 1, 1, 1, 0, 2, 2, 1, 1, 2, 2, 2, 0, 1, 0, 1
, 2, 0, 0, 0, 2, 1, 2, 0, 2, 1, 2, 8, 2, 2, 2, 2, 0, 4, 0, 1], [0, 0, 0, 0, 0,
1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 4, 0, 1, 2, 1, 2, 2, 0, 2, 2, 2, 2, 4, 1, 2, 2, 2, 2, 2, 2, 4,
2, 2, 2, 4, 0, 2, 2, 2, 2, 4, 4, 2, 2, 4, 4, 4, 4, 4, 4, 4, 2, 8, 4, 4, 4, 4,
4, 4, 2], [2, 2, 2, 1, 2, 1, 2, 2, 3, 2, 2, 3, 1, 3, 3, 1, 3, 2, 3, 3, 3, 2, 1
, 1, 2, 1, 1, 2, 2, 2, 2, 2, 2, 4, 2, 2, 4, 2, 2, 4, 1, 2, 2, 1, 2, 2, 2, 2, 4
, 1, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 4, 1, 2, 2, 2, 2, 4, 2, 2, 4, 4, 2, 4, 2, 4
, 2, 4, 2, 4, 8, 4, 4, 4, 4, 4, 1], [0, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 3, 1, 3,
3, 1, 4, 3, 2, 3, 4, 1, 1, 1, 4, 0, 0, 2, 0, 2, 2, 0, 2, 2, 2, 2, 4, 0, 0, 2,
0, 2, 2, 0, 2, 2, 2, 2, 4, 0, 2, 2, 2, 2, 4, 2, 4, 0, 0, 0, 4, 1, 2, 2, 2, 2,
4, 2, 2, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 4, 8, 4, 4, 4, 4, 3], [2, 1, 2, 2, 2, 1
, 2, 2, 3, 1, 1, 2, 2, 3, 3, 2, 3, 3, 3, 3, 3, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2
, 2, 2, 2, 2, 4, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 4, 1, 2, 2, 2, 4, 2, 2, 4, 2
, 2, 2, 4, 0, 0, 0, 0, 0, 4, 2, 2, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 4, 4, 8, 4, 4
, 4, 4], [0, 0, 0, 0, 0, 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 1, 0, 2, 0, 2, 2, 1, 4, 2, 2, 2, 4,
0, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 4, 1, 4, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4,
2, 4, 0, 4, 4, 4, 4, 8, 4, 4, 2], [2, 2, 1, 2, 2, 1, 2, 2, 3, 2, 2, 3, 2, 4, 3
, 2, 3, 3, 3, 2, 3, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 4, 4, 2, 2, 4, 1
, 2, 2, 1, 2, 2, 2, 2, 4, 1, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 4, 1, 2, 2, 2, 2, 4
, 2, 2, 2, 4, 2, 4, 0, 4, 2, 4, 4, 4, 4, 4, 4, 4, 8, 4, 1], [0, 0, 0, 0, 0, 1,
1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 4, 0, 0, 2, 0, 2, 2, 0, 2, 2, 4, 2, 4, 0, 2, 2, 2, 2, 4, 2, 4, 2,
2, 2, 4, 0, 2, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 0, 4, 4, 4, 4, 4, 4,
8, 2], [1, 1, 1, 1, 1, 1, 1, 1, 2, -1, -1, -1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1
, 1, 2, -1, -1, -1, 1, 1, 1, 1, 1, 2, 1, 2, 1, -1, -1, -1, 1, 1, 1, 1, 1, 2, 1
, 2, 1, 1, 0, 1, 0, 2, 1, 0, 2, 0, 0, 0, 2, 1, -2, 0, -2, 0, 1, 1, 1, 1, 3, 1,
3, 2, 2, 1, 3, 1, 2, 1, 3, 4, 2, 1, 2, 10]];
-
DET
2^16*3
-
PROPERTIES
INTEGRAL=1
-
MINIMAL_NORM
8
-
KISSING_NUMBER
109421928
-
GROUP_NAME
U_8(2)
-
REMARKS
The normalized center density of this lattice exceeds the
Minkowski-Hlawka bound.
This lattice comes from the algebraic geometry of the
cubic Fermat 6-fold in characteristic 2.
The same construction for the cubic Fermat surface and
the cubic Fermat 4-fold yields the root lattice of type E_6
and the laminated lattice of rank 22.
Thus this lattice can be regarded as a generalization of E_6
and is expected to be a member of an infinite series of non-trivially
dense lattices.
The discriminant group (the quotient of the dual lattice by the lattice)
is isomorphic to $Z/(2)^{\oplus 16}\oplus Z/(3)$.
-
REFERENCES
Ichiro Shimada (shimada@math.sci.hokudai.ac.jp),
"Lattices of algebraic cycles on Fermat varieties in positive characteristics",
Proc. London Math. Soc. (3) 82 (2001), no. 1, 131-172.
Available from
http://www.math.sci.hokudai.ac.jp/~shimada/preprints.html
-
LAST_LINE
Haftungsausschluss/Disclaimer
Gabriele Nebe