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 13:12:33 CEST 2014

INDEX FILE | ABBREVIATIONS

Contents of this file

NAME
DIMENSION
GRAM
DIVISORS
MINIMAL_NORM
KISSING_NUMBER
SUBGROUP_NAME
PROPERTIES
REFERENCES
NOTES
LAST_LINE

• NAME
  Ne_64
  

• DIMENSION
  64
  

• GRAM
  64 0
  12
  -3 12
  -3 6 12
  -3 -3 3 12
  -3 -3 3 0 12
  3 -6 -6 -3 3 12
  -3 6 0 0 -3 -6 12
  -6 3 0 -3 3 -3 6 12
  3 -6 0 6 3 3 -3 -6 12
  -3 6 3 -3 -3 -6 6 3 -3 12
  6 0 -3 -6 0 3 -3 0 0 -3 12
  6 0 -3 -3 -6 3 -3 -3 -3 -3 3 12
  3 -6 -3 -3 6 6 -3 0 3 -3 3 -3 12
  -6 0 3 0 6 3 -3 3 0 0 0 -3 3 12
  -3 -3 3 3 6 0 0 0 3 0 -6 -6 3 0 12
  -3 -3 3 6 3 -3 -3 0 3 -3 0 -3 0 3 3 12
  -3 6 6 3 -3 -6 3 0 0 6 -3 0 -6 0 0 3 12
  3 -6 -3 3 3 3 -3 -3 6 -3 0 -3 3 0 3 0 -3 12
  6 -3 0 0 -3 0 -3 -3 0 0 0 3 3 -3 0 0 0 0 12
  3 -6 -6 3 0 6 -3 -3 3 -6 0 3 3 0 0 0 -3 6 0 12
  3 -6 -6 -3 0 3 -3 0 0 0 3 0 6 0 0 0 -3 3 3 3 12
  -6 -3 0 3 3 0 -3 3 0 0 -3 -3 0 6 0 3 0 3 -3 3 3 12
  -6 0 0 0 3 -3 0 3 0 3 0 -6 0 3 3 6 3 0 -3 -3 3 3 12
  3 0 -3 0 0 0 0 -3 3 0 3 0 0 -3 -3 0 0 3 -3 3 0 0 0 12
  -3 0 -3 3 0 3 3 0 0 -3 -3 0 0 0 0 -3 -3 0 -3 3 -3 0 -3 0 12
  3 -3 -6 0 0 3 0 0 3 -3 3 0 3 -3 -3 0 -3 3 0 3 3 0 0 6 3 12
  3 -6 -3 3 3 3 -3 -3 6 -6 3 -3 3 0 0 3 -6 6 0 3 0 0 0 3 0 3 12
  -6 0 -3 0 0 3 0 3 -3 0 -3 0 0 3 0 0 0 -3 -3 3 0 3 3 0 3 0 -3 12
  3 0 3 3 -3 -3 -3 -3 0 0 0 3 -3 -3 -3 0 3 0 6 0 0 0 -3 0 -3 0 0 -3 12
  3 -6 -6 0 -3 3 -3 -3 3 -3 3 3 3 0 -3 3 -3 0 3 3 3 0 0 0 0 3 3 0 0 12
  -3 6 6 0 0 -3 3 0 0 3 0 -3 -3 0 0 0 3 -6 -3 -6 -6 -3 0 0 0 -3 0 0 0 -3 12
  6 3 0 0 -6 -3 0 -6 0 0 3 3 -3 -6 -3 0 3 0 3 0 0 -6 0 3 -3 0 0 -3 3 0 0 12
  -5 -3 0 3 2 0 0 2 1 -1 -4 -3 -1 1 5 3 -1 0 -1 -1 -1 3 3 -4 2 -2 2 2 -2 0 1 -5 12
  3 3 0 -3 0 -1 2 1 -2 1 1 1 -1 -2 -1 -2 1 2 1 -1 0 -3 -1 1 -1 2 -1 -2 0 -2 -2 3 -1 12
  0 -3 3 7 1 0 -3 -3 4 -4 -4 2 -2 0 3 3 3 3 2 3 -2 1 -2 -3 2 -2 0 -1 3 0 -2 -1 4 0 14
  2 -3 -3 -1 -2 0 0 -1 -1 0 -1 3 -1 -3 -1 0 0 0 0 2 3 1 -1 1 -2 1 0 -1 2 3 -2 0 2 3 0 12
  -2 6 3 -5 0 -2 1 3 -5 2 2 0 -2 1 -2 -2 0 -4 -2 -6 -4 -2 0 -1 -2 -2 -2 -1 -1 -3 3 0 1 1 -3 0 12
  -5 0 -3 0 -3 -1 2 2 -3 2 -2 1 -3 2 -2 2 0 -5 -3 -1 0 2 4 -1 1 -1 -1 4 -3 3 0 -2 1 -5 -5 0 0 14
  -1 3 0 0 -3 -4 5 2 -1 5 -3 0 -2 -2 -2 -4 1 -2 0 -1 -1 1 -2 1 2 0 -3 0 1 0 1 -1 1 0 0 0 2 0 12
  -1 4 2 -4 0 0 3 4 -5 2 0 1 0 0 0 -2 2 -5 0 -2 -1 -2 -2 -3 0 -2 -4 2 -1 -3 4 -2 -1 0 -1 -1 1 -1 1 12
  0 -1 -2 -3 4 5 -4 -1 2 -4 6 0 1 4 -2 0 -4 1 -5 0 1 0 3 2 0 2 3 1 -3 1 1 -1 0 1 -2 -2 1 1 -3 -1 16
  -4 3 6 0 2 0 -3 1 0 1 0 -2 0 4 3 2 2 -2 0 -4 -3 2 1 -4 -1 -3 -2 0 0 -2 4 -4 2 -5 1 -7 3 -1 0 3 1 16
  -3 6 3 0 -1 -4 5 1 0 4 -1 -3 -3 -1 0 -1 3 -2 -4 -5 -5 -2 1 3 1 1 -1 -1 -1 -3 5 0 -2 0 -5 -2 2 1 0 0 0 4 12
  -3 1 5 1 3 1 -2 1 1 1 0 -3 2 4 4 1 2 0 0 -1 -1 2 0 -4 0 -4 -3 0 0 -2 2 -3 2 -4 2 -5 1 -3 1 3 -1 9 -1 14
  -5 -2 -1 3 -1 1 1 1 2 2 -3 -3 -1 3 1 1 0 0 -2 -1 0 3 2 -2 2 -1 1 3 -2 1 2 -4 5 -2 1 -1 -2 4 1 -2 0 1 1 1 12
  -5 -3 0 4 3 -2 1 1 4 0 -3 -5 1 3 3 5 1 1 -3 0 0 4 5 1 -1 0 3 0 -2 2 1 -3 5 -5 0 2 0 4 1 -3 0 1 2 0 3 14
  -1 0 0 -2 -1 2 -4 -3 -3 0 0 1 1 3 0 0 -1 0 0 2 1 2 1 -2 0 -4 -1 2 -1 1 -1 1 1 0 -2 1 0 1 -2 0 1 2 -1 3 0 -1 12
  1 -3 -3 0 3 0 3 3 3 -1 1 -3 3 -2 2 -1 -2 4 -1 0 2 -1 1 2 0 3 3 -2 -2 -1 -2 -2 0 2 0 0 -3 -2 -2 -2 1 -3 3 -4 1 2 -5 12
  -5 -1 -2 1 3 1 1 4 0 -2 0 -2 1 5 -2 2 -1 0 -4 1 -1 4 2 1 2 2 1 3 -2 2 -1 -4 2 1 -1 2 2 3 -1 -3 3 -3 1 -2 2 5 0 2 12
  -4 -1 1 -1 6 3 -3 0 -1 -3 -1 -2 4 4 2 0 -4 -3 -2 0 -1 1 2 -1 3 -1 1 3 -1 -1 2 -4 0 -4 -2 -2 1 5 -2 -1 4 2 1 0 1 3 2 0 3 16
  3 -4 -5 2 0 1 1 -2 3 -2 -1 -2 3 -3 1 1 -2 5 1 4 3 0 1 4 1 5 4 0 -2 2 -5 3 -1 1 -1 1 -3 0 0 -2 -3 -7 -3 -4 -3 1 -2 2 0 -4 14
  -5 6 6 -3 4 -1 -1 1 -2 4 -1 -3 0 5 1 0 4 -3 -3 -5 -4 0 3 1 -1 -3 -3 3 -1 -4 5 -2 -1 0 -2 -2 5 -1 -1 3 0 5 5 3 -1 2 2 -2 1 3 -4 16
  5 -6 -6 -2 0 7 -2 -1 4 -3 2 3 3 -2 -1 -4 -4 3 1 3 5 0 -4 -1 1 4 2 -2 -1 3 -4 -2 -2 -1 1 1 -3 -1 0 1 2 -2 -4 -1 0 -3 -5 3 -2 -2 2 -6 16
  1 -1 -2 2 -2 1 -2 -3 3 0 -1 2 -2 0 -1 -1 0 3 1 2 1 1 0 2 1 2 0 1 1 1 -3 1 -1 1 0 -3 -4 2 1 -4 1 0 1 -1 1 0 0 0 0 -1 1 -1 0 12
  -1 -2 -1 1 2 -1 2 4 2 2 0 -3 0 1 -1 0 -1 2 -2 -1 2 4 1 1 0 3 2 -1 0 0 -1 -4 0 0 -3 1 -2 2 1 -1 1 -2 1 -1 1 1 -4 3 2 -2 2 -1 4 1 12
  -3 4 5 2 -1 -1 2 1 -1 3 -2 -2 -1 3 2 0 5 -1 1 -2 -2 -1 -1 -5 1 -3 -4 0 2 -3 2 -1 -1 1 0 -3 1 0 -2 2 -2 4 3 5 2 -3 1 -1 0 -1 -3 3 -2 0 1 14
  -3 4 -1 -5 0 -1 2 3 -3 3 2 -3 -1 -1 1 -3 0 -1 -4 -4 1 0 4 1 -1 0 -2 1 -3 -3 3 0 3 1 -4 -2 5 0 2 0 3 3 2 2 1 1 0 1 0 0 -1 2 -2 0 0 0 14
  -1 -2 2 5 0 -4 -1 -1 0 -1 -3 -2 -2 0 1 4 -1 2 1 1 0 3 1 -1 -1 -1 2 -1 3 1 -2 1 0 -2 3 0 -3 2 0 -1 -1 -1 -3 0 0 1 0 -2 -2 0 2 -3 -2 0 1 -2 -4 12
  3 -6 0 6 3 0 0 -3 9 -3 0 -4 5 -1 5 3 -2 5 3 2 2 -2 0 0 1 1 6 -4 1 3 0 0 2 -1 3 -1 -7 -5 -1 -5 0 -1 1 0 1 4 -3 6 -1 -1 1 -3 1 3 2 0 -3 -1 18
  -2 3 6 -1 1 -4 -1 3 -5 1 1 1 0 2 1 4 4 -5 1 -4 1 1 1 -3 -3 -3 -6 -1 2 -2 2 -1 0 -2 3 -1 2 -2 -1 6 -2 5 -1 5 -2 -1 0 -2 -2 -2 -3 3 -4 -5 -2 3 1 1 -3 16
  -1 0 3 -1 6 2 -4 1 -1 -1 2 0 1 4 1 0 -1 1 -1 0 -1 2 -1 1 0 -2 -1 0 2 -3 0 -3 -1 -1 1 -1 1 -4 -2 1 1 3 -1 5 -3 -1 1 -1 0 2 -2 4 -3 0 -1 0 -1 1 -1 4 14
  0 5 1 -2 -5 -5 4 0 -4 6 -1 0 -5 -3 -2 -3 3 -1 -1 -4 0 -2 1 0 -2 -3 -3 -1 0 -3 1 4 -1 3 -2 1 3 2 2 1 -2 -3 1 -2 0 -3 1 -2 -2 -5 1 2 -3 0 1 2 3 1 -5 1 -3 12
  4 4 2 -5 -2 2 -1 -3 -2 2 2 2 0 0 0 -5 1 0 4 -1 -1 -4 -3 -2 -3 -5 -1 -3 1 -3 1 4 -1 1 -1 -1 4 -4 0 2 -1 3 -1 2 -2 -2 3 -3 -5 -2 -2 3 -1 -1 -5 1 3 -2 -2 -1 1 2 16
  -2 -3 0 -1 6 2 -4 0 3 1 -1 -4 3 3 2 -1 -2 4 -1 -1 1 4 3 2 0 2 1 1 0 0 -2 -4 2 1 -1 -1 -1 -2 1 -3 3 1 0 2 1 2 1 2 1 2 1 5 0 3 3 -2 3 0 2 -2 4 -2 -2 14
  

• DIVISORS
  3^32
  

• MINIMAL_NORM
  12
  

• KISSING_NUMBER
  138458880
  

• SUBGROUP_NAME
  (SL_2(17) otimes SL_2(5))
  

• PROPERTIES
  INTEGRAL=1
  MODULAR=3
  Extremal 3-modular lattice.
  

• REFERENCES
  G. Nebe, Some cyclo-quaternionic lattices, J. Alg. 199 (1998), 472-498.
  For minimal norm see
  G. Nebe, http://samuel.math.rwth-aachen.de/~LBFM/gabi/paper.html
  G. Nebe, Some cyclo-quaternionic lattices, J. Alg. 199 (1998), 472-498
  (postscript).
  For minimal norm see
  G. Nebe,
  Construction and investigation of lattices with matrix groups
  in Myung-Hwan Kim, John S. Hsia, Y. Kitaoka, R. Schulze-Pillot (Ed.),
  Integral Quadratic Forms and Lattices,
  Contemporary Mathematics 249 (1999) p. 205-220
  (postscript)
  

• NOTES
  Densest sphere packing known in 64 dims.
  Highest kissing number known for any lattice in 64 dims.
  CycloQuaternionic lattice of type Delta2 otimes Delta2
  Official name is ^(p3) L8,2 otimes L_32,2
  The first 5 generators of L8,2 otimes L32,2 generate the subgroup of the automorphism group.
  

• LAST_LINE
  
  

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  Haftungsausschluss/Disclaimer
  
  
  Gabriele Nebe