The Lattice (4+sqrt(5)) x Leech

An entry from the Catalogue of Lattices, which is a joint project of
Neil J. A. Sloane, AT&T Labs-Research (njas@research.att.com) and
Gabriele Nebe, University of Ulm (nebe@mathematik.uni-ulm.de)

Last modified Wed Nov 29 07:18:46 EST 2000

INDEX FILE | ABBREVIATIONS

Contents of this file

NAME
DIMENSION
DET
GRAM
DIVISORS
MINIMAL_NORM
KISSING_NUMBER
GROUP_ORDER
GROUP_NAME
GROUP_GENERATORS
PROPERTIES
NOTES
LAST_LINE

• NAME
  (4+sqrt(5)) x Leech
  
• DIMENSION
  24
  
• DET
  11^12
  
• GRAM
  24
  12 -1 3 0 6 -2 -1 -2 2 -2 4 3 3 -6 4 5 0 0 -3 5 0 6 1 2
  -1 12 -1 6 -2 -1 0 6 5 -3 1 -1 0 -3 -3 0 -6 0 -6 0 -3 -3 3 3
  3 -1 12 -1 3 3 0 -3 -3 -2 -3 0 6 -5 -2 1 -2 -3 4 6 -3 3 -6 5
  0 6 -1 12 -2 0 0 3 3 -4 5 2 2 -1 -3 0 -6 2 -4 0 -3 -3 3 0
  6 -2 3 -2 12 -3 -1 -2 0 -3 -2 0 4 -6 0 5 2 -2 -3 3 -1 6 1 3
  -2 -1 3 0 -3 12 3 0 -3 3 -2 -2 -3 1 -2 -4 0 0 0 4 6 -6 -3 -1
  -1 0 0 0 -1 3 12 -3 3 -3 -3 -6 -2 1 -6 -1 -1 -6 -2 3 5 -6 -6 6
  -2 6 -3 3 -2 0 -3 12 2 3 2 -1 -3 -1 3 -5 -3 0 -4 -2 -2 -3 6 -3
  2 5 -3 3 0 -3 3 2 12 -4 5 -1 -2 0 -1 0 -4 -3 -5 0 0 -3 3 4
  -2 -3 -2 -4 -3 3 -3 3 -4 12 0 1 -3 5 6 -5 6 3 3 0 3 -2 0 -6
  4 1 -3 5 -2 -2 -3 2 5 0 12 6 0 2 3 1 -2 3 -3 -3 -2 1 6 -3
  3 -1 0 2 0 -2 -6 -1 -1 1 6 12 2 0 3 3 -1 6 3 -3 -5 6 5 -6
  3 0 6 2 4 -3 -2 -3 -2 -3 0 2 12 -3 -3 2 -3 2 3 3 -6 6 0 4
  -6 -3 -5 -1 -6 1 1 -1 0 5 2 0 -3 12 2 -5 3 3 3 -2 4 -6 -1 -5
  4 -3 -2 -3 0 -2 -6 3 -1 6 3 3 -3 2 12 -2 3 3 2 0 1 3 3 -6
  5 0 1 0 5 -4 -1 -5 0 -5 1 3 2 -5 -2 12 1 1 -3 -2 0 6 1 2
  0 -6 -2 -6 2 0 -1 -3 -4 6 -2 -1 -3 3 3 1 12 1 1 -1 5 1 -3 -4
  0 0 -3 2 -2 0 -6 0 -3 3 3 6 2 3 3 1 1 12 2 0 1 3 6 -6
  -3 -6 4 -4 -3 0 -2 -4 -5 3 -3 3 3 3 2 -3 1 2 12 0 -3 3 -4 -2
  5 0 6 0 3 4 3 -2 0 0 -3 -3 3 -2 0 -2 -1 0 0 12 3 -1 -5 5
  0 -3 -3 -3 -1 6 5 -2 0 3 -2 -5 -6 4 1 0 5 1 -3 3 12 -6 -2 -1
  6 -3 3 -3 6 -6 -6 -3 -3 -2 1 6 6 -6 3 6 1 3 3 -1 -6 12 3 0
  1 3 -6 3 1 -3 -6 6 3 0 6 5 0 -1 3 1 -3 6 -4 -5 -2 3 12 -5
  2 3 5 0 3 -1 6 -3 4 -6 -3 -6 4 -5 -6 2 -4 -6 -2 5 -1 0 -5 12
  
• DIVISORS
  11^12
  
• MINIMAL_NORM
  12
  
• KISSING_NUMBER
  37800
  
• GROUP_ORDER
  72576000 = 2^10 * 3^4 * 5^3 * 7
  
• GROUP_NAME
  2.J2 Y SL(2,5)
  
• GROUP_GENERATORS
  #3
  24
  -1 -1 0 -1 0 0 -1 0 1 0 0 1 1 -1 0 -1 -1 1 -2 -1 1 0 -2 0
  2 0 1 1 -1 0 0 -1 -1 1 -1 -2 -2 2 -1 1 1 -1 1 1 -2 0 4 1
  -1 -1 0 -1 -1 0 0 0 0 -1 0 1 1 -1 1 1 1 0 -1 1 -1 -2 0 0
  -1 0 0 1 0 -1 -1 0 0 1 1 0 0 -1 -1 -1 -1 -1 0 0 3 2 -1 0
  0 -1 1 -1 -1 1 1 -1 0 -1 -1 0 0 0 2 2 2 0 0 1 -4 -4 4 1
  -2 1 -1 0 1 -2 -2 0 0 0 2 1 1 -2 -1 -2 -2 -1 0 0 5 2 -4 -2
  0 0 -1 0 0 1 0 0 -1 -1 0 0 0 0 1 1 2 -1 1 1 -2 -2 2 1
  1 1 0 2 1 -1 -2 1 1 2 0 -1 -2 2 -4 -2 -3 -1 1 -1 5 7 -3 -1
  1 0 0 1 0 1 0 -1 0 1 -1 -1 -1 1 0 0 0 0 0 -1 -1 1 1 1
  0 1 0 0 2 -1 -1 1 1 1 1 0 0 0 -2 -2 -3 1 0 -2 4 3 -5 -2
  0 0 0 1 1 0 -1 0 1 1 0 0 0 0 -1 -2 -2 1 -1 -2 3 3 -3 0
  0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 1 -1 -1 1 1 -1 0
  1 -2 1 -1 -1 1 1 0 0 -1 -1 0 0 0 2 2 2 2 -1 0 -5 -5 2 2
  0 0 0 0 1 0 1 0 0 0 0 0 0 -1 1 0 0 2 0 -1 -1 -2 -1 0
  0 0 0 0 1 -1 -1 1 1 1 0 1 0 0 -2 -2 -3 1 -1 -2 4 3 -4 -1
  -1 0 0 -1 -1 1 1 -1 -1 -1 0 0 1 0 2 2 2 -1 0 2 -3 -3 4 1
  -2 0 0 -1 1 0 1 0 1 0 1 0 1 -1 1 0 -1 1 -1 0 0 0 -2 -2
  0 0 1 0 0 -1 0 0 0 0 0 0 0 0 0 -1 -1 1 -1 -1 1 0 -1 0
  0 -1 0 -1 0 0 1 1 0 -1 0 1 1 -1 1 0 0 2 -1 -1 -1 -2 -2 0
  -1 -1 0 -1 0 -1 -1 0 1 0 0 1 1 -2 0 -1 -1 1 -2 -1 1 -1 -3 -1
  -2 1 -1 0 1 -1 -1 0 0 0 1 1 1 -1 0 -1 -1 -1 0 0 3 1 -2 -1
  1 -1 1 -1 -1 1 1 0 0 -1 -1 0 0 1 1 1 1 1 -1 0 -3 -2 2 1
  2 1 1 2 0 0 -1 0 0 1 -1 -1 -2 3 -2 -1 -1 -1 1 -1 2 4 1 1
  1 -1 0 -1 -1 1 1 -1 -1 -1 -1 0 0 0 2 2 3 0 0 1 -5 -5 4 2
  
  24
  0 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 0 1 0 1 0 0 0 1 1
  1 0 1 0 0 0 0 0 0 1 0 -1 -1 1 -1 0 0 0 0 0 -1 0 1 0
  0 0 0 -1 0 0 0 0 0 -1 0 1 1 -1 1 0 1 1 0 -1 0 -3 0 1
  -1 0 0 0 1 -1 -1 1 1 1 1 0 0 -1 -2 -2 -2 1 -1 -1 3 3 -4 -1
  0 0 0 0 -1 0 1 0 -1 -1 0 0 0 0 1 2 3 -1 2 2 -2 -3 4 1
  -2 1 -2 -1 1 -1 -2 1 1 0 2 1 2 -2 -1 -2 -3 0 -1 -1 5 3 -6 -2
  0 0 -1 0 1 0 -1 2 1 0 0 1 1 0 -1 -2 -2 1 -1 -2 3 3 -5 -1
  0 -1 0 0 0 0 -1 0 1 1 0 -1 0 0 -1 0 -1 0 -1 0 0 1 -1 -1
  -1 -1 0 -1 1 0 1 0 0 0 1 0 0 -1 1 0 0 2 -1 0 -2 -2 -1 0
  0 0 -1 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 -1 0 0 1 1 -1 0
  -2 -1 0 0 1 -1 0 0 1 1 1 0 0 -2 0 -1 -2 2 -2 -1 1 1 -4 -1
  0 0 1 1 -1 0 1 -1 -1 0 -1 0 -1 1 1 1 2 -1 0 1 -2 -1 4 2
  1 0 1 0 -1 1 1 0 -1 -1 -1 0 -1 1 1 2 3 0 1 1 -3 -3 4 2
  -2 0 -1 0 1 -1 -1 1 1 1 1 1 1 -1 -1 -2 -4 0 -2 -1 5 5 -6 -2
  0 0 0 1 0 -1 -1 -1 0 1 0 0 -1 0 -1 -1 -1 -1 0 0 2 2 0 0
  0 0 1 0 -1 0 2 -1 -1 -1 0 0 0 -1 2 1 3 0 1 1 -3 -4 4 1
  -1 0 -1 0 0 0 0 0 1 0 0 0 1 -1 1 0 -1 0 0 0 1 1 -2 -1
  0 0 1 0 -1 0 1 -1 -1 0 0 -1 -1 0 1 1 1 0 0 1 -2 -1 3 1
  2 0 1 0 -1 1 1 -1 -1 -1 -2 0 -1 2 1 1 2 0 0 0 -3 -2 4 2
  0 1 -1 0 1 0 -1 1 0 0 1 0 0 0 -1 -1 -1 0 1 -1 3 2 -2 0
  -2 0 -2 -1 1 -1 -1 1 1 0 2 1 2 -3 0 -2 -3 1 -1 -1 4 2 -6 -2
  2 0 2 1 -2 1 2 -2 -2 -1 -2 -1 -2 2 2 3 5 -1 2 2 -6 -5 9 3
  0 -1 1 0 -1 0 1 -1 -1 0 0 -1 -1 0 1 2 2 0 0 2 -4 -3 4 1
  1 0 0 -1 0 1 1 0 -1 -1 0 0 0 0 1 1 2 1 1 0 -3 -4 2 1
  
  24
  1 0 0 1 0 0 0 0 0 1 0 -1 -1 1 -1 -1 -1 0 0 -1 1 3 -1 0
  0 0 1 1 0 0 0 0 0 0 0 -1 -1 1 0 0 0 0 0 0 0 1 1 0
  2 0 1 1 0 0 0 0 0 1 -1 -1 -1 2 -2 -1 0 0 0 -1 0 2 1 0
  -1 -1 1 0 0 -1 0 0 1 1 0 0 0 -1 0 -1 -2 1 -2 -1 1 1 -2 -1
  0 0 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 1 0 -1
  2 0 1 1 -1 1 0 0 -1 0 -2 0 -1 2 0 1 2 -1 1 0 -2 -1 4 2
  2 -1 0 1 0 1 0 1 0 0 -2 0 -1 2 0 1 1 0 1 0 -2 -1 2 2
  0 1 0 1 0 0 -1 1 1 0 0 0 0 1 -1 -1 -1 -1 0 -1 3 4 -2 -1
  0 1 0 1 0 0 0 0 -1 0 0 0 -1 2 0 0 0 -2 1 1 1 2 2 1
  -1 0 -1 -1 0 0 -1 1 0 -1 1 1 1 -1 0 0 0 0 0 0 1 0 -2 0
  0 0 1 0 -1 0 1 -1 -1 0 0 0 -1 1 1 1 1 -1 0 1 -2 -1 3 1
  0 0 1 -1 0 -1 1 -1 -1 0 1 0 -1 -1 0 0 0 1 0 0 -1 -2 1 0
  1 -1 1 -1 0 0 1 0 0 0 0 -1 0 0 0 0 0 2 -1 -1 -2 -2 0 0
  0 0 0 -1 0 0 0 0 -1 -1 0 1 0 0 1 1 1 0 0 1 -2 -3 1 1
  0 1 -1 0 0 0 -1 0 0 0 1 0 0 0 -1 -1 -1 -1 0 0 2 3 -2 -1
  -1 -1 0 0 0 0 1 -1 0 0 0 0 0 -1 1 0 0 1 -1 0 -1 -1 0 0
  -2 0 -1 -1 0 0 0 0 0 -1 1 1 1 -2 1 0 1 0 0 1 0 -1 -1 0
  -1 0 0 -1 0 0 0 0 0 0 1 0 0 -1 0 0 -1 1 -1 0 0 0 -2 -1
  1 0 0 -1 0 0 0 0 0 0 0 0 0 0 -1 0 0 1 0 0 -1 -1 0 0
  2 0 0 1 0 0 -1 1 0 1 -1 -1 -1 2 -2 -1 -1 0 0 -1 1 3 -1 0
  0 0 -1 1 0 1 -1 1 0 0 -1 1 0 1 0 0 0 -1 0 0 1 2 -1 1
  0 0 0 -1 0 0 1 -1 0 0 1 -1 0 -1 0 0 0 1 0 0 -1 -1 0 -1
  -1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 -1 -1 0 0 2 2 -1 -1
  1 -1 0 0 0 1 1 0 0 0 -1 -1 0 1 1 1 1 1 0 0 -3 -2 2 1
  
  
• PROPERTIES
  INTEGRAL = 1
  MODULAR=11
  
• NOTES
  The space of invariant forms of 2.J2 Y SL(2,5)
  contains the Leech lattice. In this space one can construct a
  sequence of modular lattices of level (n^2+5n+5) and minimum 2(2n+4)
  for n=0,1,2,... of which the density tends to the density of the
  Leech lattice.
  For n=0 one finds the well known extremal 5-modular lattice
  and for n=1, one gets this lattice.
  
• LAST_LINE
  
  

INDEX FILE | ABBREVIATIONS

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