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 13:14:34 CEST 2014

INDEX FILE | ABBREVIATIONS

Contents of this file

NAME (required)
DIMENSION (required)
GRAM (floating point or integer Gram matrix)
DIVISORS (elementary divisors)
MINIMAL_NORM
KISSING_NUMBER
GROUP_ORDER
GROUP_GENERATORS
PROPERTIES
URL
LAST_LINE (required)

• NAME (required)
  SL2(19):2,2mod
  

• DIMENSION (required)
  30
  

• GRAM (floating point or integer Gram matrix)
  36 0
  6
  2 6
  0 1 6
  -3 0 1 6
  1 2 0 1 6
  1 2 0 0 2 6
  0 -1 1 1 -1 -1 6
  1 1 1 -1 -1 -1 0 6
  1 0 1 1 1 0 0 3 6
  -1 -1 2 1 0 -2 1 2 3 6
  0 -1 0 0 -3 -3 0 2 1 1 6
  0 1 3 0 -1 0 0 1 1 1 1 6
  0 2 -1 -1 0 2 -2 0 -1 -2 0 -1 6
  0 0 0 1 -1 1 1 0 0 -2 0 0 1 6
  1 1 0 0 0 1 0 -1 -1 -2 0 0 0 2 6
  -1 1 -2 0 1 2 -2 -1 -2 -2 -1 -1 3 1 1 6
  -1 0 1 -1 -2 -2 0 0 -2 -1 0 1 0 1 2 0 6
  1 1 0 0 -1 0 2 -2 -2 -1 0 -1 0 1 1 0 0 6
  3 2 0 0 3 2 0 0 1 -1 -1 -1 1 0 0 0 -2 1 6
  1 -1 -2 -1 0 0 0 -2 -2 -2 -1 -2 -1 -1 1 1 0 2 0 6
  -2 0 0 1 -1 1 0 1 2 2 1 0 0 1 -1 0 -1 0 -2 -2 6
  0 1 0 -1 1 2 0 -2 -3 -2 -1 1 2 0 0 2 1 0 0 1 -1 6
  1 -1 -2 -1 0 0 0 -1 -1 -1 0 -1 0 1 -2 -1 -2 1 0 1 0 1 6
  1 1 1 -1 0 1 2 1 1 0 0 2 0 2 1 -1 0 2 1 -1 1 1 0 6
  1 -2 -1 -2 -2 0 0 0 -1 -2 1 1 0 1 1 0 1 0 0 2 -1 1 1 1 6
  1 2 -1 -1 0 0 2 0 -1 -2 0 0 2 0 0 1 1 1 1 0 -1 2 -1 1 -1 6
  0 3 0 -1 0 2 -2 1 0 -1 0 0 2 0 2 2 1 0 -1 -1 3 0 -2 0 -1 1 6
  2 3 1 0 2 2 1 0 1 0 -3 -1 1 2 0 0 0 2 2 -1 1 0 0 2 -2 1 1 6
  -1 -2 -1 1 -1 -2 2 -1 -1 0 0 -1 -1 -1 0 0 1 0 0 1 -2 -1 -2 -2 0 2 -2 -2 6
  0 -1 1 -1 -1 1 -2 1 2 1 0 1 1 1 -2 -1 -1 -2 -1 -2 2 -1 1 0 1 -3 0 1 -3 6
  1 1 1 0 0 1 -1 2 2 1 0 -1 1 2 1 1 0 0 0 -1 2 -2 -1 0 0 -2 2 3 -3 3 6
  -1 1 0 1 -1 1 -1 1 1 -1 2 2 0 1 2 1 0 0 0 -1 2 -1 -1 1 1 0 3 -2 -1 -1 0 6
  -3 -1 -1 2 -2 -2 1 0 -1 0 1 -1 0 1 -1 1 1 1 -2 0 1 -1 0 -1 -2 2 0 -1 2 -2 -1 1 6
  -1 -2 0 -1 -3 -2 -1 -1 -2 0 1 0 -1 -1 1 0 3 0 -3 1 0 -1 -1 -3 2 -2 1 -2 1 1 1 0 0 6
  3 1 0 -2 2 1 0 1 1 1 -1 0 0 -1 -2 -1 -2 0 2 1 -1 2 2 2 0 0 -2 2 -2 1 0 -3 -3 -3 6
  -2 -1 2 1 -3 -2 0 3 2 3 2 2 -2 0 -1 -2 1 -1 -3 -2 3 -3 -1 0 0 -2 1 -1 -1 2 2 2 2 2 -2 6
  

• DIVISORS (elementary divisors)
  1^18 2^18
  

• MINIMAL_NORM
  6
  

• KISSING_NUMBER
  

• GROUP_ORDER
  2^4 * 3^2 * 5 * 19
  

• GROUP_GENERATORS
  2
  36
  0 1 1 0 0 0 0 0 1 -1 -1 0 1 1 0 0 -1 -1 0 0 1 -1 0 1 0 0 -1 -1 -1 -2 -1 -1 1 3 1 0
  0 0 0 0 0 0 1 -1 1 -1 1 0 0 0 0 2 1 0 1 0 -1 -1 1 0 0 -1 2 0 0 0 -2 -2 -1 -1 0 3
  0 0 0 -1 0 0 1 0 0 0 0 1 0 -1 1 0 0 0 0 -1 1 -1 -1 -2 2 -2 0 1 -1 0 -1 0 3 -1 1 -2
  1 -1 -1 -1 1 0 0 1 -2 1 1 1 1 0 0 -1 0 0 -1 1 1 0 -2 -2 0 -1 -2 3 -1 0 -1 2 0 -1 -1 -1
  -1 1 1 0 0 1 0 -1 1 1 1 -1 0 1 0 1 0 -1 0 0 -1 -2 1 1 1 1 1 0 -1 -1 -2 -1 0 2 1 0
  0 0 0 -1 0 -1 1 0 0 -1 0 0 1 0 0 0 -1 -1 0 0 0 0 -1 0 0 -2 0 2 0 -1 -1 1 1 1 1 0
  0 -1 0 0 0 0 0 0 -1 0 0 2 1 0 0 -1 -1 -1 0 0 1 -1 -1 -1 1 -1 0 2 -1 -2 0 0 2 1 1 -2
  0 1 -1 0 -2 -1 0 -2 1 -1 -2 1 -1 1 -2 0 1 1 2 0 0 1 1 1 -2 0 2 -6 1 -1 4 -3 -3 -3 -2 2
  1 1 -1 -1 0 -1 1 -1 0 -1 -1 1 1 0 -1 -1 0 0 0 0 1 1 -1 0 -1 -2 -1 -2 0 -1 2 0 0 -1 -1 0
  0 1 0 0 0 1 0 -1 0 1 0 0 -1 0 -1 0 1 0 0 0 0 -1 0 0 1 1 0 -2 -1 0 1 -1 0 -1 -1 -1
  1 0 -1 0 0 0 0 0 0 -1 -1 1 0 0 -1 -1 1 1 0 0 1 1 0 0 -2 0 -1 -3 1 0 3 0 -1 -1 -1 0
  1 -1 -1 -1 0 -1 2 0 0 -1 0 1 0 -1 1 0 0 1 0 -1 0 1 -2 -3 1 -3 1 1 -1 1 -1 0 2 -3 0 0
  -1 1 0 0 -1 0 0 -1 2 -1 0 -1 -1 0 0 2 1 0 2 0 -1 1 3 2 -2 1 1 -2 2 0 1 -2 -3 0 -1 4
  0 -1 -1 -1 0 -1 1 0 0 -1 0 1 1 0 0 0 0 1 0 0 0 1 -1 -2 -1 -2 0 1 0 -1 0 0 -1 -2 0 2
  -1 -1 -1 -1 1 0 2 0 0 -1 2 1 1 0 0 0 0 0 0 1 -2 1 0 -1 -1 -3 2 3 2 1 -1 1 0 -1 1 2
  -1 0 0 0 0 1 0 0 1 0 1 -1 0 1 0 1 0 0 0 0 -1 -1 0 0 0 1 0 1 -1 -1 -2 -1 -2 1 0 2
  -1 -1 0 0 0 0 0 1 0 0 1 0 -1 -1 1 1 0 1 0 0 -1 1 1 -1 0 0 1 2 1 2 -1 0 -1 -2 -1 1
  0 -1 1 0 1 0 0 1 0 -1 0 1 2 0 1 0 -1 -1 0 0 2 -2 -1 -1 1 0 -2 2 -2 -2 -2 0 2 3 2 -1
  0 0 0 -1 0 -1 0 0 1 -1 -1 1 2 1 0 -1 -2 -1 0 0 1 0 -1 0 0 -1 -1 1 -1 -3 -1 0 1 3 1 0
  -1 0 1 0 1 1 -1 1 0 0 1 -1 1 1 1 0 -2 -1 -1 0 0 -1 -1 0 1 1 -2 3 -1 -1 -3 1 1 4 2 0
  1 0 0 0 0 0 0 0 -1 0 0 0 0 -1 -1 0 1 0 -1 0 1 -1 -1 0 0 0 -2 0 -1 0 1 1 0 -1 -1 -2
  -1 0 1 0 0 1 0 0 1 0 1 -1 -1 0 1 2 0 0 0 -1 -1 -1 1 0 1 1 1 1 0 0 -2 -1 0 1 1 1
  0 1 1 1 -1 0 -1 -1 1 0 -1 -1 -1 0 0 1 1 0 1 0 0 0 2 2 -1 2 1 -4 1 0 2 -2 -1 0 0 1
  0 -1 -1 -1 0 -2 2 -1 1 -2 -1 2 1 0 0 0 0 1 1 -1 0 1 -1 -2 0 -3 2 -1 0 -1 1 -1 1 -2 1 1
  0 0 0 0 -1 -1 0 1 1 -1 -2 0 0 0 1 0 -1 1 0 -1 1 1 -1 -1 0 0 -1 -1 -1 0 0 0 1 1 1 -1
  0 -1 0 0 0 0 0 0 0 -1 1 1 1 0 0 0 -1 -1 0 0 0 0 0 0 0 -1 0 3 0 -2 -1 0 0 1 0 1
  0 0 0 0 0 0 1 0 0 -1 1 0 0 -1 0 1 1 0 0 0 0 -1 0 0 0 -1 0 1 0 1 -1 0 0 -1 0 0
  0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 -1 0 0 1 -2 0 0 1 0 -1 1 -2 -2 -2 -1 0 1 0 0
  0 -1 0 0 1 1 -1 1 -1 1 1 0 1 1 0 -1 -2 -1 -1 1 0 0 -1 0 0 1 -2 4 -1 -1 -2 2 0 3 0 0
  1 1 0 0 -1 -1 0 0 0 0 -2 -1 -1 -1 0 0 1 1 0 -1 1 1 0 0 0 0 -1 -3 0 1 2 0 0 -2 -1 -1
  0 1 0 0 -1 0 0 0 0 0 -1 0 0 0 -1 0 1 0 0 0 1 -1 0 0 0 0 -1 -2 -1 -1 1 -1 -1 -1 -1 -1
  1 -1 -2 -1 0 -2 2 0 0 -2 0 2 1 -1 0 -1 0 1 0 0 0 2 -2 -2 -1 -4 1 1 1 1 1 1 1 -3 0 0
  0 -1 -1 0 0 0 -1 0 -1 0 1 1 0 0 -1 -1 0 0 0 1 0 1 0 0 -1 0 0 1 1 -1 2 0 -2 -2 -2 1
  0 0 1 1 0 1 -1 2 -1 1 0 -1 -1 -1 1 0 0 0 -1 0 1 -1 0 0 1 2 -2 1 -1 2 -1 1 1 1 0 -3
  0 1 1 0 0 0 0 -1 1 0 -1 -1 0 1 0 1 0 0 0 -1 0 -1 0 0 1 1 0 -2 -1 -1 -1 -1 0 1 1 1
  1 0 -1 0 -1 -1 0 0 -1 0 -1 1 -1 -1 -1 -1 1 1 0 0 1 1 -1 -1 0 -1 0 -2 0 1 3 0 0 -4 -2 -2
  36
  -1 0 0 0 0 0 0 0 0 0 0 1 0 -1 1 -1 0 0 0 -1 0 -1 0 -1 2 -1 1 1 0 0 0 0 3 0 2 -2
  0 0 -1 -1 0 -1 1 -1 0 -1 -1 2 1 0 0 -1 0 1 0 -1 1 0 -1 -2 1 -2 1 -1 0 -1 1 -1 1 -2 1 0
  -1 1 0 0 -2 -1 0 -1 2 -1 -2 1 -1 0 0 0 0 1 2 -1 0 1 1 0 0 0 2 -4 0 -1 3 -3 0 -1 0 0
  1 -1 -1 -1 0 -1 1 0 -1 -1 0 1 1 0 0 -1 0 1 -1 0 1 1 -3 -3 0 -2 -1 1 -1 0 0 1 0 -3 0 0
  0 0 0 0 0 0 -1 1 -1 1 0 1 0 0 0 -2 -1 0 -1 0 1 0 -1 0 1 0 -1 1 0 0 1 1 2 1 0 -4
  0 0 0 0 0 0 0 -1 0 0 0 1 0 0 -1 -1 0 0 0 0 0 0 0 0 1 -1 2 -1 1 -1 2 -1 1 -1 0 -1
  -1 1 1 0 -1 1 -1 0 1 0 0 -1 -1 0 0 1 1 0 0 0 1 -1 1 0 0 2 -2 -1 -1 -1 0 -1 -2 0 -1 0
  -1 0 0 -1 0 0 1 -1 1 -1 1 1 1 0 0 0 0 -1 1 0 -1 0 1 0 0 -2 2 1 1 -1 0 -1 1 0 1 1
  -1 -1 -1 -1 0 0 0 -1 0 0 2 1 0 0 -1 -1 0 0 0 1 -3 2 1 0 -1 -2 3 2 3 0 2 0 -1 -3 -1 2
  -1 0 0 -1 0 0 0 0 1 0 1 0 0 0 1 0 -1 0 0 0 -1 1 0 -1 0 0 0 2 0 0 -1 0 0 0 0 1
  0 0 -1 0 0 -1 1 -1 1 -1 0 -1 0 0 0 1 1 1 1 0 -2 2 1 0 -2 -1 2 -1 2 2 0 0 -2 -2 0 4
  -1 1 -1 0 -2 -1 0 -2 2 -1 -1 1 -1 1 -2 0 0 0 2 0 -2 2 2 2 -2 -1 4 -4 2 -2 4 -3 -2 -1 -1 2
  0 0 -1 0 0 0 0 -1 0 0 0 0 0 1 -1 0 0 0 1 1 -1 1 1 1 -1 0 2 -1 2 0 1 -1 -2 -1 -1 3
  0 0 0 0 0 1 0 -1 0 -1 1 0 0 0 -1 1 1 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 -1 -1 -1 0 2
  1 0 0 0 0 -1 0 1 -1 0 -1 0 0 -1 0 -1 0 1 -2 -1 1 0 -2 -2 2 -1 -1 1 -1 1 0 2 2 -2 0 -3
  1 0 0 0 1 0 0 0 -1 0 0 0 1 1 -1 -1 -1 -1 -1 0 1 -1 -2 0 1 0 -1 1 -1 -1 -1 1 1 2 1 -1
  0 1 1 1 -1 1 -1 0 0 1 -1 -1 -1 1 -1 1 0 -1 0 0 0 -2 1 2 0 3 -1 -2 -2 -1 0 -1 -1 2 -1 -1
  0 1 1 0 0 0 -1 0 0 0 -1 -1 0 0 1 1 0 0 0 0 2 -1 0 0 0 2 -3 -1 -1 0 -1 0 -1 1 0 1
  0 0 0 0 0 -1 0 0 -1 0 0 1 0 -1 1 -1 0 0 0 0 1 0 0 -1 1 -1 0 1 1 1 1 1 2 -1 1 -2
  1 0 1 1 1 0 -1 2 -1 1 -1 -1 0 0 1 0 0 0 -1 0 2 -1 -1 0 1 2 -3 0 -2 1 -2 1 1 2 0 -2
  -1 0 0 0 0 1 0 -2 1 0 2 -1 -1 0 -1 2 1 0 1 1 -3 1 3 2 -2 1 3 -1 3 1 1 -2 -4 -2 -2 5
  0 1 0 1 -1 0 -1 0 1 0 -2 0 -1 1 -1 0 0 0 1 0 1 0 1 2 -1 2 0 -4 0 -1 2 -2 -1 2 -1 -1
  -1 0 0 0 1 1 0 0 1 -1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 -1 0 0 0 1 0 -1 -1 0 2 1 2
  -1 1 0 0 -1 0 -1 -1 1 0 0 0 -1 0 -1 1 1 0 1 1 -1 1 3 2 -2 1 1 -2 3 0 3 -1 -3 -1 -2 2
  -1 2 1 2 -1 1 -1 -1 1 1 0 -2 -3 0 -1 2 2 -1 2 1 -2 0 5 5 -2 3 3 -5 3 2 3 -2 -2 1 -1 1
  0 0 0 0 -1 0 -1 0 0 0 -1 0 0 1 -1 0 0 0 0 0 1 -1 0 0 0 1 -1 -1 -1 -2 0 -1 -2 0 -1 0
  0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 -1 0 1 1 0 0 2 -1 1 0 1 -1 -1 -1 -1 1
  -1 0 0 -1 0 1 0 -1 0 0 1 1 0 0 0 0 0 0 0 0 0 -1 0 -1 1 0 0 1 0 -1 0 -1 0 -1 0 1
  1 0 1 1 -1 0 -1 1 -1 1 -1 -1 0 0 0 0 0 -1 -1 0 2 -2 -1 0 1 2 -3 0 -3 -1 -1 1 0 1 0 -3
  -1 0 -1 0 0 1 1 -2 1 0 2 0 -1 0 -1 1 1 0 2 1 -4 2 3 2 -2 -1 5 -1 4 1 2 -2 -2 -2 -1 5
  -1 0 0 -1 1 1 1 -1 0 0 2 0 0 0 0 0 0 0 0 0 -2 0 0 -1 1 -1 2 2 1 1 -1 0 1 -1 1 2
  0 0 0 0 0 -1 0 -1 0 -1 0 0 0 0 -1 0 0 0 0 0 -1 1 1 1 -1 -1 2 -1 2 0 2 0 -1 -1 0 1
  1 -1 0 -1 1 0 0 1 -1 -1 0 0 2 1 0 0 -1 0 -1 0 2 -1 -3 -2 0 0 -4 2 -3 -2 -3 1 -1 1 0 1
  0 1 1 1 0 1 0 0 0 1 0 -2 -1 0 0 1 0 -1 0 0 -1 -1 1 2 0 2 0 -1 -1 1 -1 0 0 2 0 0
  -1 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 1 0 0 -1 1 1 1 0
  -1 0 0 -1 0 0 1 -1 1 -1 1 0 0 0 0 1 0 0 1 0 -2 1 1 0 -1 -1 2 0 1 0 0 -1 -1 -1 0 3
  

• PROPERTIES
  INTEGRAL = 1
  MODULAR = 2
  

• URL
  Second 2-modular lattice related to SL2(19):
  2-modular lattice
  

• LAST_LINE (required)
  

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  Haftungsausschluss/Disclaimer
  
  
  Gabriele Nebe