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 Sat Oct 17 06:01:59 PM CEST 2026

INDEX FILE | ABBREVIATIONS

Contents of this file

NAME
DIMENSION
DET
GRAM
DIVISORS
MINIMAL_NORM
KISSING_NUMBER
HERMITE_NUMBER
SUBGROUP_ORDER
SUBGROUP_NAME
SUBGROUP_GENERATORS
PROPERTIES
REFERENCES
NOTES
LAST_LINE

• NAME
  X139
  

• DIMENSION
  88
  

• DET
  1
  

• GRAM
  88 0
  28
  7 32
  12 -12 42
  5 -4 18 32
  -5 13 4 3 38
  6 0 -10 -1 1 36
  1 -11 -1 -10 0 -5 42
  2 11 -8 -5 -2 -12 12 36
  8 3 -4 0 -6 6 9 14 34
  7 14 -7 -3 3 -3 -10 -6 5 40
  2 -7 0 5 -3 2 15 -2 -2 -14 34
  -13 9 -12 0 8 -13 -9 1 2 19 -18 44
  -10 -17 2 4 -12 -12 -4 -11 -9 -5 6 8 38
  11 12 1 -4 3 -4 6 8 10 7 -2 -6 -6 30
  -1 -5 -11 -1 -1 8 6 -1 -5 -5 4 0 3 -10 26
  6 13 2 -4 15 4 -11 -8 -1 9 1 -2 -4 1 -9 36
  0 11 2 8 5 -3 -2 11 8 -9 2 5 -10 -5 -7 7 34
  1 7 5 3 11 -8 -11 -9 -15 4 7 -3 10 7 -8 12 -9 36
  8 11 -5 -8 -10 9 -10 -1 -6 -6 -2 -17 -2 13 -6 9 3 5 44
  2 13 -4 8 18 1 -5 5 -6 -4 1 5 -6 4 -1 11 13 5 1 36
  -2 12 5 13 24 1 -13 -10 -7 13 -13 12 -1 9 -1 2 0 10 1 11 46
  2 -2 -4 1 4 16 -10 -17 6 6 3 6 5 -3 7 5 -3 4 -4 -5 3 38
  9 0 -2 -13 -11 -10 12 12 -3 0 9 -5 0 -1 1 3 -3 1 4 -2 -25 -11 36
  6 11 -11 -8 -3 4 3 14 9 -7 1 -4 -8 1 7 0 6 0 5 -2 -11 3 4 30
  10 -8 -3 4 -18 4 -4 -5 7 10 -6 2 5 5 0 -7 -14 1 -1 -3 -6 2 2 -5 38
  -8 8 -17 -3 -5 -7 -4 3 4 19 -6 17 2 -3 2 1 -2 -3 -8 -7 -1 -5 2 2 1 34
  12 18 1 -7 -3 -9 -13 -3 -5 10 -4 2 3 9 -12 18 6 19 18 2 3 -4 10 10 3 7 46
  -12 -13 6 -8 9 -5 21 -6 4 -7 4 2 5 -1 -1 1 -4 -1 -11 -8 -2 -2 -3 -5 -7 -3 -11 38
  13 2 11 16 6 9 -7 -10 9 5 -5 -2 2 5 2 6 5 -2 2 7 20 12 -15 1 1 -3 5 -7 40
  13 -1 3 9 5 21 12 -11 15 0 3 -8 -11 7 7 -1 5 -9 4 1 12 13 -13 0 4 -7 -9 10 21 50
  -7 -7 -13 -15 -16 -2 12 4 -6 -7 1 0 5 -5 2 -7 1 -12 6 -2 -19 -13 13 -1 0 8 2 2 -17 -9 36
  10 -11 19 17 4 2 8 -9 -6 -3 13 -8 -3 -4 8 -4 0 -7 -14 4 5 3 -4 -10 -1 -6 -11 -2 7 7 -4 44
  -7 7 0 -9 2 -6 5 -3 -3 6 -9 8 1 6 -9 0 1 -2 10 -6 5 -9 -2 -3 -9 7 7 11 0 2 5 -15 38
  -11 -13 -13 2 -11 9 4 -7 4 -2 -2 3 12 -11 10 -8 -2 -11 2 -6 3 -2 -9 0 7 9 -10 10 7 13 8 -6 5 40
  7 -2 -2 -11 -7 12 -6 -3 10 -2 1 -16 0 9 0 7 -11 3 10 -8 -6 14 0 5 8 -6 4 0 4 1 -6 -4 -10 -2 36
  10 8 9 4 3 -4 -12 6 -3 7 -1 -1 -8 -7 -4 11 5 8 -2 0 -6 -2 12 6 0 -2 9 -11 -5 -13 -9 -2 -8 -13 -3 32
  7 -6 10 -5 6 9 -2 -4 -5 3 -2 -16 -8 3 -5 4 -18 11 2 -7 1 5 1 -2 5 -12 -5 1 -3 -5 -12 4 -8 -11 14 10 42
  -13 1 -11 4 14 -3 13 9 -5 -10 8 9 3 -6 7 -5 10 -5 -10 19 6 -9 -4 0 -15 0 -15 7 -3 2 7 1 2 8 -21 -7 -12 38
  -12 -11 -7 -11 -11 -13 0 -8 -4 14 -12 20 21 -6 7 -4 -13 -7 -4 -14 1 4 1 -9 3 14 -3 11 -1 -7 7 -4 10 15 0 -11 -5 -3 46
  4 -3 11 8 4 4 -4 -10 -11 -9 13 -13 -3 -1 3 2 -6 5 1 1 -1 0 1 -2 -6 -7 -4 -1 -4 -2 -4 18 -11 -11 0 5 7 -2 -12 32
  13 -4 17 16 6 -2 -9 1 -7 2 -7 1 -4 -11 12 -6 2 1 -14 3 11 6 -2 2 6 -10 -1 -16 13 -2 -15 19 -22 -9 -6 19 11 -4 -7 4 52
  4 -12 -2 0 -7 18 5 -17 -3 6 9 -16 2 1 10 -9 -22 -1 -2 -14 3 11 -7 -4 7 1 -10 -1 4 10 2 15 -5 9 11 -9 12 -7 0 10 1 40
  5 -2 -4 7 -5 -1 -8 -7 7 6 11 -1 7 -3 -2 10 -1 9 -9 4 -7 10 1 4 12 5 6 -5 7 2 -11 2 -17 3 8 5 -1 -4 0 4 0 1 40
  4 -6 7 0 -15 7 -9 2 -7 -12 6 -22 5 0 -2 -9 -8 3 13 -8 -9 -4 3 -1 -1 -13 -4 -11 -9 -13 5 0 -6 -6 12 5 11 -8 -11 9 1 11 -8 40
  -2 -8 -5 -2 -17 4 -10 -3 -11 -8 -2 -11 14 -4 10 -12 -8 -4 11 -5 2 -5 -4 -1 3 -2 0 -12 2 -7 9 -1 -4 10 6 -4 -2 -2 9 1 7 11 -3 22 38
  -8 9 -18 -17 12 -8 6 13 8 12 -9 17 1 5 0 3 -4 0 -14 6 6 0 -1 7 -5 10 1 4 -2 -13 1 -12 3 -1 4 -3 3 11 10 -13 -4 -4 4 -11 -3 44
  -13 -18 16 -2 -6 -20 3 -5 0 4 1 0 15 -7 -14 -1 -10 4 -18 -20 -8 -8 2 -14 -1 4 -4 15 -10 -18 -4 -4 8 0 1 1 5 -7 15 -4 -9 1 9 4 -2 5 54
  -1 -4 12 3 -3 -16 3 4 -3 5 -12 2 -2 7 -11 -11 1 0 0 -6 6 -10 -3 -7 2 2 2 -1 -2 -3 3 2 7 -5 -10 -2 4 -3 5 -5 7 -2 -7 -3 -1 0 11 32
  1 3 -3 7 5 -7 -7 0 2 12 -2 6 10 1 -3 12 2 8 -2 8 10 2 -4 -6 5 5 5 -7 11 -2 -5 -4 -7 4 -2 4 -3 4 5 -8 6 -4 15 -7 0 12 5 3 32
  4 -3 4 -9 -4 1 -4 -4 -1 10 -7 -1 5 0 -8 12 -9 7 8 -6 -2 1 8 -2 4 -3 8 2 -2 -8 0 -7 0 1 8 7 17 -12 11 -4 -2 2 -1 3 -2 4 8 1 3 30
  -12 -13 -12 4 3 17 6 -2 -5 -12 0 4 6 -18 17 -11 -1 -10 -9 4 2 6 -9 -2 2 -4 -22 1 1 5 9 3 -6 20 -8 -8 -1 17 1 -8 8 9 -12 1 8 0 -11 -8 2 -3 46
  -3 7 -2 -2 2 4 -6 -10 4 16 1 1 1 11 -9 2 -8 4 1 -10 12 9 -14 -7 -6 8 1 -1 3 -1 -6 1 12 -5 7 -9 1 -7 3 2 -16 17 -2 6 -1 6 11 2 1 2 -11 36
  -14 10 -12 -14 6 -10 -10 2 -14 2 -14 12 0 -1 0 7 2 3 6 9 1 -8 1 1 -8 10 10 -1 -9 -19 11 -15 11 -3 -3 -2 -9 5 11 -7 -8 -14 -10 -6 4 9 -2 2 -3 -2 -3 -2 36
  -5 -9 0 -7 0 -9 2 -9 -11 -6 -4 11 15 -10 9 -3 -4 3 -10 -1 1 5 2 2 1 -2 8 4 2 -2 5 -1 -2 2 -3 -6 -7 2 15 -3 10 -4 6 -12 7 8 5 1 1 -1 4 -11 8 36
  0 -1 9 1 1 -1 0 -1 -9 0 -7 -2 -7 -7 -4 0 7 -3 7 -4 4 -9 0 -4 -4 -3 4 1 -3 -1 4 3 8 2 -9 7 6 -1 -1 -5 10 -2 -15 1 1 -10 0 11 -5 5 4 -5 3 -3 24
  -4 11 6 0 6 -4 -19 -9 0 17 -20 18 -6 5 -14 2 3 -1 -3 -3 17 6 -16 -9 -4 5 7 -3 5 -6 -4 -4 10 -13 -1 -1 -2 -9 7 -1 -1 -3 -8 -2 -2 3 6 10 -2 1 -12 19 11 1 3 42
  2 5 9 -2 18 9 1 -8 8 9 -10 -2 -18 2 -4 10 6 -9 -3 -4 15 8 -12 -5 -15 -1 -6 8 13 18 -8 4 6 -3 3 -3 4 -5 -4 2 0 5 -5 -9 -12 5 5 5 4 0 -4 12 -5 -4 3 15 42
  -14 10 -19 -3 5 -2 -4 -4 1 17 2 11 -2 2 0 -2 0 0 -3 -1 15 0 -9 -6 -5 14 -4 4 -2 -1 -3 -11 10 12 -5 -5 -11 8 8 -3 -14 6 4 -7 1 11 6 -3 6 -7 -4 17 6 -8 -5 11 7 44
  -6 8 -6 -6 11 -3 -1 -4 4 15 -11 12 -9 7 0 0 -8 -2 -6 -2 14 0 -6 -1 -3 10 -2 8 4 4 -10 -6 9 2 1 -7 -1 -1 9 2 -9 1 2 -15 -10 12 7 1 -3 -2 -13 10 8 1 -7 13 15 20 32
  11 3 -6 3 -8 13 -4 -1 -1 8 0 -6 -6 -2 11 0 -9 -3 7 -4 0 -2 7 4 9 9 1 -13 2 5 0 7 -8 5 1 6 6 -7 -1 7 7 12 0 3 5 -10 -14 -6 -4 6 4 -2 -6 -7 1 -9 -4 -3 0 30
  -1 -6 12 -2 -9 -17 9 10 6 2 -3 5 4 0 -9 -9 5 -3 -8 -13 -8 -6 5 1 -1 -1 2 7 -8 -11 3 -2 7 -3 -6 6 0 -4 6 -10 4 -6 -7 2 -4 1 18 14 -4 7 -6 1 -3 0 9 5 -3 -4 -6 -8 30
  -2 -4 0 -9 10 11 8 -12 5 -6 4 -3 2 7 2 5 -9 3 -4 0 4 15 -11 2 -1 -12 -3 16 4 11 -5 1 1 -1 15 -13 3 -1 0 3 -12 9 1 -3 -6 7 1 -12 -7 2 1 10 -1 8 -11 3 7 -1 8 -8 -5 34
  9 -5 13 7 -6 11 6 3 -2 -13 3 -14 -8 2 -6 -11 -2 -5 7 0 -9 -11 3 -1 5 -16 -7 -4 -6 1 4 7 5 -8 -9 3 10 2 -21 9 2 5 -16 19 0 -18 -6 3 -16 0 7 -1 -12 -14 10 -4 -8 -17 -13 6 4 -2 44
  4 16 5 11 6 -1 -22 -6 -9 15 -7 11 -2 -4 -1 6 2 7 2 4 12 1 -2 -3 -6 7 7 -11 6 -6 -9 -2 9 -7 -8 13 -5 -2 -2 5 5 -3 -5 5 -1 -6 -5 -5 1 -2 -6 9 5 -9 3 16 3 10 5 5 -4 -11 2 36
  5 -9 20 -2 5 3 2 -13 -6 2 1 -9 -4 -4 -2 1 -5 2 -5 -12 2 6 2 -7 -5 -12 1 8 1 2 -7 10 3 -8 2 9 13 -10 0 6 13 9 -8 6 0 -9 13 4 -10 5 -3 3 -5 2 10 6 11 -3 0 -2 9 6 7 2 32
  -8 7 -6 -6 10 -6 -9 3 -16 3 -14 15 -4 -9 10 -6 2 2 -6 3 13 0 -5 3 -6 3 1 -6 -8 -14 0 4 -3 -5 -6 6 4 3 6 -3 20 -7 -12 -2 7 11 -9 5 -3 -3 8 -5 16 11 9 8 1 0 2 1 2 -6 -8 6 2 40
  -4 -9 -2 -2 -17 4 1 4 4 -2 6 -15 -6 -1 -2 -8 -7 -7 1 -11 -16 -7 3 -1 4 6 -10 0 -14 -11 9 6 -9 5 11 -2 6 -9 0 6 -9 14 -2 16 5 -10 8 0 -12 5 -3 7 0 -18 1 -1 -7 4 -1 8 6 -3 6 -5 1 -7 40
  4 2 -9 -5 2 17 11 8 16 -8 3 -13 -10 10 8 0 4 -12 6 4 -3 4 -4 5 -2 -4 -11 6 4 20 5 -2 -4 6 12 -10 -6 3 -10 -3 -8 7 -9 4 4 0 -18 -9 -3 -7 10 0 -3 -12 -6 -9 8 0 -4 1 -8 9 1 -9 -3 -9 6 36
  -13 -17 14 -2 3 -3 2 -13 -3 -11 -3 0 12 -2 -7 -3 -3 1 -1 -13 3 9 -12 -12 -6 -10 -9 19 -1 3 -1 -2 9 2 5 -10 5 -4 10 -2 -6 0 -8 2 0 -5 18 7 -5 1 2 5 1 8 5 9 9 -5 -1 -17 7 12 -1 -9 12 -1 -3 -2 34
  3 12 -8 7 6 -7 -12 4 8 7 5 6 4 11 -8 16 9 13 4 18 9 2 -2 -1 7 5 12 -6 6 -1 -9 -5 -14 -1 7 3 -6 2 -6 1 -3 -13 23 -10 -6 10 -7 -5 19 -2 -15 -2 2 -3 -12 -3 -5 10 4 -3 -10 -3 -20 -1 -17 -5 -6 1 -9 44
  1 13 -3 10 1 -13 -13 -3 -13 16 -9 13 -2 -1 2 -2 6 6 1 10 16 -11 -5 -5 5 12 13 -12 3 0 -2 2 4 2 -19 2 -15 4 6 -3 12 -9 3 -9 6 -5 -6 11 9 -10 -8 -2 12 4 7 8 -5 13 6 2 -3 -17 -8 15 -5 13 -7 -11 -13 9 44
  4 -3 1 2 -14 7 -1 -9 10 11 -10 -1 -8 -3 -5 -7 3 -9 2 -17 -2 4 -10 4 9 9 3 0 10 18 0 -5 11 14 -3 -4 -1 -12 4 -10 -1 8 4 -7 -1 -8 4 13 -2 0 -3 6 -7 -1 9 9 11 4 4 2 8 -6 1 -1 3 -5 4 -3 3 -11 9 40
  0 5 -8 0 2 6 -3 8 8 -1 -11 11 -5 -1 7 -3 0 -10 -2 1 -2 4 -3 7 4 5 -6 -3 2 3 1 -6 -1 -1 -1 0 -2 3 0 -2 4 -6 -6 -5 -3 4 -14 -4 -1 -7 9 -7 3 0 -3 3 0 -7 1 4 -6 -1 3 3 -9 3 -6 6 -2 -2 -4 -1 24
  7 -1 11 1 10 10 1 -12 7 9 -5 2 -4 -2 -3 11 -4 -3 -9 -1 8 9 -7 -6 2 -8 -2 8 14 13 -13 4 5 -2 4 0 7 -6 1 -2 3 5 2 -6 -9 2 10 -6 3 9 1 8 -11 3 0 9 17 -3 6 -2 0 15 3 3 11 -8 -10 -1 5 -9 -7 5 0 32
  -8 -6 6 12 13 4 0 -9 5 3 4 6 3 1 -5 -2 1 -2 -12 0 14 9 -17 -11 -8 -1 -15 8 9 8 -11 9 1 2 -9 -4 2 12 -1 6 0 6 1 -5 -5 1 7 2 3 -7 4 12 -9 -8 -2 8 10 11 6 -8 -2 4 3 2 8 -6 -3 1 12 2 -4 0 0 5 34
  -7 -6 5 14 14 12 -14 -18 -9 0 3 -4 -2 -9 5 1 -3 5 -8 4 14 13 -16 -10 -2 -6 -13 -1 9 3 -16 11 -9 4 1 2 6 2 -7 10 12 11 7 2 2 -9 2 -5 2 -9 10 4 -2 -6 1 3 9 12 6 -2 -12 2 -1 9 12 4 2 0 6 2 4 -1 -4 2 19 40
  7 6 -3 -4 2 4 2 -1 -7 -2 0 0 -1 -3 6 7 7 3 9 6 3 0 5 5 -3 -1 13 -4 5 4 7 3 0 -1 -3 0 -3 0 0 -5 8 -4 -8 -4 3 -2 -18 -3 -1 4 7 -8 5 8 8 -4 -2 -10 -10 7 -4 -3 0 1 0 11 -10 2 -5 -1 5 -3 0 -1 -11 -7 26
  4 -14 11 6 0 -1 8 -3 -8 -9 6 -5 7 -10 10 -5 -8 -3 -9 0 -7 -3 10 -4 2 -6 -7 1 0 1 3 12 -5 0 -6 3 3 5 -1 8 14 5 -1 2 2 -4 3 -2 0 -3 8 -13 -8 10 1 -15 -4 -14 -6 2 -4 -2 9 -2 6 -4 -6 -4 1 -9 -3 -8 4 3 0 2 0 28
  -2 14 2 -8 7 -13 -15 -1 -13 8 -11 7 0 7 -9 6 -5 19 4 3 10 -7 2 0 -3 3 20 -2 -7 -18 -4 -10 5 -15 3 5 4 -10 3 2 -2 -11 -1 4 3 5 5 4 -2 5 -17 5 16 6 0 17 -2 3 8 -4 2 0 -6 11 2 15 -3 -12 -1 5 13 -6 -5 -2 -10 -2 3 -11 34
  -5 -14 11 8 6 -11 2 -3 5 3 2 14 8 -14 3 -2 0 -1 -30 -6 0 11 -2 -3 -2 -1 -13 9 1 -2 -14 11 -14 -1 -5 7 4 2 10 2 18 -4 14 -12 -9 9 19 0 4 2 -1 -6 -9 15 -5 0 3 -1 6 -7 7 2 -14 -5 7 5 -5 -13 8 5 -4 -2 -2 6 12 6 -7 10 -6 44
  -9 5 -1 -11 2 -6 -14 1 -9 -4 -3 -4 0 -2 -4 9 -3 9 9 -8 -7 0 5 7 -14 4 8 0 -11 -20 -3 -12 1 -11 8 9 9 -9 4 9 -5 -7 0 8 5 -1 9 -1 -6 4 -14 2 15 0 0 7 0 2 5 -2 -1 -3 -7 4 4 6 7 -6 6 -1 -4 -6 0 -8 -2 2 -5 -6 16 0 34
  3 4 -3 -13 1 -7 -2 1 -9 2 6 -8 -3 1 -6 17 -3 11 4 3 -14 -4 15 0 1 -2 11 -4 -16 -16 4 -2 -9 -17 6 12 10 -7 -2 7 -2 -6 13 2 0 5 7 -1 6 7 -14 -5 7 5 -2 -3 -4 -4 -4 -2 -4 -2 -7 -6 0 2 -1 -7 -4 10 0 -10 -7 -3 -10 -4 2 -1 11 2 14 34
  1 4 3 4 2 -3 -4 -4 -13 -3 10 -10 -4 0 -7 8 8 6 9 14 1 -11 5 -10 -5 -6 3 -5 -4 -2 2 5 -2 -5 -6 3 -3 7 -9 8 -5 -5 6 6 3 -11 0 0 2 -3 -8 -2 5 -5 3 -1 -3 6 -3 -2 -6 -7 2 4 0 -2 3 -4 -5 9 12 -5 -13 -5 -1 6 1 -4 6 -7 2 12 28
  9 11 -8 7 8 5 5 12 12 6 9 5 -5 5 6 0 9 1 -10 10 7 10 1 6 1 0 0 -10 9 8 -10 6 -13 -5 -1 6 -4 8 -11 -7 13 -1 4 -5 -3 5 -16 -8 6 -6 10 -2 -11 -5 -5 -5 -5 2 -8 4 1 -1 -4 0 -2 1 -7 11 -14 8 0 -7 3 2 5 1 4 -5 -5 2 -11 -4 -2 38
  11 -1 -8 3 2 26 -3 -15 10 8 7 -11 -7 7 7 1 -16 2 -5 -1 7 17 -13 1 15 0 -4 -4 11 22 -11 10 -14 2 16 -7 12 -7 -10 10 2 26 16 0 4 2 -10 -12 -1 -5 1 11 -15 -2 -12 -3 5 3 7 11 -20 15 1 -5 2 -10 2 12 -6 9 -4 5 3 9 11 14 -3 3 -8 2 -9 -1 -6 7 50
  1 -4 11 -6 -5 -11 -5 -2 4 10 -16 6 5 6 -7 -1 -8 -4 -1 -14 5 -2 -4 -3 5 -1 6 4 1 -8 -4 -2 5 -1 7 2 5 -14 14 -3 4 1 -3 1 5 6 18 8 1 9 -14 7 2 6 3 15 6 1 10 -7 11 4 -5 -1 8 3 1 -7 10 -3 0 5 -2 7 2 -4 -9 -1 7 10 7 2 -8 -11 -3 32
  0 -15 14 5 7 4 -4 -14 -17 -4 0 -14 11 -3 -1 2 -16 10 -1 3 12 0 -7 -14 -1 -11 -7 -3 9 -2 -3 10 -6 -1 3 -1 22 4 1 6 12 14 0 11 13 2 10 6 10 8 8 2 -4 8 3 -2 4 -11 -6 -1 -8 3 4 -4 8 2 -4 -8 11 -5 -5 -6 -6 7 9 9 -1 14 -1 5 0 4 5 -8 9 4 44
  3 -2 11 13 -6 1 -2 6 17 -5 -6 -1 2 9 -4 -11 4 -12 -3 -6 6 2 -16 1 2 1 -5 -4 17 8 -7 3 6 1 4 -9 -7 -5 -2 -4 -1 3 -6 3 4 -3 4 5 -2 -8 1 10 -9 -6 -5 10 5 -4 -1 -4 6 4 6 1 -2 -11 2 9 6 -6 -7 7 8 6 9 -3 -9 -2 -8 -3 -6 -19 -13 5 3 8 -2 38
  

• DIVISORS
  1
  

• MINIMAL_NORM
  8
  

• KISSING_NUMBER
  168498000
  

• HERMITE_NUMBER
  8
  

• SUBGROUP_ORDER
  2^2*5*23
  

• SUBGROUP_NAME
  C115 : C4
  

• SUBGROUP_GENERATORS
  2
  88 88
  26 73 82 -115 20 -56 103 28 121 -40 -102 -72 32 -170 -23 -43 -32 80 45 49 -12 90 -60 -96 45 66 21 -49 -1 -32 -47 19 -13 10 -26 15 -21 52 8 42 -41 26 5 4 18 -29 -36 -12 -22 0 -2 -9 2 -5 18 -2 6 28 13 -10 2 -15 -1 -4 -1 0 -8 1 -7 -3 -7 -8 -1 0 -5 -7 1 -4 -6 3 0 3 1 -4 7 4 -4 -3
  -72 67 92 -64 -90 7 71 53 56 -11 -69 -28 4 -63 23 9 -33 79 6 6 16 61 -27 -74 9 9 16 -30 14 -19 -2 27 10 6 -14 18 -4 47 -13 20 -16 -4 31 -14 10 -11 -29 -1 -21 -6 7 -3 1 -8 9 -5 13 13 5 -11 -10 -7 3 -1 2 -4 -1 4 -9 5 -3 -5 0 8 -7 -4 3 4 -3 4 6 2 2 -3 7 8 -9 -5
  52 -35 14 -51 36 -9 41 9 36 -29 -37 0 6 -37 4 -12 29 40 11 15 -10 16 -33 -33 19 34 9 -34 -11 -16 -31 -4 -11 7 -3 7 -5 13 19 20 -22 24 -5 10 7 -16 -11 -7 -8 -1 -7 -1 4 -2 9 -4 6 15 8 -12 7 -7 0 7 -6 -5 -4 -5 -6 -3 0 0 4 3 0 -5 1 1 1 0 1 0 2 0 0 -5 2 -1
  -4 8 21 -16 -48 -7 45 13 14 -22 -44 -5 -18 -22 4 15 -10 36 -4 6 9 35 -17 -38 0 11 14 -15 -4 -19 -14 7 2 8 0 13 -4 29 1 7 -10 4 7 3 6 -11 -15 -5 -9 -1 -1 2 -1 -1 2 -7 9 8 8 -8 -1 -9 2 -3 -1 -2 -5 5 -5 2 -2 -1 1 6 1 -3 2 3 2 2 3 0 -2 -2 2 -2 -3 1
  -11 -37 4 47 -138 85 44 -4 -89 -41 -20 78 -75 132 53 67 33 23 -61 -24 35 9 15 -12 -50 -10 13 3 -15 -37 -5 -24 1 26 9 7 20 9 3 -10 9 -14 13 -2 5 1 14 8 9 -9 -11 8 -1 -7 -6 -17 13 -4 -8 -15 -7 0 5 11 -6 -8 2 4 -3 8 3 3 7 12 -2 0 4 10 1 5 3 -4 2 3 1 1 -2 3
  9 25 1 6 -8 5 12 -19 -11 -7 -3 4 -5 7 2 -11 2 -13 0 12 0 10 9 0 -11 10 2 16 -4 -10 -11 -6 -11 13 -6 0 7 -9 -7 -4 -1 -4 -4 -1 6 -1 10 3 11 3 -4 3 3 3 -5 0 -3 -2 -4 2 -2 1 4 1 1 -1 3 0 6 -1 1 -2 5 -5 -2 1 0 -3 -5 2 -6 4 0 1 2 1 2 4
  -41 -67 -23 55 16 85 -101 -11 -88 62 109 78 -15 139 44 12 42 -53 -28 -43 -1 -95 51 97 -28 -50 -22 35 12 30 36 -10 0 -12 18 -25 24 -53 -11 -25 21 -29 -2 -11 -3 19 33 19 29 1 -2 5 9 6 -8 1 -7 -13 -17 3 0 12 5 11 -3 -5 9 -6 13 0 5 3 5 -5 -5 2 -3 0 -3 -2 -11 1 4 5 -6 0 5 3
  -60 -75 2 35 24 42 -96 49 -70 80 95 49 11 69 11 20 2 -41 0 -49 -7 -68 25 78 -9 -55 -23 20 21 51 52 3 15 -25 24 -26 7 -33 -29 -17 19 -30 11 -26 0 23 11 8 10 -1 8 4 2 -1 -7 10 -6 -7 -8 9 -5 9 2 3 8 3 8 -5 -1 -2 7 0 -3 1 -4 1 -2 -1 2 -6 -2 -3 2 3 -2 -1 -2 -3
  -7 13 16 -24 14 -8 -5 4 34 0 -7 -8 5 -31 6 -15 -18 14 16 5 -3 10 1 -5 3 9 -1 0 8 2 2 16 1 -4 -5 -3 2 5 -13 7 -7 -9 5 -9 12 -6 -1 2 4 5 3 1 5 0 3 1 -3 9 -1 -2 1 -2 4 2 3 -1 2 -2 4 -2 0 -3 0 -4 -6 -4 -1 -4 -5 1 -9 3 2 1 1 4 -1 0
  33 128 59 -91 -40 -63 158 -21 153 -100 -164 -100 0 -168 -24 -25 -36 105 20 63 13 124 -57 -138 40 79 33 -62 -11 -63 -54 15 -6 17 -31 35 -23 80 41 50 -35 40 2 26 3 -36 -35 -17 -33 -4 -5 -13 -5 -3 20 -19 9 26 14 -18 8 -20 -4 -7 -11 1 -14 9 -8 2 -11 -5 -3 3 2 -6 4 1 -1 8 7 2 -1 -5 5 9 -5 -4
  32 -25 15 -66 83 -15 -56 39 52 30 32 -13 31 -76 8 -52 -21 15 52 12 -14 -9 -10 6 25 22 -10 19 25 9 0 29 -1 -24 -18 -7 -13 -8 -26 8 -22 -13 5 -9 11 -7 -6 7 2 12 12 3 8 3 12 13 -6 12 5 -1 4 1 -2 -3 3 -3 2 -4 9 -9 0 -2 -3 -10 -11 -4 -7 -8 -12 -1 -11 7 3 0 1 6 3 1
  -11 45 12 13 -99 -12 66 -7 18 -44 -65 -19 -40 -18 -8 31 -36 36 -16 10 24 57 -13 -53 -14 12 18 -12 -9 -29 -7 1 12 8 -1 15 -7 42 7 7 3 1 9 4 0 -7 -12 -6 -13 -5 2 -3 -8 -3 1 -13 8 3 2 -5 -4 -7 -1 -7 -2 4 -7 10 -7 7 -5 -2 -5 9 4 1 2 5 4 4 8 -4 -3 -2 0 5 -6 -2
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  18 -18 -30 48 26 -17 -25 -21 -32 17 33 4 -7 2 -37 1 -21 -47 9 4 -5 -17 7 40 -8 -4 -9 16 -6 17 13 -19 2 -10 17 -14 -4 -14 1 -10 16 -7 -14 0 -4 9 11 -3 7 -1 0 3 -7 5 -10 1 -7 -7 -5 11 2 1 -1 -8 5 10 0 2 -1 1 -1 -1 -4 -3 5 5 -1 -4 2 -3 -1 -1 -3 5 -4 2 -2 0
  -9 -22 14 24 -5 46 -20 19 -68 32 43 32 -12 34 -2 2 -3 -31 -6 -6 -3 -18 10 36 -8 -7 1 23 3 3 -1 -19 -9 -3 4 -18 -2 -18 -4 -6 5 -17 0 -4 2 6 10 7 9 -4 -2 4 1 -3 -4 2 0 -3 -9 3 -2 3 -2 -2 2 2 -1 3 2 4 -3 1 -1 -3 -2 0 0 -2 -6 -1 -6 1 -1 2 -1 5 3 1
  50 -133 30 20 69 36 -35 33 -105 57 105 96 -15 46 -11 11 13 -52 13 -17 -28 -50 -30 68 -12 -7 -11 -7 -14 22 12 -62 -10 -1 42 -44 -3 -31 -10 -9 6 -17 -6 -22 18 11 13 9 16 -5 -9 6 -6 -8 -9 4 -4 0 -11 10 -11 -2 3 4 10 12 8 -5 -12 -2 0 -1 1 4 -1 0 -1 -4 2 -6 1 -7 3 9 -3 0 -4 0
  -103 79 68 -40 -79 21 31 65 38 24 -52 -47 18 -47 22 -1 -25 54 -13 1 6 38 -1 -49 15 -18 25 -6 26 -4 -5 38 9 1 -20 17 -7 28 -8 18 -17 -1 21 -5 -2 -6 -29 -1 -18 -5 9 -3 2 -2 10 0 12 7 8 -7 -1 -2 0 -7 -3 -5 -6 3 4 6 -2 -6 2 0 -4 -3 4 1 -8 1 3 8 2 -8 5 3 1 -4
  -87 23 52 5 -155 61 57 34 -21 -24 -45 40 -52 53 54 52 -11 47 -50 -11 38 38 9 -30 -29 -19 31 -8 2 -22 -5 14 9 24 -4 17 22 24 -7 3 -3 -12 16 -10 13 -9 -4 4 1 -7 -9 5 9 -5 -8 -14 23 4 -7 -17 -7 -5 14 10 -6 -14 -1 4 3 12 1 -2 5 4 -7 -1 5 7 -4 2 -2 6 6 2 3 0 -2 -3
  2 -134 -119 149 102 56 -184 -16 -182 108 176 68 18 178 -23 22 63 -149 -14 -73 -8 -143 79 174 -16 -78 -60 75 14 69 71 -35 1 -37 37 -39 11 -89 -12 -53 51 -33 -23 -8 -29 53 48 10 26 -5 4 15 -8 13 -21 10 -23 -37 -13 22 3 21 -10 -3 10 5 16 -7 5 -10 10 7 -2 -3 6 5 -5 -3 14 -11 -1 -5 0 1 -3 -6 2 4
  139 -3 1 -61 115 -83 48 -43 90 -57 -51 -44 16 -123 -46 -59 -16 19 65 41 -28 53 -51 -47 26 83 -6 -27 -24 -30 -36 -12 -14 2 -11 -5 -21 25 5 18 -24 23 -14 3 21 -18 -7 -12 0 12 -4 -8 -5 -1 12 2 -12 19 7 -1 1 -9 -5 -2 4 11 1 -5 -8 -7 -3 -6 -4 2 -1 -7 -4 -8 -1 2 -1 -5 1 4 1 2 -6 1
  33 -6 1 -29 56 -43 -27 5 40 10 13 -24 2 -74 -17 -35 -37 -2 35 16 -9 8 -15 1 18 19 -2 9 7 3 -2 9 6 -18 -4 -10 -18 6 -9 6 -12 -3 -3 2 5 -6 -6 -1 1 10 5 -4 1 2 5 6 -5 5 -1 7 1 -2 -2 -8 2 5 -3 1 1 1 -5 1 -8 -4 2 -1 -2 -5 -6 -1 -4 1 -3 5 -4 6 0 -1
  15 3 9 -33 49 -22 -9 13 41 4 -8 -25 23 -57 -13 -30 -20 6 27 12 -15 7 -9 -7 16 17 2 -1 5 5 -10 14 -1 -12 -10 -3 -10 2 -5 10 -13 4 1 0 5 -7 -8 -4 -2 5 5 -4 3 0 7 7 -2 10 3 -1 4 0 -2 -2 2 3 -1 -3 1 0 1 -1 -3 -4 -3 -3 0 -6 -8 -1 -5 2 -1 1 -1 1 3 -1
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  -14 16 -58 35 68 -48 -81 -40 -10 40 55 -29 42 14 -28 -27 0 -66 19 -10 -17 -51 32 55 -1 -27 -29 34 4 39 27 12 0 -15 4 -6 4 -35 -17 -30 21 -6 -9 -4 -13 19 9 -1 9 8 10 1 0 7 -10 16 -13 -19 -1 20 -2 10 -1 -6 12 3 5 -4 4 -3 2 3 0 -3 3 5 -3 -4 2 -2 -3 -1 -5 1 -1 -4 1 3
  -186 36 8 59 -145 91 -59 60 -82 68 31 11 0 127 48 59 9 -10 -71 -60 29 -45 66 32 -21 -97 7 39 34 28 36 42 21 -12 -2 16 14 -9 -19 -26 20 -22 28 -3 -26 21 -11 9 -5 -7 14 2 7 3 -7 -1 15 -19 3 -3 -4 9 -1 -4 -2 -12 -2 4 10 10 7 5 4 4 0 2 6 8 1 -1 4 4 -5 -10 1 -4 5 0
  -52 33 39 -34 -33 20 33 19 52 -5 -31 -10 1 -26 25 7 -3 47 -12 8 11 11 -3 -27 12 -4 14 -28 11 -1 -1 16 7 -1 -5 10 4 16 15 23 -10 7 9 6 -1 -12 -10 2 -10 -5 0 -4 6 -1 6 -7 7 10 0 -11 7 -3 4 5 -9 -4 -4 0 3 3 -1 0 1 -4 -3 0 3 2 -3 1 0 5 2 -2 -1 4 2 -3
  15 70 -7 -15 -2 -81 41 -24 62 -33 -61 -69 10 -94 -40 -17 -39 11 21 27 9 57 -10 -38 16 24 2 -6 -7 -11 -10 13 3 4 -13 10 -12 25 1 7 -5 10 -6 5 0 -5 -13 -13 -10 2 1 -4 -5 -1 1 3 -3 0 4 8 0 -4 -1 -10 4 4 -5 6 -4 1 -6 -2 -6 1 5 2 0 -1 0 0 4 0 -2 -1 5 1 -4 -1
  -185 134 49 -28 -155 39 21 31 70 10 -61 -46 -3 11 69 28 -14 82 -55 -16 36 18 32 -55 2 -58 24 -9 40 -3 22 67 25 3 -22 35 14 35 2 13 -2 -5 31 4 -16 -9 -23 6 -19 -4 9 -6 10 1 8 -10 17 -3 4 -12 1 -1 7 0 -12 -15 -9 7 12 9 -4 1 2 -2 -4 0 4 11 -5 6 3 10 1 -10 2 3 2 -2
  -37 -67 6 1 97 16 -93 27 -10 81 104 30 34 18 6 -6 7 -33 20 -34 -16 -77 10 72 19 -43 -28 -4 20 53 50 3 12 -33 27 -24 -1 -28 -12 -6 11 -17 6 -12 -4 14 4 9 4 3 9 -1 5 3 -1 7 -10 -3 -2 7 1 0 0 2 5 4 8 -7 -1 -8 4 4 -3 -4 -2 -2 -2 -5 6 -6 -1 -2 2 -1 -4 1 -1 1
  -108 87 72 -46 -24 -23 11 39 61 38 -22 -56 33 -90 -7 -19 -57 26 18 4 -2 22 -2 -19 24 -18 10 -10 19 17 8 42 11 -15 -10 4 -10 23 -15 8 -9 -5 21 -8 3 -6 -30 -4 -13 3 11 -8 8 -3 2 7 7 7 4 3 -5 -5 4 -9 5 1 -6 5 0 11 -5 -1 -4 -2 -2 -2 3 -3 -8 -1 -2 4 -4 -1 2 4 -1 -2
  72 -30 9 -35 81 -31 14 21 33 -20 -9 -11 24 -78 -35 -19 -26 -1 35 18 -16 24 -30 -12 26 38 1 -17 -6 -2 -18 -10 -4 -11 -5 -8 -17 9 2 17 -16 8 -3 4 10 -11 -10 -4 -6 2 2 -1 -3 -5 9 5 -3 16 7 -3 3 -6 -9 -3 5 9 3 -3 -6 -5 1 1 -8 -3 -2 -6 1 -6 -3 -4 -3 -2 1 0 1 3 -2 -2
  -88 82 5 1 -128 46 39 -8 19 -39 -71 -17 -1 75 57 24 33 47 -49 -15 24 21 33 -50 -25 -19 14 2 6 -21 -6 41 4 25 -27 39 29 13 -4 -8 4 6 16 -1 -6 -2 -2 1 -2 0 1 -1 8 4 1 -6 13 -5 5 -16 -3 8 5 13 -8 -18 2 -2 8 5 7 -2 12 4 -5 -1 2 9 1 6 1 6 4 -8 5 -7 0 -1
  -3 13 42 5 -78 50 99 -1 -34 -36 -51 27 -17 34 11 25 36 21 -21 14 12 32 -13 -33 -17 22 20 -21 -27 -32 -36 -26 -18 27 0 15 11 6 21 8 -4 18 5 9 1 -6 0 -4 -2 -11 -12 0 7 0 -5 -13 12 8 1 -13 -3 -6 6 8 -8 -5 -1 3 -5 4 3 -1 12 7 2 1 5 4 4 3 1 0 0 1 4 -3 3 1
  105 -54 -35 39 33 -23 18 -7 -48 -24 -1 17 -32 -28 -52 4 -25 -36 17 19 -4 29 -15 12 -6 36 5 15 -20 -14 -21 -40 -9 3 8 -19 -16 -2 5 4 1 -4 -20 4 10 0 12 -6 5 -4 -8 7 -11 -4 -6 0 -6 4 -5 6 5 -3 -6 -9 3 11 -3 5 -5 -2 -3 -3 -8 -2 3 2 -1 -6 -1 -3 -2 -4 -2 4 0 3 1 1
  185 -67 -78 69 73 -41 27 -103 -86 -57 22 50 -45 52 -46 2 34 -73 0 21 -25 0 -22 23 -32 44 -13 2 -55 -31 -25 -92 -29 39 23 -28 0 -23 21 -18 18 7 -39 5 9 8 37 -1 22 -3 -23 7 -25 -9 -9 1 -19 -14 -11 18 -6 2 -6 0 6 16 4 1 -9 -6 -7 0 -1 4 9 4 -2 0 9 1 6 -14 1 7 0 -4 -3 8
  60 -164 -78 70 168 36 -187 -20 -128 98 202 94 20 140 0 -22 47 -126 26 -53 -41 -148 31 149 -24 -42 -58 55 5 62 59 -39 -1 -35 39 -51 8 -88 -31 -48 36 -35 -9 -21 -6 36 47 19 37 12 8 5 -2 7 -11 20 -27 -23 -16 24 -9 20 -4 8 14 11 21 -13 0 -9 8 9 -1 -2 0 1 -8 -6 4 -6 -7 -9 0 9 -10 -1 0 5
  -119 -115 -81 105 37 88 -227 51 -163 150 187 71 35 209 38 23 57 -112 -34 -97 -12 -168 92 159 -27 -125 -42 84 42 93 81 23 18 -42 25 -23 24 -91 -44 -59 43 -43 7 -22 -26 49 29 18 23 3 19 11 9 12 -19 20 -6 -36 -10 17 -3 31 -1 6 9 -10 15 -13 11 -1 15 10 5 -2 -1 5 -2 0 3 -9 -5 1 -2 -1 -6 -10 8 0
  -41 -33 28 -2 -47 34 5 49 -31 21 6 34 -14 30 26 34 8 22 -23 -21 2 -6 -6 -6 -6 -28 13 -10 5 4 3 3 7 4 12 0 1 8 -7 2 -4 -4 14 -5 5 -2 -13 3 -6 -4 0 3 3 -6 -1 -3 11 4 5 -6 -2 -5 3 0 -1 -4 -3 -1 -4 1 2 0 3 4 -1 -2 3 3 2 -1 4 -1 -1 -3 1 -4 2 1
  -92 19 34 -24 36 -35 -69 47 34 85 52 -35 50 -61 -8 -29 -43 -13 36 -8 -12 -34 2 32 20 -39 -11 10 28 53 39 36 19 -33 7 -6 -9 -11 -28 -7 3 -14 15 -16 -8 11 -19 -1 -8 5 20 -2 6 5 -3 15 -1 -4 2 14 -5 2 5 -10 11 3 2 -1 -1 3 -1 0 -3 -2 0 3 -1 -5 -4 -5 1 6 -2 2 -1 2 -1 -4
  19 -106 -60 121 10 60 -74 -30 -180 50 112 106 -35 172 5 41 49 -100 -28 -48 -9 -83 41 107 -52 -49 -27 45 -14 25 34 -56 -10 9 39 -34 22 -59 -16 -46 41 -33 -11 -18 0 33 43 12 29 -6 -9 18 -8 -2 -26 4 -11 -25 -18 16 -10 10 3 4 11 4 13 -2 -1 0 4 2 5 3 1 4 -3 1 9 -4 -2 -7 0 6 0 -5 0 8
  -45 -82 -36 95 7 42 -116 0 -147 79 129 83 -14 152 14 27 7 -94 -17 -63 -14 -85 44 105 -50 -72 -27 52 2 45 56 -17 7 -3 35 -32 19 -51 -47 -55 42 -49 7 -35 5 35 29 13 28 5 4 12 -3 -4 -23 15 -8 -26 -15 21 -19 12 6 3 18 3 14 -3 -2 4 5 4 2 5 -3 1 -2 -1 3 -4 -3 -6 1 5 -1 -5 -4 6
  47 -47 -40 4 107 -18 -67 -11 4 17 47 -2 29 -7 -16 -29 10 -38 30 -8 -16 -43 8 45 17 4 -24 13 7 24 15 -1 1 -26 0 -12 -4 -27 -1 -7 1 3 -11 1 -6 8 13 0 6 7 4 -1 0 6 0 9 -12 -2 -3 4 4 7 -5 1 3 5 8 -7 2 -3 5 3 -5 -4 1 -1 -4 -5 -1 -4 -3 -2 0 5 -4 -1 1 -3
  -57 62 98 -102 -59 -1 74 52 107 -17 -69 -34 3 -102 28 -11 -42 96 10 29 1 54 -44 -86 18 22 35 -43 8 -20 -27 38 9 0 -14 20 -11 48 1 34 -33 14 28 4 12 -30 -39 -2 -18 5 5 -13 13 -2 16 -5 18 26 13 -16 0 -15 8 0 -8 -4 -12 3 -1 6 -2 -3 2 0 -4 -4 4 -1 -11 6 -2 3 -4 0 -3 5 1 -3
  59 -29 -33 26 120 -27 -55 -17 -28 32 65 -2 24 -19 -49 -26 -7 -64 30 -4 -26 -30 -2 53 19 2 -26 14 3 24 21 -29 -6 -17 15 -30 -15 -25 1 -4 12 -9 -17 -2 -4 13 16 2 8 2 1 0 -10 0 0 8 -21 -6 -5 19 2 2 -10 -8 8 15 5 -1 -3 -9 -4 1 -10 -6 4 0 -4 -5 4 -4 -1 -5 1 0 -1 4 -2 1
  -9 56 47 -40 -38 -33 67 14 60 -30 -55 -28 -3 -88 -7 1 -52 38 7 34 1 55 -26 -54 7 21 25 -25 -8 -15 -24 8 5 9 -9 9 -8 32 0 18 -15 6 9 5 11 -18 -24 -3 -11 0 2 -2 -1 -6 6 1 9 13 7 -3 -2 -10 0 -8 1 3 -6 5 -2 5 -5 -2 -3 -3 1 0 4 -3 -5 1 -1 1 -3 -1 1 5 -1 0
  49 -26 17 -20 58 -23 10 31 9 13 -10 -18 17 -66 -40 -15 -17 4 32 2 -26 24 -31 -15 22 20 0 -9 0 -3 -11 -8 -3 -10 4 -12 -25 18 -6 13 -12 4 -1 -4 10 -3 -15 -7 -9 0 4 -2 -9 -3 10 2 -6 13 10 4 0 -7 -12 -11 4 11 -2 -1 -9 -7 -1 -6 -6 2 1 -6 -1 -4 4 -3 3 -5 -1 -5 2 0 -3 1
  177 9 -11 -48 122 -105 60 -32 70 -54 -62 -69 11 -167 -82 -66 -33 7 66 58 -35 77 -65 -57 38 92 4 -15 -25 -42 -52 -30 -20 5 -13 -11 -43 31 17 24 -28 29 -27 13 14 -18 -13 -18 -9 7 -6 -6 -18 -4 16 4 -14 15 12 9 4 -11 -15 -18 4 18 -9 3 -10 -7 -10 -7 -10 -1 8 -6 -4 -8 -2 0 4 -4 -1 0 1 3 -3 2
  153 33 -45 -27 108 -107 50 -108 90 -86 -46 -55 7 -101 -60 -57 -16 -13 52 59 -15 41 -35 -32 16 86 -11 -22 -37 -31 -35 -27 -18 8 -13 0 -13 11 28 7 -6 30 -29 19 3 -14 9 -13 3 10 -7 -9 -10 2 6 2 -16 4 0 6 4 -4 -5 -1 2 13 -1 2 -4 -3 -7 0 -6 -1 6 1 -5 -5 1 4 0 -5 -2 6 -3 2 -4 1
  -6 -9 40 12 -72 44 66 -1 -46 -19 -19 49 -38 37 21 36 20 15 -25 4 18 18 -6 -7 -17 7 12 -18 -17 -22 -16 -30 -12 24 9 -1 11 6 12 5 -1 -2 2 1 7 -3 3 3 3 -12 -13 7 2 -6 -6 -13 8 5 -6 -9 -3 -8 7 5 -3 -5 0 4 -2 3 -3 -2 6 4 -1 1 3 3 4 1 0 1 2 2 5 1 1 3
  -71 -56 -117 179 -44 58 -151 -10 -225 95 135 55 -9 226 -3 57 44 -143 -64 -79 3 -113 93 132 -53 -112 -32 98 6 51 57 -20 7 -9 28 -20 15 -75 -28 -73 59 -37 -4 -5 -37 56 32 9 23 -4 7 13 -9 6 -26 15 -9 -51 -8 30 -10 23 -7 -12 13 0 6 3 6 3 7 10 1 5 13 10 1 4 11 -5 5 -8 -9 -3 -1 -11 7 9
  -91 -11 38 27 -13 61 -56 70 -69 96 64 16 10 27 7 9 -10 -25 -10 -29 -8 -36 22 46 6 -53 2 29 29 29 26 3 6 -20 13 -18 -10 -15 -15 -2 6 -26 14 -8 -7 14 -6 9 -2 -7 9 5 1 1 -1 3 0 -4 -1 7 -1 1 -4 -13 1 3 -2 3 3 1 -2 0 -1 -4 0 -1 3 -3 -3 -4 -3 4 -2 -5 0 5 5 2
  17 19 38 3 -28 -17 43 20 -25 2 -9 -11 -19 -62 -48 3 -62 -15 7 25 -5 44 -27 -12 1 22 21 2 -16 -15 -29 -23 -6 2 3 -15 -24 19 -1 6 -8 -4 0 4 8 -6 -19 -5 -2 -3 -2 -5 -6 -5 2 2 6 6 1 8 -5 -11 -4 -17 6 13 -11 12 -5 7 -8 -4 -8 2 5 2 2 -6 -7 0 -1 -5 -8 1 -2 5 0 1
  -157 -45 41 72 -131 124 -42 66 -130 86 74 87 -33 148 56 76 19 -17 -56 -59 33 -59 47 67 -34 -86 1 22 21 32 47 1 15 -8 30 -6 23 -20 -18 -20 27 -41 26 -16 -9 22 7 15 4 -16 4 13 10 2 -17 -11 14 -11 -10 -6 -6 2 9 3 -2 -11 4 5 1 6 3 3 7 5 -2 3 5 6 7 -3 -1 4 0 -2 1 1 1 1
  

• PROPERTIES
  Unimodular=1
  

• REFERENCES
  I. Gauvin, Reseaux ideaux pairs unimodulaires de rang 88 sur Q(zeta_115), preprint, 15 October 2026 (arXiv identifier pending).
  Data, programs and certificates: https://github.com/Katabolanga/rang88-reseaux-ideaux (input data: commit f20d0bd714a94e0dbb2bf562807dbed1a393dd53).
  

• NOTES
  The two generators span a subgroup of the automorphism group, hence the catalogue sections SUBGROUP_ORDER, SUBGROUP_NAME and SUBGROUP_GENERATORS; the full automorphism group is not known.
  SUBGROUP_NAME: C115 : C4 = <zeta> : <g2>, zeta of order 115, g2 of order 4 with g2^2 = -1 and g2^-1 zeta g2 = zeta^24 (C4 acts through C2); equivalently C23 x Dic5.
  X139 = (P, b), an ideal lattice over L = Q(zeta), zeta = zeta_115 (rank 88): P = (139, zeta^2 + zeta + 77) is a prime ideal
  of Z[zeta] of residue degree 2, and b(u, v) = Tr_{L/Q}(alpha u conj(v)), alpha = beta_0/nu totally positive (see the paper, section 6).
  Minimum 8 is established by the computer-assisted theorem of the paper, in the execution model of section 5.4 (M1, M2, H1-H3).
  The same enumeration program was run on two computers; the attached Python scripts check the exported matrices and generators, not the minimum.
  The kissing number 168498000 follows from that minimum-8 theorem and the extremal theta series.
  Basis: an LLL-reduced Z-basis of P; X139_construction.gp gives it in the power basis of Z[zeta], together with alpha.
  Generators (row convention, g*GRAM*g^T = GRAM): g1 = multiplication by -zeta, of order 230;
  g2 = the semilinear map u -> lambda*sigma_24(u), sigma_24(zeta) = zeta^24, lambda a unit of Z[zeta] (in X139_construction.gp);
  g2^2 = -1, and g2 o g1 o g2^(-1) = g1^139 as maps (in matrices: g2^(-1)*g1*g2 = g1^139).
  <g1, g2> has order 460 = 2^2*5*23: an extension of the Galois stabiliser <sigma_24> (order 2) of the class of P by the cyclic group
  <-zeta> (order 230). By the normaliser argument and the exact Galois stabilisers established in sections 2.4 and 6.4 of the paper
  (class-group results, not only these matrices), it is the normaliser in Aut(X139) of the group of multiplications by roots of unity.
  The full automorphism group has not been determined: |Aut(X139)| >= 460.
  A vector of norm 8 (the witness of the paper, transported to this basis), as a row of coordinates in this basis:
  NORM8: -36 30 34 -28 -53 9 31 18 25 -8 -32 -4 -3 -9 27 4 -2 39 -8 4 8 23 -6 -34 -6 0 12 -10 4 -11 -6 16 3 12 -8 12 7 14 -8 6 -8 -1 12 -7 7 -7 -10 2 -4 -1 1 0 4 -2 2 -2 8 4 2 -6 -4 -1 6 4 -1 -7 0 0 1 2 0 -3 5 2 -4 -1 1 2 -3 2 0 4 3 -2 3 1 -1 -1
  

• LAST_LINE
  

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  Haftungsausschluss/Disclaimer
  
  
  Gabriele Nebe