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 06:02:04 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
  X599
  

• DIMENSION
  88
  

• DET
  1
  

• GRAM
  88 0
  30
  -4 36
  -8 -16 44
  5 9 10 38
  -11 7 -3 -1 32
  5 -8 -9 0 -9 38
  1 16 -23 1 15 11 50
  8 1 -5 10 11 -10 -6 42
  11 1 -5 6 -14 0 -10 1 34
  -7 8 -1 -5 6 -3 -9 9 8 36
  -4 7 14 21 -8 -12 1 -7 14 4 56
  7 -3 -14 -18 4 10 18 3 -11 -11 -12 50
  -14 -3 -10 -16 15 6 4 2 -3 13 -11 15 42
  10 -10 13 -8 -18 2 -4 -20 7 -9 3 4 -17 44
  5 2 2 -4 -15 -5 -1 -14 16 -1 23 8 -9 26 44
  0 -9 0 -17 -8 12 -5 -10 6 -3 -17 4 -1 13 -2 42
  -7 7 -11 3 14 -11 13 18 -5 -4 7 11 12 -21 -7 -4 44
  6 -2 -1 13 6 5 -5 2 11 -5 -2 -9 -4 -11 -9 10 -5 46
  -8 -13 8 -4 -7 7 -7 -18 -5 -2 -1 -11 -4 13 -1 -1 -22 -8 40
  -10 -3 -8 -4 10 1 5 3 2 12 5 -1 22 -16 -5 -7 7 0 0 30
  -4 14 -22 0 3 12 14 2 -2 -2 -14 6 10 -10 -6 7 6 9 -10 0 38
  -8 6 -4 -4 5 4 12 -13 -3 7 5 -7 2 -3 0 3 -1 -3 8 1 5 26
  4 -11 -12 -21 -4 19 6 2 -7 -8 -17 24 13 -7 -10 23 12 -4 -6 0 3 2 58
  8 15 5 23 9 5 0 7 10 6 11 -2 -8 -11 -4 -7 -2 21 -8 -4 -8 2 -3 62
  10 -16 20 10 2 -15 -2 5 8 0 21 -5 -5 15 10 -5 2 7 1 -2 -17 -4 -13 -3 60
  15 -17 1 -5 -18 8 -7 -14 19 -6 14 10 -5 21 22 10 -12 9 11 2 -9 1 7 4 17 52
  4 7 9 20 -4 -3 4 0 10 -1 29 -6 -15 3 18 -13 1 0 -7 -1 -13 0 -12 21 8 1 36
  -10 15 8 3 1 -13 -2 4 5 18 25 -8 -8 5 11 2 3 -8 0 -5 0 9 -7 5 6 0 5 46
  -3 -1 -13 0 -6 18 -6 -1 0 -6 -11 -1 3 -10 -5 10 6 12 -6 -2 20 -3 15 -2 -21 -6 -3 -8 44
  11 16 -8 -1 -15 -5 -14 1 15 5 -4 5 -11 7 11 6 -11 -1 -4 -14 4 -3 -9 12 -15 6 2 8 -2 48
  -4 12 7 14 -7 -7 -5 4 11 3 19 -10 -15 6 10 -7 3 -3 0 -11 -6 -1 -10 14 -1 -3 13 19 -4 10 38
  -1 14 7 3 -3 -6 -11 -8 13 8 8 -16 -14 14 13 12 -9 10 0 -17 1 5 -10 17 -3 -2 6 21 4 14 20 46
  -10 -1 17 -5 4 4 3 -12 -1 5 0 2 6 9 8 14 -7 5 -1 1 -2 1 -1 2 9 1 -1 8 -6 -5 -2 9 42
  13 15 1 3 -13 -6 -8 -8 14 -1 3 1 -8 15 16 6 -5 -4 -12 -11 1 -7 -6 13 -5 8 6 4 -1 25 1 19 3 48
  -5 -7 -11 -16 -6 10 2 -8 1 -6 -17 1 1 0 1 12 -9 3 9 0 8 4 13 -18 -18 3 -11 -8 10 0 -2 -4 2 -15 34
  8 15 -8 20 13 8 31 -1 6 -7 12 11 -6 1 10 -11 -1 11 -6 -1 3 4 -13 28 7 7 22 1 -9 2 4 0 3 4 -9 58
  -8 -7 13 2 -10 7 -2 -11 -1 -11 -4 -1 -3 6 -2 10 -3 -1 1 -3 0 -2 -2 -8 -6 -5 4 -11 4 1 4 -5 12 -1 8 -6 34
  1 -3 -12 -5 -11 16 0 4 -5 -1 -17 4 1 -3 -2 -4 -11 -6 4 2 16 -2 9 -15 -22 -6 -9 -9 16 -1 0 -9 -7 -9 18 -12 7 42
  -4 3 -10 -10 -8 1 6 -5 9 -4 -5 7 1 4 7 19 5 5 -6 -5 10 -1 7 -16 1 0 -6 0 3 7 6 5 8 0 14 -4 13 3 38
  -1 -12 -7 -18 -13 3 -7 0 2 -7 -10 7 2 6 0 21 1 -7 7 0 6 -1 24 -25 -9 8 -12 -1 6 4 -4 -3 -6 -3 17 -21 7 11 14 38
  9 -16 -4 5 -2 23 10 4 8 -4 7 15 11 -6 -2 2 2 10 0 8 -1 2 17 9 9 18 7 -9 3 -12 -7 -16 -1 -16 4 16 0 0 0 1 46
  12 -16 9 -3 -20 10 -10 -6 8 0 7 5 -11 23 18 7 -10 -9 4 -5 -12 -7 3 -8 11 23 4 -1 1 3 1 -3 2 6 0 -5 2 3 -2 3 5 36
  -5 -17 16 -9 10 -5 -17 2 -9 5 -13 4 14 -2 -6 -4 -12 0 11 8 -11 0 -4 -1 4 2 -8 -9 -11 -1 -14 -9 5 -7 -2 -6 1 1 -13 -1 1 -3 38
  12 3 -2 -8 3 -4 2 3 -8 -9 -8 15 -8 6 6 9 7 3 -18 -15 1 -3 12 4 -3 -2 -3 1 4 2 -7 11 5 12 -4 -2 -9 -6 -3 -3 -4 2 -5 40
  14 14 -19 8 9 -13 7 22 6 11 3 -2 0 -20 -12 -11 18 11 -17 7 3 0 -3 12 9 2 -2 1 -8 4 -1 -4 -14 3 -12 2 -18 -8 -4 -6 -1 -8 -10 4 50
  13 -4 1 -6 -12 -7 -15 9 11 10 10 0 -1 -2 5 -5 -6 -10 0 -1 -15 1 16 2 7 11 6 5 -10 12 4 1 -11 4 -3 -11 -9 3 -3 11 5 4 6 -3 10 48
  -11 4 -13 -8 7 -5 1 11 3 5 -5 5 18 -18 -8 -5 8 -6 2 16 6 -4 7 -2 -15 -6 -8 0 -2 1 -1 -10 -7 -7 10 -4 -4 7 3 11 2 -14 3 -13 2 4 36
  10 -6 2 -2 -13 8 -6 -2 6 2 2 0 -10 14 3 5 -6 -12 7 -7 -9 -2 7 4 1 14 2 5 4 5 4 4 -7 9 -5 -1 -3 0 -6 3 4 17 -6 -1 -2 10 -6 28
  -3 15 10 23 0 -14 -12 8 7 1 12 -20 -14 -9 0 -11 5 10 -7 -7 0 -3 -21 21 0 -9 13 10 4 11 17 15 3 11 -10 5 3 -5 -5 -15 -13 -10 -5 -2 9 -2 -2 -2 36
  -6 -7 -1 1 -11 9 2 -9 1 1 20 6 10 -7 1 -10 -1 -8 8 16 -5 3 4 -8 -4 8 6 -4 -2 -4 -1 -20 -9 -11 0 -3 8 9 1 6 16 6 3 -18 -5 9 8 -1 -11 42
  -2 15 -5 -8 3 -1 1 -2 -11 9 -5 6 7 -7 -6 4 0 -7 -7 1 7 4 13 2 -15 -5 -10 8 0 7 -8 3 4 15 -6 -8 -10 0 -6 2 -12 -4 0 11 9 7 1 0 -3 -2 34
  -11 1 3 2 8 -11 -10 11 -10 11 0 -4 7 -18 -8 -7 12 2 -6 12 3 0 -9 -2 0 -10 -5 0 1 -4 -6 -7 -1 -6 -8 -15 -1 5 -3 -4 -12 -4 10 -2 13 -5 5 -10 9 5 6 36
  11 -7 6 0 2 -4 -2 13 -1 -4 -7 11 -7 5 1 6 2 7 -14 -9 -5 -8 5 0 14 4 3 -5 -6 0 -2 -5 0 -1 -3 6 1 -5 6 0 5 8 3 10 6 3 -10 0 -4 -9 -2 -2 28
  6 2 -12 -8 12 -8 -4 24 0 2 -17 18 10 -5 1 6 16 4 -14 -2 12 -10 10 4 0 0 -10 4 5 5 -8 5 -3 9 -5 -1 -17 -4 -1 7 -3 -7 4 18 13 -2 10 -3 0 -24 6 5 10 46
  2 -15 -5 -17 2 14 3 -8 -7 2 -7 10 12 2 -1 9 -9 0 10 9 -3 4 20 -11 3 9 -6 -7 8 -13 -21 -9 5 -11 6 -5 -8 1 -4 9 16 5 9 3 -8 7 -3 3 -22 6 6 -8 0 1 36
  -4 8 -11 1 3 6 -4 1 -5 2 -16 4 12 -6 -5 -8 -6 3 3 1 16 -7 -7 -1 -16 -9 -8 -9 13 7 -4 -2 -6 4 3 1 -3 10 -4 -2 -7 -8 4 -5 -3 -11 6 -7 1 -3 6 2 -4 10 0 34
  13 -10 13 9 -4 -4 -4 2 -7 -14 10 7 -14 10 9 -9 -4 1 -2 -8 -13 -6 1 2 16 12 13 -5 -4 -6 -3 -7 -5 2 -10 10 -3 -4 -14 -3 6 10 6 12 -1 8 -16 3 0 1 0 -4 14 0 7 -3 34
  -14 -12 -4 -21 -6 18 -12 -10 6 10 -9 8 23 5 6 20 -4 -1 11 8 10 3 22 -11 -10 12 -14 4 21 -4 0 7 11 -6 13 -16 3 12 11 17 8 6 5 -8 -20 3 7 6 -13 5 2 -3 -8 6 17 8 -13 50
  -6 16 10 24 4 -9 -2 12 -9 -5 10 -7 -18 -4 -7 -4 5 8 -1 -14 11 1 -16 13 -2 -14 8 20 0 12 16 12 -2 -1 -11 7 3 -3 -4 -3 -11 -11 -5 1 5 -9 -7 -8 21 -7 3 8 3 5 -18 5 5 -14 52
  12 -3 -14 -14 9 -10 2 2 0 -2 -12 0 -6 -2 -9 1 -2 6 7 0 -14 3 10 6 -1 11 -11 -3 -7 1 -7 -1 -14 -3 7 0 -20 -9 -8 2 -3 -5 3 9 17 8 4 6 -8 -10 3 -8 1 6 9 -5 3 -9 -14 50
  1 -2 7 6 -6 -3 -5 7 10 9 12 -10 -5 -3 5 -2 -1 -1 -1 -2 -3 5 -4 0 9 0 8 10 -5 3 6 4 4 -2 1 -2 3 2 5 4 3 2 -1 -7 2 12 0 0 6 4 -6 2 1 -4 -4 -8 -2 -2 3 -7 22
  5 -8 -13 4 0 -4 6 13 -4 -21 -6 14 -6 -4 -4 3 23 3 -1 -2 6 -8 5 -11 5 6 -5 -8 7 -1 -4 -16 -16 -8 2 2 1 0 8 11 4 0 -5 2 8 -11 4 -2 1 -1 -14 8 6 16 -6 1 5 -6 10 6 -4 48
  -1 3 -8 -6 3 -13 2 -3 -9 -7 -11 -12 -9 5 -1 -11 0 -9 9 -9 6 3 -13 -19 -4 -15 -10 -5 4 -4 1 7 -14 -3 7 -14 -5 10 -4 3 -27 -6 -3 3 2 -6 -1 -2 2 -12 -4 3 -9 2 -8 7 -4 -8 1 12 -1 7 46
  6 6 -13 -12 -11 10 2 -18 15 4 -3 -1 2 12 15 6 -10 -9 7 -3 -1 7 9 6 -12 14 0 -3 5 13 2 12 1 17 7 4 -5 1 1 2 -4 8 -8 -1 -6 10 1 14 -4 -5 4 -17 -11 -3 4 1 -10 14 -22 10 -1 -14 5 40
  -11 -16 6 -9 -9 -2 -16 5 9 14 14 4 11 4 18 5 5 -7 3 14 -1 -5 -5 -22 12 15 -3 14 2 -5 -1 -7 10 -9 3 -15 1 4 11 10 7 16 5 -7 -8 -1 7 0 -5 14 -8 16 -1 7 5 -3 -4 23 -8 -17 10 9 -11 -9 54
  14 -6 2 -2 4 10 7 0 -8 -3 -17 13 4 0 -4 4 -9 8 -8 -6 0 2 8 4 10 1 -5 -17 -7 -2 -20 -9 12 1 -1 9 -1 1 -1 -9 12 2 12 16 1 1 -12 -7 -9 -6 7 1 13 2 10 1 11 -10 -9 1 1 -5 -9 -3 -9 42
  -13 -4 8 -16 11 -20 -7 6 -1 17 -5 -9 14 -6 -9 -1 3 -14 -1 9 -7 6 -7 -17 7 -12 -10 5 -19 -5 -3 -5 -1 -7 3 -16 2 -6 4 6 -10 -8 12 -10 3 9 15 -4 -6 1 0 8 2 -2 -4 -6 -12 -1 -13 4 8 -11 7 -5 8 -4 46
  -13 -11 8 0 -1 9 0 -16 3 4 12 -14 11 -2 -3 0 -7 4 16 17 -5 5 -1 -7 6 10 -4 -3 -1 -22 -3 -5 10 -14 5 -4 3 0 -1 -2 10 4 3 -18 -9 -4 6 1 -5 16 -4 1 -12 -20 9 -7 -7 14 -16 -5 0 -11 -6 3 11 -6 4 36
  -4 16 -6 8 15 -17 4 9 3 4 -4 -11 6 -12 -3 -16 6 5 -6 3 9 4 -17 8 3 -12 0 1 -5 2 4 6 -3 5 -2 9 -6 0 -4 -11 -12 -21 4 -5 14 -2 10 -12 18 -14 -1 6 -5 9 -15 8 -6 -11 6 3 1 -2 13 0 -12 -1 11 -5 36
  11 -7 2 7 -8 -4 -10 5 5 -13 -8 -9 -14 11 4 2 -5 3 5 -10 -4 -9 -1 -4 9 5 1 -7 2 3 2 2 -7 4 4 1 -2 1 -2 5 -4 4 0 3 -5 0 -4 3 5 -14 -9 -7 4 7 -4 4 11 -6 1 7 -2 13 9 1 -5 3 -9 -7 5 30
  13 11 7 2 -18 -16 -22 -7 14 7 2 -21 -18 22 9 9 -18 0 3 -15 -3 -3 -17 -2 10 10 -5 13 -10 27 6 26 3 32 -10 -16 -5 -10 2 3 -25 6 -5 6 11 12 -13 9 13 -16 14 -3 -1 1 -9 2 1 -8 9 5 2 -13 7 10 -6 0 1 -9 6 13 62
  -6 8 -12 9 5 -6 5 0 -5 -2 4 -10 6 -12 -3 -25 5 -1 10 7 11 1 -16 -7 -1 -8 -5 -4 5 -8 -1 -6 -14 -6 -1 -2 -10 9 -8 -7 -9 -12 -2 -11 8 -8 9 -8 7 4 -3 9 -16 1 -6 15 -5 -5 4 1 -3 7 23 0 -3 -10 -4 6 17 3 -2 38
  11 -18 -1 2 6 9 8 9 3 -8 -8 8 7 -2 -5 4 4 14 -9 4 3 -5 4 -4 21 6 -1 -20 2 -18 -16 -16 3 -11 1 9 3 -3 3 -1 23 4 3 5 3 -7 -4 -4 -11 -3 -13 -4 15 7 11 -3 8 -2 -13 -2 2 8 -7 -9 5 18 2 3 -2 3 -17 -6 38
  4 -3 7 10 8 -3 7 -3 -5 -15 7 -1 -13 -1 -2 0 8 18 -6 -4 -10 -1 -1 15 7 2 10 -6 1 -11 1 4 3 -5 -3 9 2 -13 -5 -11 3 -2 -4 14 4 -13 -12 -6 6 -6 -8 0 6 -3 -2 -9 11 -16 6 11 -5 9 0 -9 -13 5 -13 -1 -2 3 -7 -3 6 32
  -4 -8 -10 -1 -4 -3 -6 14 -4 -3 0 2 10 -20 -12 -3 21 -7 -2 10 3 -4 16 -16 -2 -5 -7 -7 10 -9 -9 -23 -18 -12 3 -21 -1 7 3 14 6 -2 -3 -5 8 6 15 -4 -4 15 0 15 -3 6 2 2 -1 2 -3 0 3 24 2 -11 15 -4 4 1 -4 3 -13 11 4 -5 42
  5 7 10 20 -3 -17 -15 20 0 -2 5 -16 -17 0 -3 -13 6 -1 -10 -10 -5 -12 -17 4 7 -17 11 1 -3 4 16 6 -12 2 -14 -7 0 -1 -5 -7 -12 -2 -6 4 11 2 -7 -1 20 -9 -5 7 10 3 -18 0 10 -20 22 -9 1 4 7 -15 -7 -6 -2 -12 8 13 16 4 -4 4 4 44
  1 15 5 14 -12 11 3 -13 10 -2 4 -9 -16 15 10 8 -17 0 7 -13 5 6 -8 21 -9 4 12 10 -1 19 16 19 9 18 1 19 9 1 3 -5 -2 2 -10 -5 -14 -6 -5 8 13 -7 1 -13 -7 -10 -13 1 -4 -1 13 -13 1 -11 -9 13 -12 -1 -13 -2 4 8 16 -7 -13 -4 -19 2 42
  -14 -4 -6 -22 -1 8 -5 -4 2 12 -14 14 21 -5 0 14 2 -5 1 9 9 1 11 -9 -15 1 -14 3 9 9 -5 -4 12 -3 13 -13 9 8 14 10 0 1 7 -8 -10 -4 15 0 -7 7 0 7 -7 5 3 4 -21 26 -12 -7 2 -1 -7 6 21 -6 11 5 -5 -14 -11 -5 -5 -14 4 -21 -3 40
  -14 -8 3 3 3 12 -8 10 -5 8 -8 -11 6 -12 -9 -1 -4 -1 11 8 5 4 3 -3 -12 -15 -1 0 14 -11 5 -2 2 -25 13 -9 6 21 -1 4 8 -5 8 -11 -15 -2 9 -4 2 4 -8 8 -8 -4 5 3 -8 17 4 -13 5 -3 -1 -8 13 -4 1 9 1 2 -19 2 2 -7 7 4 3 7 42
  -2 -8 7 -9 -4 9 -14 -10 -2 -5 -13 10 10 -1 -4 17 -10 15 3 -5 6 -2 12 5 -4 8 -12 -9 7 9 -14 3 14 7 4 -8 6 -3 7 6 6 -3 12 8 -10 -1 0 -6 -3 -1 8 1 1 7 9 6 -2 14 -1 -5 -3 -5 -13 2 2 14 -6 1 -7 -3 4 -8 5 -2 -2 -12 2 13 -1 36
  -13 12 15 -5 -9 4 -11 -24 -4 10 4 -8 -8 8 8 2 -18 -5 8 -10 -9 7 -2 18 -20 -5 9 9 4 13 14 21 14 12 4 -8 10 2 -1 -9 -16 3 -2 -2 -14 -1 -8 6 11 1 10 -1 -9 -19 -6 2 -8 7 2 -6 0 -29 -1 16 -10 -6 -3 3 -4 -12 10 -6 -28 1 -19 -2 20 12 -1 9 60
  12 -16 -8 -9 -4 3 -12 6 0 -6 -7 13 -2 -6 3 -6 -4 5 5 4 -11 -9 11 2 -7 17 -1 -14 8 6 -10 -13 -14 -5 7 -1 -10 6 -9 5 8 8 13 4 1 13 4 6 -7 6 -5 1 3 8 13 3 12 4 -15 21 -3 13 0 4 6 1 -10 -4 -10 7 -13 1 3 4 7 -10 -16 5 0 5 -7 44
  -9 -16 -6 -22 7 -7 -16 14 -3 5 -22 9 19 -3 -6 8 6 1 6 6 7 -4 8 -23 1 2 -23 -1 6 -4 -5 -4 -3 -20 15 -22 -3 9 7 20 3 -6 16 0 0 0 14 -5 -9 -5 -7 8 3 22 8 9 -6 24 -4 7 -2 13 8 -9 22 -5 16 -2 6 5 -5 3 7 -10 12 -1 -19 17 15 7 -15 9 52
  -14 19 -8 13 5 -12 12 9 6 10 13 -18 -1 -19 -3 -17 11 -9 -1 9 7 10 -16 -1 -8 -14 6 13 -9 -1 15 -6 -7 -5 6 7 3 6 7 -3 -9 -13 -12 -22 11 -2 19 -9 14 7 -3 8 -8 -9 -22 0 -15 -12 8 -5 14 2 7 -3 4 -16 18 4 20 -6 -7 15 -9 -9 12 4 6 1 7 -21 -2 -15 -7 54
  3 0 4 -3 -5 -11 0 -1 6 -3 -1 2 -6 7 7 0 -8 1 0 0 -6 -6 -10 -4 10 8 -2 -2 -20 8 1 -5 9 4 6 1 7 -1 13 1 -4 1 2 -4 3 0 8 -5 3 0 -5 2 6 -3 -10 -7 -3 -12 -5 1 4 1 -3 -3 5 5 10 2 6 3 10 -3 2 -1 -4 2 5 4 -8 3 1 -2 1 10 30
  14 5 -14 -1 6 2 13 16 -8 -5 -19 19 -7 -3 -3 -1 3 6 -14 -4 5 -10 8 6 -1 -5 -2 -8 0 5 -5 -10 -1 -1 0 11 -5 10 3 -3 -4 0 -2 12 12 -4 -3 -4 -2 -10 2 4 15 14 -1 4 7 -12 3 9 -4 14 1 -6 -8 14 -12 -18 1 5 -7 -2 6 7 0 5 -4 -1 -5 -4 -7 11 -1 -4 7 38
  9 13 -12 10 -12 4 2 5 5 -10 -6 -4 -18 -3 1 -4 -5 6 -5 -9 12 -7 -2 8 -18 -4 4 -3 12 14 11 4 -3 7 10 1 5 19 5 0 -6 -4 -16 3 5 0 3 1 17 -7 -2 -4 -3 0 -12 6 -3 -8 10 -2 1 5 6 4 -10 -5 -16 -10 6 7 8 6 -7 1 -2 11 14 -3 4 -3 6 3 -8 10 7 13 38
  -8 -7 -4 -4 9 15 11 -1 2 11 6 14 15 -4 2 4 -2 4 10 14 -1 9 8 8 4 14 2 8 -4 -8 -2 -9 4 -18 3 20 -5 -1 2 -1 24 2 9 -14 -6 -2 8 0 -14 14 -5 -1 1 0 13 -2 -3 18 -5 0 0 2 -22 -2 16 1 -1 12 -8 -11 -27 -7 6 -4 -1 -22 0 13 11 2 -5 6 6 1 -4 -1 -17 42
  

• DIVISORS
  1
  

• MINIMAL_NORM
  8
  

• KISSING_NUMBER
  168498000
  

• HERMITE_NUMBER
  8
  

• SUBGROUP_ORDER
  2^3*5*23
  

• SUBGROUP_NAME
  C115 : C8
  

• SUBGROUP_GENERATORS
  2
  88 88
  -62 -107 -125 64 35 -12 -16 -20 -27 16 23 -15 10 32 -26 0 -29 -24 -44 -42 -14 -13 18 4 -4 18 15 5 -15 15 -4 1 19 12 0 -3 -1 6 6 -2 2 4 -3 -1 -5 -1 3 -7 7 -6 6 0 -4 3 -4 -7 2 -4 3 -5 4 -2 1 -10 -9 0 0 2 9 -9 4 2 0 -3 -6 2 -5 2 1 2 3 5 -1 1 2 -1 1 -2
  40 68 54 -29 -34 28 -31 -11 44 -36 -37 44 0 -16 -35 -10 24 3 7 37 -15 29 -39 -2 11 -2 27 19 5 -18 -13 1 8 -36 -18 -3 -30 11 18 27 -7 12 -8 6 -18 -4 -8 -14 3 -11 15 -1 1 3 -9 1 1 -7 -10 -1 -1 0 1 0 -10 -11 -1 0 3 -1 2 3 3 -3 0 -5 4 -4 0 -7 4 8 2 8 -4 -6 -3 7
  13 -33 -106 -38 11 66 -13 -7 -54 -67 -4 -29 105 -43 -61 6 -60 -12 -58 -65 -79 13 -66 -5 19 67 93 46 1 -5 25 -5 29 12 -23 -13 -41 27 18 42 -27 20 1 -7 -22 -18 3 -18 13 -19 19 -5 -5 6 -24 -26 -9 -9 15 -9 1 3 -7 -5 -3 -7 12 4 3 4 -5 2 3 4 10 12 -5 -2 1 -1 3 2 0 2 -4 -5 -1 -5
  -34 -44 -108 7 -34 37 -13 31 -56 -65 -3 -21 93 -19 -39 34 -34 -1 -55 -59 -69 23 -65 1 13 43 72 40 -16 -12 7 -5 18 6 -18 -11 -33 21 9 24 -19 21 2 -3 -15 -5 7 -12 6 -17 13 -4 -9 -4 -9 -9 -4 -2 0 -4 -3 -5 5 -6 -13 -6 -2 1 7 -6 0 -2 2 -1 -3 0 -4 3 -2 1 6 4 0 4 2 -1 2 -1
  29 115 159 -35 -25 -4 21 9 77 5 -29 42 -71 -19 44 1 45 20 53 64 40 8 11 -4 2 -49 -53 -28 13 -6 -8 0 -16 -19 6 -2 8 -10 -20 -16 11 -18 -5 2 9 13 -12 17 -21 19 -13 7 8 4 10 19 8 -1 -9 11 -3 6 -1 12 10 -3 -2 0 -5 5 5 4 0 0 -1 -5 5 -4 2 -4 -1 0 3 0 -1 -3 -4 7
  -30 -171 -192 95 103 -32 -91 -81 -38 54 43 2 -44 65 -56 -59 -27 -58 -62 -33 25 -17 68 1 -6 11 13 12 -30 14 -17 26 24 25 27 18 28 -7 21 -10 -1 4 8 -16 13 -18 16 -14 16 -11 4 -10 -7 -3 -1 -15 -6 7 16 -12 4 -12 -7 -7 2 8 3 8 -2 -3 -5 1 -2 4 3 6 -3 0 0 7 -4 -6 -4 -2 -1 8 5 -7
  11 30 66 35 36 -20 -70 -47 97 38 -5 74 -132 36 -2 -33 52 -29 32 78 74 7 51 2 -2 -63 -57 -22 -9 -8 -33 13 1 -26 23 14 21 -16 4 -22 17 -14 -9 0 11 6 -8 3 -13 11 -1 5 8 2 11 15 11 -1 -13 1 2 -3 0 4 -2 -2 -8 0 5 -1 7 7 -3 0 -5 -9 1 -2 -2 -4 0 4 3 1 -1 1 -1 7
  -8 -41 -88 -15 -28 13 25 31 -82 -39 8 -47 117 -9 -29 30 -42 18 -34 -62 -71 8 -50 5 -3 51 62 24 9 5 20 -14 20 9 -25 -10 -33 24 10 25 -10 19 -4 -1 -23 -3 0 -12 10 -16 13 -4 -8 3 -19 -15 -5 -11 4 -8 2 4 1 -6 -7 -8 8 1 4 -2 0 0 5 -1 -1 5 -3 2 0 0 3 3 -2 4 2 -4 -1 -1
  -44 -61 -13 73 -36 -87 64 18 -30 58 4 -15 19 11 27 23 -13 7 4 -26 24 -17 21 2 -20 4 -15 -24 1 3 9 -4 -17 12 4 -3 10 -14 -13 -11 5 -1 6 11 6 7 6 10 2 3 -14 -1 -3 -10 15 4 -6 9 -4 7 -4 -4 5 0 -1 8 -14 -5 -2 -6 -2 -10 -5 -1 -8 -5 0 5 2 7 -2 -2 -1 -4 7 4 3 -2
  22 41 56 -36 -32 2 22 23 -15 -22 -15 -30 15 -6 47 10 33 27 25 17 4 -6 -15 16 -4 -19 -23 12 -3 -2 -7 -2 -19 -10 -3 3 15 1 -4 -6 6 5 -3 -1 10 11 4 8 -4 7 -2 -2 3 -4 8 12 3 5 -4 4 -1 1 4 2 2 4 -2 -2 -2 -1 -1 -6 2 -2 -1 -1 6 3 -2 0 -1 1 -2 -1 -1 1 -2 1
  -14 41 24 -23 -68 23 27 59 -17 -55 -21 -21 65 -27 22 50 9 25 5 -11 -37 15 -61 8 4 8 17 22 -6 -16 3 -13 -10 -11 -18 -9 -18 9 -3 9 -4 11 -4 6 -7 8 3 3 -5 -2 5 2 0 -9 4 6 4 0 -11 0 -3 -2 13 -2 -13 -3 -9 -7 9 -5 3 -7 1 -4 -4 -8 -2 4 -4 -1 6 7 1 0 2 0 3 3
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  -121 -181 -199 142 142 -30 -33 -103 60 57 6 30 -81 17 -56 -42 -59 -80 -72 -58 17 -49 76 -25 7 36 21 -12 -28 32 -6 18 33 14 -6 -29 1 4 -9 0 7 -12 -5 2 -9 -13 -11 -6 11 -1 2 6 -5 11 -6 -12 7 -5 14 -3 8 -9 -9 -14 -1 5 4 6 8 -10 5 3 -6 -3 -7 5 -5 -3 7 7 8 0 -5 1 5 -3 -3 0
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  77 133 119 -99 -23 68 -27 51 8 -76 7 -7 23 -48 49 43 35 19 45 33 8 29 -51 10 13 -43 -17 14 -5 -30 5 -9 -15 -4 16 13 13 4 -12 -10 -11 0 -7 -15 14 11 -4 18 -23 18 -7 3 9 3 8 10 4 2 -7 11 -10 8 2 9 6 2 -2 -1 -2 7 4 1 2 5 4 -1 2 2 -4 -4 2 1 6 -3 2 4 2 -1
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  31 -70 -64 18 -35 -58 48 -9 -70 27 -3 -25 87 -15 -42 -1 -44 -3 -31 -56 -41 -7 -21 -1 -9 66 54 9 18 1 17 0 -1 3 -17 3 -24 6 15 34 -16 14 9 7 -18 -22 7 -21 24 -22 8 -9 -10 -6 -8 -16 -21 9 9 -10 0 -5 -5 -6 -2 -1 -1 2 -6 0 -12 -7 -4 2 3 6 1 2 6 5 -3 -7 -4 4 -3 0 2 -7
  -66 -136 -209 7 -72 2 85 55 -155 -32 8 -64 211 -32 -105 22 -136 14 -114 -155 -141 21 -87 -24 6 148 156 34 25 1 59 -15 23 52 -48 -25 -72 19 23 61 -37 34 38 15 -28 -27 25 -35 34 -54 17 -14 -27 -19 -24 -38 -34 -1 27 -15 1 -6 -3 -3 -3 -8 7 3 -9 10 -21 -3 -2 9 8 12 -5 3 2 6 -2 -12 -5 1 0 0 7 -8
  -13 10 20 26 45 -13 -34 -34 59 38 -19 45 -107 39 4 -50 26 2 16 49 40 -6 62 -16 -2 -27 -33 -16 -5 8 -17 11 -1 -9 1 -3 19 -14 0 -20 20 -14 3 4 9 -2 3 3 0 3 -1 1 0 -6 4 9 5 -3 0 -5 9 -2 -2 7 6 -1 5 2 -4 2 1 4 0 -2 0 -6 4 -4 -1 0 -2 -2 -3 1 -6 -2 -5 2
  -44 3 -41 -1 1 54 -13 6 45 -72 -29 6 4 -49 -4 31 -13 -18 -26 -20 -34 2 -40 1 24 9 35 27 -25 -5 -2 -9 11 -16 -21 -21 -19 19 -10 12 -5 6 -14 1 -15 0 -14 4 -10 -2 14 7 -3 8 -4 8 11 -2 -8 2 2 1 3 -10 -10 -4 1 5 20 -14 8 2 -10 -2 -10 0 -9 0 1 3 8 5 2 3 1 -5 -8 2
  57 87 164 -16 17 -33 9 -26 87 61 8 29 -97 14 56 -9 74 -3 92 87 83 -26 54 14 -15 -68 -97 -46 19 7 -21 1 -24 -31 25 19 32 -14 -12 -24 20 -22 -17 4 2 14 -26 12 -10 25 -12 9 14 15 9 15 14 -1 -13 10 -2 5 0 -4 2 6 -1 -4 2 -3 10 0 1 -7 -3 -3 2 -4 -1 -4 0 6 2 -1 5 0 -1 6
  32 106 218 45 75 -54 -67 -67 145 120 4 96 -260 97 100 -63 130 13 120 180 171 -13 153 5 -28 -165 -187 -76 -3 16 -57 21 -18 -31 52 24 78 -40 -17 -83 51 -45 -19 -9 41 31 -8 38 -31 47 -28 5 25 7 22 32 32 -9 -17 14 4 9 1 18 16 3 -2 -6 -5 3 17 8 11 -7 -3 -16 9 -9 -4 -9 -4 9 6 -5 -1 -2 -2 10
  36 193 202 -129 -28 85 13 58 74 -92 -36 -40 -57 -15 136 37 128 61 93 84 12 -23 -17 28 4 -81 -91 11 -16 11 -39 -9 -32 -52 -7 -5 31 14 -33 -39 25 -17 -31 -10 13 31 -22 34 -30 41 -6 14 11 14 12 44 35 -1 -25 9 2 8 11 -8 -9 5 7 -3 11 -16 18 -2 8 -20 -8 -11 7 -3 -2 -3 5 12 0 7 -1 -2 -12 11
  -19 58 155 38 32 -53 8 -9 106 106 22 61 -171 56 77 -18 69 13 90 100 115 -16 105 -3 -23 -102 -139 -77 9 11 -23 -2 -24 -4 35 14 52 -37 -25 -61 35 -38 -5 7 23 27 -9 27 -22 32 -26 11 17 2 22 24 19 -4 -13 13 1 7 4 10 9 5 -7 -6 0 2 14 5 1 -4 -7 -10 4 -2 -2 -5 0 2 5 -8 3 -1 2 4
  55 155 197 -48 -3 28 -46 10 97 5 7 51 -118 10 79 7 90 12 93 123 87 16 33 14 1 -116 -110 -36 0 -13 -25 1 -16 -26 38 23 39 -18 -20 -43 17 -23 -17 -12 18 25 -18 31 -33 33 -16 7 20 13 12 23 23 -4 -18 18 -5 11 4 6 6 2 -2 -4 5 -1 14 7 2 0 -2 -9 0 -6 -6 -5 1 8 7 -4 3 -1 0 7
  -74 55 0 -83 -47 113 1 87 -21 -84 -4 -41 0 11 38 9 21 45 -16 6 -52 12 -35 4 24 -8 -16 28 -16 3 -9 -4 -12 19 -5 -4 10 -5 -8 -18 8 -3 21 1 17 16 18 2 -9 -1 2 5 -2 -14 6 15 10 2 2 0 4 0 13 4 -4 0 6 -1 2 0 0 4 6 -5 0 0 0 2 -9 -7 1 -2 -5 -3 -6 -1 2 1
  21 113 127 -48 38 64 -57 -33 124 -43 -45 32 -126 -2 56 -13 94 -2 54 94 47 -16 16 13 11 -72 -63 14 -24 2 -41 4 -9 -51 0 -3 23 2 -7 -25 21 -13 -25 -3 11 16 -13 16 -23 27 5 11 12 10 8 29 29 -3 -19 3 8 4 4 -4 -8 1 0 -2 16 -11 14 6 3 -11 -6 -7 2 -7 -1 -6 5 12 2 4 -5 -5 -10 9
  -13 -79 -52 115 121 -70 -50 -115 71 76 -12 37 -115 28 5 -41 17 -61 1 19 75 -52 91 -7 -7 -20 -37 -20 -21 19 -25 27 10 -23 13 -8 27 -2 -11 -17 18 -19 -19 -7 5 -7 -14 9 2 22 -7 2 4 13 3 2 10 1 0 1 5 -7 -9 -8 0 9 -2 0 1 -12 5 -5 -1 -6 -5 -4 3 -3 6 8 2 5 -4 5 4 0 -6 1
  43 58 126 15 67 -28 -43 -57 87 98 22 62 -149 44 38 -40 47 -12 77 106 105 -12 98 -4 -15 -81 -104 -59 16 10 -15 5 -1 -13 32 23 38 -27 -1 -39 27 -30 -12 -2 10 9 -11 16 -13 26 -13 5 17 14 0 5 14 -12 -6 6 5 13 -3 6 7 -1 1 -6 3 3 9 7 5 1 3 -3 0 -8 -2 -8 -3 4 5 -6 -2 -4 0 0
  7 118 122 -83 -21 64 35 49 24 -30 0 -17 -10 -25 62 30 25 38 45 32 -7 0 -13 -2 9 -27 -40 -18 13 6 6 -14 -16 -3 -7 -5 4 2 -18 -16 9 -15 -1 1 5 14 -6 17 -16 16 -6 5 6 6 -1 11 12 -6 -3 10 0 13 4 4 5 -2 5 -6 3 3 4 3 5 -3 1 0 -1 -1 -3 -8 0 2 3 -2 2 -5 -1 -1
  -11 71 144 33 23 -43 19 -23 121 74 -19 61 -151 -2 70 -18 52 -6 66 76 94 -24 74 -12 0 -70 -101 -59 3 1 -15 9 -30 -14 25 2 37 -30 -38 -34 21 -37 -1 5 16 13 -18 27 -20 32 -24 11 11 5 21 21 12 8 -8 18 -4 2 -2 7 11 9 -8 -3 -5 -2 7 -3 -5 -1 -4 -9 3 -2 3 1 0 -4 0 -5 6 -2 0 2
  15 44 65 -50 -8 27 6 18 -16 -2 15 -44 -23 45 57 3 63 33 43 47 6 -21 21 23 -13 -33 -59 1 3 23 -23 -2 -18 -14 2 13 28 2 8 -31 23 -5 -4 1 13 14 8 3 0 7 1 -1 3 -1 4 16 14 -3 -5 -4 7 2 10 -6 -6 3 7 -5 4 -4 3 1 9 -14 -3 0 1 -1 -7 -4 -3 7 0 3 -4 3 -3 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 : C8 = <zeta> : <g2>, zeta of order 115, g2 of order 8 with g2^4 = -1 and g2^-1 zeta g2 = zeta^22 (C8 acts through C4).
  X599 = (P, b), an ideal lattice over L = Q(zeta), zeta = zeta_115 (rank 88): P = (599, zeta^2 + 87 zeta + 322) 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; X599_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_22(u), sigma_22(zeta) = zeta^22, lambda in Q(zeta) generating the fractional ideal
  P sigma_22(P)^(-1) (in X599_construction.gp); g2 o g1 o g2^(-1) = g1^137 as maps (in matrices: g2^(-1)*g1*g2 = g1^137);
  g2 has order 8, g2^4 = -1, and g2^2 lifts sigma_24 = sigma_22^2.
  <g1, g2> has order 920 = 2^3*5*23: an extension of the Galois stabiliser <sigma_22> = {sigma_1, sigma_22, sigma_24, sigma_68}
  (cyclic of order 4) 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(X599) of the group of multiplications by roots of unity. The full automorphism group has not been determined: |Aut(X599)| >= 920.
  A vector of norm 8 (the witness of the paper, transported to this basis), as a row of coordinates in this basis:
  NORM8: 32 66 105 -22 1 -7 1 7 25 24 9 2 -59 15 64 1 49 20 55 57 48 -7 36 9 -8 -58 -71 -26 1 5 -13 1 -19 -9 18 14 33 -11 -13 -31 16 -15 -6 -5 16 12 -3 20 -13 20 -12 1 8 4 10 15 10 3 -6 8 -2 6 2 4 7 4 0 -3 -1 0 4 -1 1 -2 -1 -4 1 0 -2 -1 -3 0 2 -2 1 1 -1 -1
  

• LAST_LINE
  

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  Haftungsausschluss/Disclaimer
  
  
  Gabriele Nebe