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 31 05:03:14 PM CET 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
  X1151s24
  

• DIMENSION
  88
  

• DET
  1
  

• GRAM
  88 0
  30
  0 34
  -8 14 40
  -13 0 19 48
  15 5 -14 -24 48
  -14 12 5 -6 14 52
  1 -12 -21 -4 17 -10 54
  -8 -12 -9 -14 -9 -4 -11 48
  -5 13 14 -9 -1 17 -21 16 42
  10 -8 7 -1 8 -17 1 1 6 38
  2 8 15 14 7 4 0 -9 12 14 38
  4 1 -3 -17 -5 11 -14 -2 13 1 -13 46
  11 9 8 5 3 -3 -15 -10 9 15 16 11 38
  -6 -5 11 -8 6 0 -2 9 15 10 -1 4 5 42
  0 4 -1 -17 -7 0 3 -1 9 -1 -15 27 -6 8 44
  10 -5 1 -11 15 1 0 9 -3 1 -1 -3 -10 1 2 42
  14 7 3 -7 11 4 -15 -15 7 3 12 13 12 -3 -4 8 44
  15 12 -4 -5 8 -16 18 -3 -8 5 0 -5 4 -7 7 11 -3 48
  -2 -1 -12 6 7 14 -2 5 -9 -6 3 -2 -5 -16 -10 7 0 -5 40
  -2 8 19 4 0 9 -13 -13 0 -6 10 4 -1 -4 1 14 22 -9 -2 44
  -4 6 7 -13 4 18 5 8 12 -2 3 10 -8 1 18 18 -1 16 4 11 44
  12 9 13 -9 3 -17 -3 6 6 10 1 -10 3 7 0 4 -7 13 -19 0 2 42
  -2 16 11 -9 4 -3 -15 2 9 -1 -6 5 1 9 5 4 10 -2 -6 11 -4 15 38
  4 -14 -6 -16 -8 -6 -3 11 -7 -7 -25 16 -16 7 18 13 -6 -1 -7 -1 5 6 0 42
  0 -6 4 -13 3 6 -8 13 5 -3 -4 4 -4 11 -2 17 3 -3 -2 4 7 3 4 16 28
  8 -8 6 8 9 -16 11 1 3 25 12 -8 2 11 -7 4 -1 8 -1 -9 -3 10 -2 -4 1 40
  9 -13 -18 -15 -3 -23 19 11 -17 0 -15 -2 -5 2 7 5 -15 15 -6 -16 1 12 -9 15 4 -4 44
  -4 16 10 0 -11 1 -2 -16 3 -3 3 5 5 -12 15 -9 -2 8 -14 7 8 6 3 -3 -6 -18 2 44
  -5 10 -5 1 -6 10 10 5 5 -9 3 -6 -2 -9 1 -11 -11 13 8 -18 12 -1 -10 -10 -5 -7 7 4 40
  -3 10 10 12 -20 10 -21 5 2 -13 2 9 11 -9 2 -7 -4 7 7 8 11 -3 -8 3 -1 -14 -5 1 12 50
  10 -7 -15 1 1 -16 -1 2 -5 2 4 0 8 -4 -3 -1 12 -3 6 -1 -17 -4 2 -2 -4 -1 3 -7 -7 -2 32
  1 -14 -17 -17 3 -16 4 29 6 6 -6 -3 1 16 -7 -1 -6 1 -4 -22 -6 8 3 9 14 11 16 -16 3 -11 11 48
  3 -16 7 -6 -4 -1 -7 5 -2 11 3 3 5 12 1 1 -1 -13 -1 2 2 5 -4 6 8 1 7 -2 -6 1 0 5 38
  3 11 5 5 -11 -14 -8 -3 -9 -4 -12 2 -5 -7 6 -1 -7 11 1 2 -6 5 7 10 -2 -1 5 7 -5 10 -5 -10 -11 36
  13 12 6 -20 3 0 -21 -6 3 -2 -1 17 11 0 14 9 19 -3 0 19 4 7 15 8 10 -15 2 12 -9 6 7 -5 11 6 44
  -10 5 15 1 9 28 -3 -6 19 3 8 8 3 15 -1 -1 8 -10 0 4 12 -8 0 -11 4 4 -18 -3 5 -8 -15 -9 5 -13 -7 42
  -6 4 -8 -3 -13 14 2 12 1 -17 -12 13 -12 -11 18 0 -13 13 13 -6 19 -8 -7 12 -2 -15 4 2 23 24 -5 -5 -2 6 -3 -2 48
  9 -11 -17 -2 14 -4 29 -2 -12 5 4 -15 -11 -16 -8 3 -13 9 4 -9 4 4 -22 -2 -6 12 9 -6 9 -7 0 2 -8 -7 -18 -9 -1 44
  7 -3 -12 -9 21 17 -7 -7 -6 1 5 10 8 -9 -15 9 20 -7 18 6 -1 -18 -2 0 6 -6 -8 -5 -9 0 6 0 1 -6 7 2 -6 0 46
  11 -14 -22 -5 13 -15 27 -7 -13 7 -9 -4 -5 -3 -5 -9 -3 3 -1 -19 -15 -1 -8 -1 -7 13 10 -12 0 -23 10 11 -7 -3 -15 0 -10 21 -2 42
  4 -11 7 5 8 -6 5 -14 -13 15 -3 -4 -3 4 -5 11 -1 -7 -4 4 -10 2 1 6 4 11 0 -2 -18 -20 -5 -10 6 5 -1 4 -18 4 5 8 36
  2 -1 -2 -14 -5 -1 -19 17 2 -2 -3 -1 -6 -8 -3 5 3 -11 8 3 1 7 6 9 6 -9 2 -2 -1 5 3 9 12 1 18 -11 2 -5 6 -11 -5 34
  -13 0 -6 15 -2 15 14 4 -8 -18 -4 -20 -16 -6 -16 -5 -18 4 10 -11 -2 -6 -10 -7 -3 -5 1 -9 20 4 -8 -3 -11 -2 -26 2 20 11 -8 3 -4 -11 50
  3 -13 7 12 -19 -29 -4 2 -14 13 -1 0 -1 -6 1 1 -4 0 -6 8 -11 6 -1 10 0 11 7 0 -19 3 7 0 9 14 2 -16 -6 1 -7 0 10 5 -17 42
  5 -7 9 19 -7 5 6 -9 -8 -8 0 -11 -6 -5 -10 5 -6 6 -1 -4 2 5 -12 1 2 8 1 -7 9 6 -7 -11 6 -5 -18 8 6 10 -8 5 6 -13 22 -2 48
  -8 2 8 13 -5 8 4 -14 -9 -3 -1 0 -6 -11 5 6 1 -1 9 9 7 -8 1 -2 -2 -4 -6 8 5 -1 -6 -18 -2 0 0 7 6 -3 -2 -2 11 -5 4 -1 7 30
  -6 1 -11 -17 6 8 -1 2 -2 0 -8 13 7 -3 0 -15 -5 -6 -3 -3 -4 -5 4 2 2 -15 1 10 -4 2 -2 9 1 0 6 -5 1 -1 15 1 -2 5 -5 1 -22 -7 40
  14 15 -11 -27 11 1 -10 -4 0 -8 -9 6 1 -11 3 -2 17 5 -1 -1 -2 10 13 5 3 -13 5 9 6 -4 3 7 -3 4 23 -8 -1 -3 8 1 -7 19 -11 -10 -13 -5 12 46
  -6 -12 -1 -6 8 6 -6 21 1 2 -2 -2 -2 19 -3 13 -12 -13 10 -1 2 -2 -1 13 12 6 -1 -23 -12 5 1 14 10 -6 1 0 -3 -2 6 -8 1 10 -6 3 -8 -9 3 -11 44
  4 -13 2 12 -15 -10 -14 -4 -4 4 6 2 5 -7 -12 -2 13 -21 3 5 -15 -4 -7 4 0 0 -1 0 -9 -5 13 4 12 -5 5 -3 -12 -2 10 1 7 12 -12 12 2 1 -7 2 -2 40
  8 -11 -13 -23 5 -17 -3 25 1 7 -13 6 -4 8 2 17 7 9 -6 -2 0 6 8 14 14 3 15 -14 -12 -7 8 23 1 1 8 -11 -2 -4 2 4 -2 11 -9 8 -16 -11 4 7 9 -2 42
  -14 10 9 6 -16 12 -19 0 -1 -13 0 11 1 -3 6 -9 5 -10 9 19 3 -8 8 4 -3 -17 -7 7 1 27 -1 -11 4 10 12 -4 16 -20 4 -26 -13 10 1 4 -12 5 7 3 5 2 -6 46
  13 3 -10 -26 3 0 0 0 -1 -6 -10 9 -3 -10 7 -3 7 4 -4 -1 3 9 2 6 7 -10 15 10 7 -4 0 6 8 1 19 -4 3 3 -1 8 -5 13 -6 -3 0 -3 9 28 -14 1 8 -1 40
  7 0 -13 -12 6 12 5 -6 -9 -16 -13 16 0 -10 6 6 0 2 7 -3 5 -3 -3 12 4 -14 11 1 4 5 1 -4 0 -3 8 -1 11 4 11 3 -2 2 1 -8 8 3 6 11 1 1 -2 0 11 32
  -12 5 15 11 -15 -1 -23 11 19 4 6 0 2 12 -3 -5 5 -16 -4 5 -7 -2 11 -8 0 2 -12 -5 -6 2 3 1 -3 4 -2 6 -9 -17 -6 -12 -1 5 -3 3 -8 -6 -7 -6 2 8 3 7 -11 -16 36
  8 -8 -16 -11 -2 -4 2 -6 -11 -7 -10 8 -4 -11 -1 -13 6 -6 -4 -1 -11 -6 -8 10 -4 -11 5 4 -7 0 4 1 4 4 5 -12 2 5 9 10 -3 8 -1 5 -5 -11 16 14 -9 8 4 3 16 6 -9 34
  -8 -17 -6 2 9 18 22 -8 -12 -7 -3 3 -16 0 1 13 -2 -9 13 6 14 -22 -16 6 4 1 -2 -11 -4 -8 -9 -11 5 -12 -11 16 2 11 11 7 12 -9 6 -5 10 14 -4 -16 10 0 -6 -5 -7 12 -12 -1 46
  -2 11 2 -12 -8 4 -4 3 -1 -18 -4 -5 -11 -15 5 6 -2 0 -2 8 9 6 6 6 6 -21 8 17 12 5 -8 -3 -3 4 17 -11 6 -1 -3 -13 -8 19 -3 -5 -5 6 3 20 -5 0 -1 9 19 8 -4 3 -4 42
  7 6 10 -10 13 2 2 -17 1 5 1 10 8 14 15 3 8 2 -12 11 7 8 7 5 0 3 -5 3 -12 -3 -6 -9 7 -1 14 11 -10 -8 -3 0 5 -3 -22 1 -5 1 0 5 7 -7 -3 0 0 4 -8 -2 6 -3 36
  7 4 -15 -19 5 -6 -3 17 5 3 -13 5 -7 -1 5 -1 -7 4 5 -20 -4 11 4 11 -1 4 16 -8 8 -5 1 14 -10 10 1 -10 4 6 -1 7 -3 14 0 -4 -11 -11 2 18 5 -2 12 -2 12 5 6 3 -14 6 -9 42
  -5 5 -12 -5 9 5 18 2 7 -6 -4 4 3 11 11 -9 -8 11 -5 -14 3 -2 -1 -2 -6 3 3 -2 10 -3 2 11 -12 -6 -14 7 9 0 -9 9 -13 -16 7 -15 -2 -3 2 -2 3 -17 -3 -2 -4 4 -6 -7 1 -9 7 3 38
  -2 11 -10 -2 9 7 8 -5 -3 -7 6 -2 1 -17 2 -2 0 10 11 4 10 -9 0 -14 -10 -8 -6 11 4 8 4 -5 -11 0 0 -9 5 7 10 -3 -12 -1 -2 -6 -9 2 11 5 -5 -11 -7 4 0 3 -6 1 1 7 -3 -4 6 32
  -3 1 -11 5 -3 -5 17 -6 -1 -7 -3 7 5 0 9 -9 -2 9 -2 -6 3 -12 -5 -6 -12 -4 4 4 7 2 6 1 -14 -1 -8 -1 3 3 -2 10 -10 -16 -2 -7 -5 0 0 -9 -8 -6 -5 -4 -7 4 -4 0 8 -4 0 -6 18 11 34
  1 -16 3 6 -12 -10 -15 0 -6 2 -15 15 7 14 6 -3 1 -15 -6 3 -15 -4 5 18 8 2 -3 -9 -21 4 8 5 13 5 7 -2 -5 -16 1 3 9 -3 -13 17 2 -2 2 -10 11 9 5 3 -7 -1 2 6 0 -16 11 -12 -3 -15 0 44
  14 -12 2 -8 -8 -19 -9 1 -2 15 -11 7 -1 4 5 -3 4 -6 -11 -2 -6 11 -3 16 -1 9 6 -2 -11 -6 -3 1 11 8 8 -6 -12 4 -6 10 12 10 -18 11 -2 -7 -4 9 -1 15 8 -7 11 -6 1 15 -5 -1 5 11 -16 -17 -6 12 44
  -13 12 1 -13 -11 20 -27 4 17 -19 -11 19 4 5 5 -20 13 -26 -5 4 -8 -6 15 1 3 -25 -9 9 3 4 -1 5 5 -5 16 11 2 -28 2 -13 -16 15 -5 -15 -15 -4 15 24 -4 10 0 22 16 8 14 11 -11 13 1 7 4 -3 -4 3 0 58
  11 7 -4 -15 18 -11 9 -4 3 5 8 -4 2 2 -5 10 17 11 -12 8 -1 10 8 -2 7 8 6 4 -8 -19 4 10 -12 3 7 -1 -18 5 6 7 2 -2 -13 1 -12 -9 1 14 -7 -1 12 -11 7 -2 -1 1 -9 5 5 5 4 2 1 -10 -1 -3 38
  10 -13 -22 5 14 -22 35 -14 -29 2 -9 -10 -9 -2 -6 2 -4 15 4 -12 -11 -4 -9 2 -9 17 16 -16 -4 -18 8 8 -11 6 -19 -11 -7 22 1 30 10 -17 10 6 8 -2 -6 -5 -5 -1 3 -18 -6 4 -13 7 12 -18 0 4 9 2 13 2 1 -25 10 52
  11 -6 -13 -13 -3 -22 -8 4 -10 -2 -23 6 -3 4 3 -7 9 2 -9 -5 -17 2 10 16 -2 -4 6 -8 -12 0 11 14 -2 15 7 -15 -5 -8 1 15 -2 7 -12 8 -11 -12 7 17 -1 4 20 2 9 -4 0 19 -13 -2 4 9 -3 -4 3 18 23 10 4 12 50
  0 -2 2 -11 16 1 4 12 16 9 12 -7 3 18 -7 7 6 -3 -9 3 5 13 10 -7 8 9 -1 -7 -5 -14 4 16 4 -16 -2 9 -13 0 -1 2 -2 -2 -2 -9 -4 -10 -1 -4 6 -5 9 -12 -4 -8 6 -8 -5 -8 2 -2 5 -2 -3 -4 -6 -1 15 -2 -6 32
  -12 -5 -8 3 -13 15 -12 6 8 -18 -10 8 -6 -3 -2 -12 1 -21 3 -4 -7 -16 -4 6 4 -14 -9 2 3 4 3 7 -3 -4 -4 1 6 -6 7 -2 -7 3 10 -12 1 -1 6 2 -5 11 -4 6 2 3 7 11 3 2 -14 -2 2 0 6 9 -1 24 -10 -8 5 -4 36
  0 -1 9 17 -1 -10 8 -6 -6 13 19 -18 7 3 -16 -9 -7 1 4 -8 -7 10 -4 -11 -6 16 3 -5 7 -1 0 2 5 -4 -10 -3 -10 7 -3 2 4 -1 6 4 8 -1 -7 -7 3 5 -13 -1 -9 -8 2 -7 -4 -5 -1 2 -3 -5 -5 -7 0 -15 -3 6 -13 3 -12 34
  4 8 -5 -10 1 4 2 1 -5 -14 -17 11 0 3 17 3 -12 12 1 -3 8 5 5 9 1 -11 15 2 7 12 -6 -4 -1 10 8 -6 18 -5 -7 -6 -8 -6 6 -5 6 -2 2 3 2 -18 -1 6 7 15 -10 -2 -3 5 6 5 10 3 5 5 -7 2 -6 3 1 -7 -4 -8 34
  -4 4 8 17 7 -6 14 -17 -8 10 18 -12 13 1 -12 -8 2 7 -1 3 -6 1 3 -21 -6 14 -9 1 -1 -5 1 -2 -1 -8 -8 4 -13 2 -1 4 2 -14 0 5 1 7 1 -9 -4 -4 -8 -2 -12 -7 -6 -9 1 -12 9 -19 6 6 4 0 -14 -11 4 10 -12 5 -12 17 -7 36
  -12 2 0 -13 16 28 3 -9 7 -4 -1 12 -2 13 10 4 4 -17 -1 7 9 -16 3 -3 4 -12 -9 2 -2 -10 -10 -8 3 -15 5 19 -3 -4 13 -8 7 -3 -3 -17 -11 4 11 0 8 -5 -5 2 -5 7 4 -4 17 2 10 -4 1 3 1 -4 -10 15 -1 -6 -12 5 4 -9 2 -1 44
  -13 12 22 -5 -17 -4 -29 26 18 4 -2 7 9 17 0 -4 -1 1 -10 10 10 21 27 2 7 -5 1 2 0 19 -10 13 8 8 11 3 4 -30 -9 -26 -14 15 -8 7 -16 -8 9 2 9 -7 15 26 -1 -12 18 -9 -24 6 2 6 -4 -7 -8 6 1 20 -1 -27 8 12 -6 2 6 -1 -7 68
  15 -14 -1 -19 17 -9 3 -4 -9 17 -12 5 -2 11 8 19 3 1 -8 -1 -1 8 6 14 14 11 8 -3 -20 -22 -4 2 18 0 12 1 -17 1 5 10 25 1 -18 8 4 2 1 3 6 -2 14 -21 9 2 -11 2 7 -6 14 -2 -12 -10 -13 16 18 -12 5 8 9 5 -11 -8 3 -2 8 -10 48
  3 3 -4 -7 -7 -9 -9 -4 -6 -11 -8 6 1 -8 7 4 14 0 -8 16 -4 -2 10 3 -1 -17 1 12 -11 -1 9 -1 -1 3 19 -12 -2 -11 0 -7 -5 9 -12 7 -11 -1 5 15 -7 8 11 9 11 3 -1 11 -9 13 5 -7 -5 5 2 7 2 15 8 -3 15 -5 4 -16 2 -2 1 5 2 34
  -8 -10 -8 4 -5 -3 15 -1 -11 -9 -15 1 -11 5 12 6 -12 4 -5 -4 0 -9 -5 14 2 -4 10 4 -3 -4 0 0 -5 2 -10 -6 7 1 -5 2 6 -13 8 0 4 9 -3 -12 2 -4 2 0 -8 5 -6 -3 12 -1 -3 -5 12 0 12 7 -5 -6 -4 11 1 -7 7 -9 7 -2 5 -13 3 3 30
  -11 2 17 27 -29 -11 -25 9 -1 1 8 -8 11 -7 -17 -10 -5 -7 2 6 -10 1 1 -9 -2 -1 -5 2 -1 23 2 2 2 8 -2 -10 -3 -10 -3 -18 -7 8 -1 20 0 -2 1 -10 4 14 -3 17 -11 -12 17 -5 -18 2 -15 -5 -15 0 -6 6 -1 4 -8 -13 -1 -6 2 9 -6 5 -14 26 -17 3 -8 46
  3 -7 -19 6 -12 -2 -1 14 -1 -7 -2 9 -1 -17 1 2 -3 6 18 -10 2 -18 -16 4 -5 -5 6 -7 11 17 11 2 -12 6 -8 -14 23 8 9 -1 -12 1 10 0 1 -5 -4 -7 -1 5 3 4 -5 6 2 6 -1 -2 -22 11 1 7 11 -2 -4 -9 -7 5 -2 -10 11 -3 4 -14 -10 -9 -18 -1 4 7 44
  4 -14 -9 -9 -5 -6 0 -2 -14 -4 -6 9 -2 -7 3 -2 1 -12 -1 9 -5 -6 -7 15 3 -9 5 4 -17 5 4 1 14 3 12 -16 -2 1 8 -1 2 12 -13 13 -9 -5 15 5 9 12 6 11 9 7 -10 22 7 7 6 -6 -10 4 -5 11 12 7 -3 0 14 -8 5 -6 -1 -5 0 -5 8 14 2 1 1 38
  -6 16 17 0 -16 0 -1 -7 13 -4 6 10 -2 4 16 -14 3 3 -11 11 13 9 9 -4 -7 -5 -2 15 9 8 -9 -11 1 8 10 7 11 -12 -19 -9 -11 2 -2 -1 -5 0 -3 4 -17 -3 -7 14 9 -4 10 3 -7 9 8 -1 -1 -1 4 -9 3 13 0 -11 -3 0 -3 4 3 0 -1 20 -16 4 -9 0 -5 -1 44
  5 9 -9 -24 10 -5 12 -2 -9 -1 -16 4 -12 -9 20 6 -6 17 -7 5 9 8 10 8 -1 -11 10 15 2 -3 -5 -8 -7 13 13 -11 9 8 -5 3 2 4 -6 2 -15 2 13 16 -8 -20 11 0 16 3 -10 6 -6 16 5 10 -1 11 -1 -9 4 -3 8 2 10 -5 -12 -15 12 -7 7 0 13 10 5 -14 -5 4 5 46
  13 0 -13 -8 21 15 1 -1 8 1 7 3 6 -6 -7 4 14 -3 12 -3 2 -8 -7 -8 -6 0 -8 -13 7 -1 5 -3 -5 -12 -5 9 1 10 17 10 -5 -3 1 -18 4 -2 -4 3 0 -1 -5 -9 -1 5 -7 2 6 -5 -1 4 5 5 6 -7 0 -3 2 2 0 6 2 -3 -1 -7 4 -15 -2 -10 -7 -14 11 -7 -13 -6 38
  2 10 -4 5 -9 2 -9 2 -5 -14 -4 6 2 -20 2 -2 2 10 10 4 2 -5 1 0 -4 -9 -2 0 7 27 2 -8 -10 12 2 -12 19 -4 2 -9 -17 3 2 6 2 3 2 5 -4 -8 -1 17 1 7 -4 2 -8 7 -4 1 3 13 2 -1 -11 3 -5 -4 4 -12 0 -7 13 -2 -12 7 -14 3 -3 14 15 3 -1 5 5 36
  5 11 -4 -4 -2 -3 -9 -5 -6 -12 -2 -4 7 -15 -11 -12 7 4 -4 5 -10 2 2 -5 -3 -16 0 12 0 13 2 -1 -6 11 10 -10 -4 -3 5 -1 -10 7 -4 1 -9 -6 15 19 -10 0 2 10 14 1 -6 16 -13 13 0 -2 -4 10 1 -3 2 14 7 -6 19 -7 5 -7 0 -1 -11 8 -9 14 -8 10 -2 13 2 7 0 14 36
  0 17 -1 0 16 26 8 -7 8 -5 17 1 10 -5 -8 -2 6 17 11 -5 14 -4 -6 -15 -4 -3 -8 0 25 13 -3 -5 -7 -11 -9 18 19 5 12 -5 -14 -8 18 -23 9 1 1 3 -5 -11 -13 6 -2 11 -8 -7 2 -4 -3 1 15 9 3 -24 -25 1 2 -2 -21 5 -5 7 2 8 7 -1 -20 -13 -8 -6 8 -17 4 -5 15 7 -1 52
  

• 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
  -25 -16 -72 -34 -33 -16 -18 -52 20 28 2 4 -17 20 -28 31 -20 7 2 13 -8 15 0 -4 -16 0 -24 14 -2 27 -5 9 -1 12 12 7 -3 -9 3 2 -13 -6 11 10 4 6 -7 1 7 -7 -4 -9 4 1 0 -4 7 6 -6 0 0 -7 0 -3 0 4 2 -2 -5 1 -3 5 3 4 4 -4 -1 -2 0 0 -8 -4 4 -7 6 3 0 0
  26 -60 157 97 96 9 63 160 -95 -58 -10 51 16 -17 142 -47 81 -10 8 -37 -44 72 -56 -85 28 -44 -26 19 -23 -7 -9 40 -35 23 -16 -1 7 2 36 -14 14 20 5 -18 -1 -5 -6 20 6 -1 -9 1 8 -3 9 6 6 7 -4 -7 -12 -3 -4 -4 12 -2 6 8 0 1 -5 2 -1 1 3 -1 -1 0 0 -6 2 -2 -2 4 2 -3 2 2
  -15 -46 68 -3 44 -2 2 -3 24 -29 -21 12 11 -26 -5 -17 -15 24 6 13 4 -19 2 7 -6 -3 12 -12 21 -14 14 -5 7 1 -5 -20 -3 -6 -6 4 3 -10 -7 0 -11 -11 -11 8 -1 7 2 1 -7 -5 -1 5 -7 -3 0 5 1 2 -5 -1 -4 1 0 -2 -6 -2 3 4 -6 -3 -1 0 6 1 -2 0 -1 1 -1 -1 1 0 4 1
  19 143 -37 -13 -70 -40 98 68 -21 43 -17 -4 17 93 12 62 50 -97 42 -17 -39 14 -46 -17 -9 -16 -40 68 -53 63 -45 23 -31 -42 -2 11 17 20 40 17 13 37 26 27 39 15 -12 15 -20 -28 -10 -3 11 4 3 -9 2 -9 6 -8 -5 0 -1 7 5 3 2 6 4 -6 -8 -2 3 1 -4 7 3 2 2 -3 0 1 -3 -1 -4 2 2 2
  -18 11 23 45 -24 -5 84 55 -8 5 -49 -16 38 12 113 -9 76 -15 33 -29 -52 81 -52 -68 47 -50 -37 7 -71 16 -18 36 -15 11 -1 31 11 23 54 -20 4 29 18 -4 16 8 -6 10 2 -13 -14 0 3 -4 2 -2 10 15 -3 2 -11 -5 -8 -10 15 -8 7 10 8 6 -11 0 10 9 -3 3 -5 0 1 -1 7 5 0 0 9 -1 -1 1
  -4 -189 232 159 249 -5 -27 143 -95 -129 4 68 5 -106 179 -109 59 48 -27 -35 -7 65 -45 -83 46 -19 1 -22 40 -63 31 14 -17 50 -28 -17 1 -24 4 -29 19 -3 -17 -49 -23 -31 26 10 22 26 9 10 9 0 14 15 8 24 -12 -5 -14 -11 3 -8 12 -3 3 4 -1 9 -5 -1 -1 -1 1 -3 -5 -3 -2 -5 4 -3 2 5 2 -5 2 3
  -9 127 -131 -77 -179 0 72 -73 53 82 -48 -72 29 81 -23 70 31 -59 48 -8 -19 27 -11 1 9 -17 -34 13 -66 52 -29 3 7 -28 -1 36 6 24 28 1 -3 16 26 30 15 15 -9 -5 -11 -17 -8 -8 1 2 -14 -11 -4 0 4 11 -1 6 0 -2 -4 -11 8 2 7 -3 -1 0 8 7 -11 7 -1 1 3 3 8 8 -3 -1 -1 5 -2 -2
  63 205 -165 -37 -188 73 64 -13 -10 134 10 -37 -12 86 -116 75 -9 -67 -17 -4 22 -52 45 63 -28 4 2 16 -34 35 -30 -4 14 -24 31 13 -2 31 -6 25 -11 -5 6 39 11 17 -28 -3 -15 -19 -4 -7 0 6 -8 -5 -12 -17 14 -4 9 17 -1 9 -14 7 -9 -6 4 -2 11 5 -3 -5 0 3 1 11 4 3 -1 1 -4 -4 -3 2 2 -9
  -1 30 -33 -17 -13 5 -8 -2 7 43 17 27 -27 11 -72 18 -33 16 -18 16 8 -38 25 37 -28 -3 4 2 18 6 -14 -8 16 -4 23 -18 -9 1 -13 25 -7 -20 1 16 -5 4 -29 6 5 -2 -7 -3 1 4 2 4 -6 -8 2 -5 8 7 -2 6 -11 8 -6 -4 -7 1 12 9 -5 -8 9 -7 1 7 0 3 -9 -5 1 -7 1 4 7 -3
  -16 91 -142 -76 -116 2 -5 -67 53 82 13 -26 -7 42 -93 41 -46 8 -1 28 0 -38 38 54 -22 1 4 -3 -19 23 -27 -10 14 -23 34 9 -1 15 -3 20 -14 -4 9 21 14 19 -26 -7 -3 -8 -15 -5 -3 2 1 -3 2 -13 4 -2 5 5 -3 3 -9 2 -1 1 0 -3 7 6 -2 -1 2 -4 1 4 4 7 -9 1 0 -5 3 6 4 -3
  0 141 -97 6 -70 -44 87 51 -6 89 -18 15 -12 97 -5 75 38 -74 25 -11 -42 24 -38 -16 -12 -32 -53 66 -50 76 -56 26 -19 -30 20 15 10 14 39 23 4 26 37 35 36 16 -27 17 -4 -28 -20 -7 19 9 7 -7 7 1 0 -12 -5 -1 -1 6 3 6 4 5 1 -1 -3 7 4 4 2 0 0 5 1 -1 -7 0 -1 -9 2 7 7 0
  -36 -329 45 -41 146 48 -271 -206 62 -104 74 17 -73 -179 -93 -105 -164 201 -82 70 104 -61 103 55 -21 75 80 -120 156 -114 104 -62 56 75 -10 -39 -44 -75 -109 -20 -22 -89 -63 -55 -84 -37 38 -32 37 60 38 6 -27 1 -14 17 -8 14 -23 18 18 -9 10 -2 -19 0 -9 -15 -19 7 13 4 -7 -8 13 -18 -1 -7 -4 4 -13 -11 10 0 3 0 -3 0
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  36 125 -69 -19 -161 22 134 85 -58 96 -22 2 18 106 45 56 68 -68 40 -25 -68 74 -39 -57 4 -67 -71 50 -96 71 -63 54 -36 -5 15 32 7 33 69 9 -6 28 42 37 25 30 -54 23 0 -33 -29 -17 12 -2 -3 -6 6 -6 3 -9 -9 8 -8 -3 4 3 10 7 -2 1 1 10 2 0 7 -1 -2 6 8 -5 -6 -3 -1 -7 5 6 7 -1
  -11 -38 128 36 111 -26 11 91 -57 -56 1 62 16 -3 42 -16 32 -10 1 -13 -14 10 -34 -19 -6 -2 -7 16 30 -20 6 -9 -10 3 -25 -29 12 -4 1 4 8 5 3 -7 -7 -10 -8 15 3 5 -6 -6 16 0 9 8 -1 -1 5 -6 -8 5 2 -1 -5 -2 2 -2 -2 -2 8 2 -9 -4 -1 5 2 -1 0 0 1 -3 -6 3 -6 1 7 -4
  -66 -149 120 -74 94 -34 -72 -105 83 -109 -42 -54 71 -80 -5 -61 -35 52 35 33 28 -51 30 46 -7 40 45 -42 62 -47 39 -34 14 3 -33 -32 7 -12 -33 -13 4 -23 -24 -12 -21 -28 13 -15 -8 29 25 -2 -15 -7 -3 7 -5 -9 4 20 1 1 -3 -11 -12 -12 2 -13 -10 -2 0 -9 -10 -7 -12 6 7 -5 -3 3 3 8 1 6 -3 -5 -6 6
  26 48 72 62 12 2 76 127 -62 -30 -7 21 39 11 91 -32 58 -28 5 -28 -29 30 -35 -45 34 -31 -8 7 -52 -10 -12 20 -20 -5 -8 17 12 32 40 -13 12 35 6 -13 15 12 -8 14 -8 -7 -12 5 0 -11 3 3 3 -2 6 -2 -12 1 -8 -4 7 -7 1 7 10 0 -3 -2 1 0 -6 4 1 1 4 -3 3 2 -5 7 1 -3 -1 0
  -6 -94 256 97 178 -39 65 215 -118 -114 -16 72 51 -31 191 -70 112 -12 25 -52 -68 80 -82 -97 35 -48 -28 30 -19 -21 -14 36 -47 19 -29 -14 23 16 50 -16 23 29 11 -24 7 -3 -11 24 8 -1 -14 -1 16 -8 20 10 10 3 3 -13 -18 0 -3 -9 14 -1 7 10 -1 2 -4 -1 -4 -3 -1 3 -5 -1 0 -3 3 -1 -1 7 1 -5 1 3
  -43 -116 114 -38 98 -45 -67 -43 42 -109 -3 30 13 -40 -18 -36 -32 55 8 23 10 -30 2 9 -3 29 32 -23 55 -47 33 -41 20 -4 -29 -33 0 -9 -26 -2 6 -5 -11 -17 -23 -8 14 -3 -1 22 8 3 -10 -10 -4 4 -10 -4 7 11 4 8 2 -4 -7 -7 0 -5 -2 -6 5 -4 -7 -3 -7 3 2 -10 -9 1 6 0 -9 8 -6 -2 -3 2
  37 90 -63 27 -127 59 117 26 -22 84 -58 -59 26 52 63 44 82 -77 29 -35 -24 68 -30 -44 24 -38 -49 23 -71 49 -25 40 -13 9 0 35 6 13 39 -10 0 11 21 21 8 -2 -9 6 -2 -17 -3 -10 7 7 -8 -5 1 11 -1 6 -9 1 0 -4 3 -4 8 3 7 3 -8 5 9 8 -5 6 -1 5 5 -2 7 3 1 -3 2 1 1 -1
  35 37 -48 81 18 30 6 59 -58 50 28 25 -56 8 17 27 27 -27 -35 -25 -3 30 -16 -21 4 -3 -20 26 1 16 -14 16 -4 11 18 2 -4 -17 -4 8 -1 -4 1 -1 7 -2 16 1 10 -8 -2 5 14 15 11 -2 6 13 -4 -15 2 -7 11 6 9 9 -4 6 7 5 -5 1 7 5 9 -3 -8 1 -2 1 3 -6 4 -4 -1 -3 2 -3
  -10 -8 108 44 63 -28 60 112 -38 -35 -21 19 43 -9 108 -40 73 -9 27 -29 -47 44 -43 -42 31 -43 -13 10 -39 -7 -25 21 -21 0 -9 1 17 24 38 -8 12 25 15 -9 15 7 -16 10 7 -4 -18 -1 9 -5 14 4 7 -1 5 -9 -13 3 -3 -6 8 -3 6 8 3 1 -1 -1 -2 1 -1 4 -4 2 3 2 3 1 -2 5 0 -1 3 2
  -49 -105 72 -58 149 -91 -116 -73 62 -97 26 8 0 -64 -67 -35 -69 38 2 36 32 -85 28 64 -39 57 52 -2 80 -24 25 -33 -1 -18 -13 -39 7 -24 -49 4 4 -14 -22 -14 0 -12 25 -16 -6 15 16 6 -3 -2 12 -1 2 -14 2 -3 6 -8 3 4 -2 5 -6 -8 -10 -7 -2 -11 -8 -7 -2 1 7 -7 -4 2 -6 0 3 3 -8 -5 -9 8
  -55 -138 92 -29 164 -87 -119 -45 36 -121 36 53 -10 -69 -24 -51 -54 68 -6 32 15 -37 -1 24 -15 40 32 -14 76 -49 31 -35 15 1 -27 -38 -2 -25 -36 -2 6 -10 -16 -28 -11 -5 23 -8 12 17 10 7 -2 -5 9 2 1 -1 -1 -4 1 -7 5 -1 -1 -1 -5 -5 -4 -1 0 -7 -5 -5 2 -1 0 -10 -6 3 -2 -10 1 5 -8 -2 -4 3
  26 81 -175 -20 -148 77 4 -110 38 127 -8 -49 -49 35 -100 73 -38 -18 -23 9 44 -12 36 34 -18 13 -12 -12 3 29 11 -5 28 10 21 15 -22 -18 -22 11 -16 -32 -5 27 -13 -8 0 -5 1 -6 14 -1 -4 16 -19 -2 -11 12 -10 8 14 0 4 10 -9 5 -7 -6 -2 7 3 8 8 2 6 -5 0 7 0 1 -1 0 6 -10 4 4 6 -7
  14 16 -8 72 32 15 16 54 -26 29 10 17 -27 -18 31 3 15 -3 -25 -16 -5 17 -7 -13 11 -16 -8 2 -4 6 -13 12 1 5 19 3 -4 -9 4 6 1 -4 1 -4 3 -4 4 4 9 -1 -5 8 8 6 9 3 4 6 -3 -10 -2 -3 3 2 7 7 1 6 5 5 -1 6 5 0 5 -5 -3 3 1 1 0 -4 2 -4 1 -2 4 -1
  25 196 -106 1 -157 7 165 56 3 133 -55 -37 24 101 16 76 83 -98 49 -31 -52 35 -31 -12 5 -67 -56 53 -100 85 -72 46 -13 -26 21 28 12 42 58 17 3 24 41 46 37 20 -46 16 -4 -36 -23 -12 14 5 1 -9 -1 -5 8 -7 -7 8 -6 2 3 6 6 4 2 0 -2 7 6 3 -1 6 -1 12 5 2 -1 2 0 -8 1 4 6 -2
  -9 34 -84 -55 -82 -11 8 -48 21 64 3 -18 -18 58 -52 63 -6 -25 22 8 -14 11 -4 8 -29 4 -28 32 -6 44 -22 7 -13 -6 11 2 3 -7 6 17 -10 -5 14 27 9 8 -12 2 -1 -17 -2 -8 9 16 -3 -5 1 -2 -4 -4 8 3 2 8 -2 8 -1 0 -7 -1 2 4 2 -3 11 -2 0 2 -2 0 -7 -2 4 -5 2 4 5 -1
  

• PROPERTIES
  Unimodular=1
  

• REFERENCES
  I. Gauvin, Even unimodular ideal lattices of rank 88 over Q(zeta_115), preprint (October 2026), and its addendum (October 2026),
  archived on Zenodo, concept DOI 10.5281/zenodo.23638084. The data of this lattice are in version 1.4 of the deposit.
  Data, programs and certificates: https://github.com/Katabolanga/rang88-reseaux-ideaux
  

• 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).
  X1151s24 = (I, b), an ideal lattice over L = Q(zeta), zeta = zeta_115 (rank 88): I = (1151, zeta^2 + 540*zeta + 177) = (1151, zeta - 6)(1151, zeta - 605)
  is the product of two prime ideals of Z[zeta] of residue degree 1, exchanged by sigma_24, and b(u, v) = Tr_{L/Q}(alpha u conj(v)),
  alpha = beta_0/nu totally positive, with nu a generator of I conj(I) in the real subfield (as in the paper, section 6).
  Minimum 8 is established by the computer-assisted method of the paper (same certifying program, same execution model of section 5.4:
  M1, M2, H1-H3): both sides of the criterion were swept completely, each side twice on two different machines (Java 11 and Java 21),
  with identical receipts once sorted by task (equal canonical fingerprints) and no hit. The attached Python scripts check the exported matrices and generators, not the minimum.
  The kissing number 168498000 follows from that minimum-8 statement and the extremal theta series.
  Basis: an LLL-reduced Z-basis of I (the one in version 1.4 of the deposit); X1151s24_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
  I sigma_22(I)^(-1) (in X1151s24_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 <sigma_22> = {sigma_1, sigma_22, sigma_24, sigma_68} (cyclic of order 4) by <-zeta> (order 230).
  <g1, g2> is the full normaliser in Aut of the group of multiplications by roots of unity, without assuming GRH: the Galois group acts on the
  331-part of the class group of L, cyclic of order 331, through a character with kernel <sigma_22> (paper, Proposition on the Galois action),
  and the 331-component of the class of I is non-zero (certified without GRH), so the Galois image of the normaliser lies in <sigma_22>;
  its kernel consists of the 230 roots of unity (Kronecker). The full automorphism group has not been determined: |Aut| >= 920.
  A vector of norm 8 (found by a G6K sieve, checked exactly), as a row of coordinates in this basis:
  NORM8: -3 66 13 -6 -20 -40 50 84 -43 6 8 35 4 56 38 19 39 -32 24 -16 -51 33 -40 -37 4 -29 -31 40 -42 30 -38 20 -23 -22 3 7 11 22 39 10 8 30 23 8 25 21 -19 12 -6 -17 -20 -3 9 -2 7 -3 8 -7 4 -10 -3 2 -3 2 8 -1 5 8 2 -2 -2 0 0 1 2 0 -4 1 1 -2 -1 0 -4 0 2 3 2 0
  

• LAST_LINE
  

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  Haftungsausschluss/Disclaimer
  
  
  Gabriele Nebe