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 06:02:06 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
  X4049
  

• DIMENSION
  88
  

• DET
  1
  

• GRAM
  88 0
  30
  13 34
  -3 -13 34
  -13 -12 18 40
  -12 -5 7 14 34
  -8 -12 -1 15 10 44
  -14 0 4 -1 -7 -20 50
  -13 8 -16 4 13 15 10 52
  12 17 8 -10 1 2 -7 -11 50
  11 -2 6 -7 6 0 -8 -16 10 36
  -2 11 5 -4 14 -6 4 1 24 8 42
  -9 7 -13 -8 1 -11 0 13 -9 -6 -6 44
  10 15 -15 -4 -12 -11 9 7 -4 -2 -6 13 38
  -12 -11 -3 2 4 -4 6 11 -11 -13 6 5 -10 40
  -2 -2 -13 -16 -4 3 10 1 10 9 -2 -6 -3 -7 48
  6 -3 -8 5 -13 5 2 -2 -2 4 -8 -11 10 4 5 40
  -1 10 9 10 8 4 -11 9 4 1 1 15 -6 -9 -17 -15 44
  -15 -5 10 17 8 12 11 2 0 -9 9 -7 1 -4 -7 -11 -7 44
  -8 7 -15 8 0 -9 19 6 -21 -1 0 1 10 -5 12 7 1 4 60
  9 1 -10 -20 -11 6 -1 -5 14 8 3 0 11 -5 9 7 -17 -2 -11 42
  -12 -12 0 -8 -3 -2 7 7 -10 -2 5 7 -12 21 0 0 2 -12 -9 -4 42
  -4 9 8 0 3 -7 2 -5 8 5 9 0 -11 -5 0 -13 9 2 -4 -7 -4 36
  10 -7 6 -13 6 1 -8 -10 14 15 0 1 -11 -7 10 -5 0 -14 -22 12 6 1 46
  -2 -13 18 12 13 17 -21 -16 13 6 13 -23 -30 -4 4 -6 7 0 -12 -6 7 6 17 58
  2 -2 6 3 7 14 -17 1 12 -2 7 -11 -19 7 1 -6 7 -2 -11 -4 -2 3 2 22 30
  15 16 -10 -13 -8 -6 -3 5 -2 10 -8 13 15 -17 -6 -5 13 -7 4 5 -10 7 7 -22 -7 42
  -5 1 1 7 7 4 -5 -4 1 9 4 7 6 -14 2 -6 7 12 7 -5 -6 3 0 4 -5 3 26
  12 19 -14 -7 6 -17 -2 12 -8 4 11 2 8 2 -8 -3 4 -16 14 -5 1 6 -3 -5 -9 11 -3 56
  10 16 -4 -11 -3 -7 9 1 16 0 11 -9 12 -17 5 -4 -4 3 0 5 -12 5 4 -2 -1 11 0 3 38
  -10 2 -14 -5 -5 7 11 7 -1 -11 2 4 6 -1 10 -1 -8 17 10 13 -5 -3 -10 -4 -5 -3 3 -8 3 36
  11 4 -10 -21 -16 -13 13 5 -7 3 -5 10 13 0 5 -3 -2 -13 7 12 4 -8 5 -22 -5 18 -6 3 11 -2 42
  -6 11 -12 3 8 15 -16 19 -6 -15 0 14 6 -1 -18 -13 9 10 0 5 -9 0 -14 -8 4 6 1 5 1 8 -4 44
  -14 2 1 11 9 6 7 14 -8 0 5 5 3 -8 1 -5 13 6 11 -10 4 6 -8 9 -3 -3 15 2 5 7 -5 8 34
  0 -2 15 20 9 2 -7 -16 -2 0 7 -12 -1 -4 -19 -5 2 22 5 -10 -16 4 -9 10 5 -9 8 -9 0 -1 -14 7 1 42
  1 -10 -6 -9 3 -1 -7 5 -8 -8 5 -4 -12 17 0 -3 -9 -7 -7 -3 9 -3 4 3 9 -6 -12 6 -1 -3 6 -2 -12 -5 38
  15 10 -15 -21 -9 1 -8 0 7 -1 -4 4 6 -12 7 -1 -8 -2 -4 17 -11 -4 17 -9 -2 16 -1 1 10 8 11 9 -12 -6 8 38
  -7 -9 17 18 -3 8 -3 -10 7 -13 7 -7 -4 -4 -18 -2 6 21 -5 -5 -3 -4 -7 16 7 -9 6 -17 6 1 -8 4 2 15 3 -3 48
  -3 -7 16 17 0 9 0 1 -4 -2 -8 -6 -6 0 -7 3 7 2 -8 -6 -1 5 -1 11 7 -6 -3 -11 -9 -5 -9 0 6 8 -8 -8 4 28
  0 -4 -4 -2 -8 8 -7 -12 -4 0 -2 -14 0 -9 -1 8 -12 3 6 10 2 -8 -9 6 -2 -5 2 1 2 2 -5 8 1 1 -1 2 9 -4 36
  -3 0 6 10 8 0 1 1 6 4 -3 -4 2 3 -3 9 -1 -6 -9 -4 0 3 -2 0 -4 -7 -4 5 0 -9 -14 -5 0 -1 -12 -18 -6 4 -2 32
  -3 7 0 2 9 14 -12 2 6 13 0 2 -3 -8 0 -1 7 8 -6 0 -8 14 -2 2 6 5 7 -6 -1 -3 -12 10 6 8 -14 -2 -9 7 -1 8 42
  9 12 -9 -12 -7 -7 7 12 9 -4 11 -2 13 -1 2 5 -6 -6 -4 12 3 -1 -2 -7 -4 7 -7 16 14 3 8 -1 -2 -15 5 5 -3 -8 3 0 -8 30
  -5 -16 17 14 6 -7 -6 -6 -21 -7 -2 -4 -19 16 -26 -9 12 -5 -1 -18 16 5 -5 12 6 -8 -7 10 -15 -19 -5 -1 -4 8 13 -15 12 5 5 0 -10 -5 54
  2 12 0 5 -5 -13 10 1 -5 0 4 -1 9 3 -10 10 1 -3 17 -2 -8 10 -18 -14 -3 3 -4 11 0 -7 4 2 0 8 -7 -8 -4 5 0 4 6 4 6 32
  -13 -6 5 4 16 -4 19 6 -5 9 16 -3 -7 1 12 -7 -2 4 20 1 7 4 4 12 -5 -11 2 9 1 4 0 -6 15 1 -3 -12 -7 0 -3 1 -4 -2 3 2 40
  -10 -11 14 20 15 -4 -4 -3 -10 -7 -9 -1 -15 7 -11 -7 8 1 -3 -27 -1 4 -1 10 4 -7 1 5 -10 -13 -19 -6 -1 5 5 -15 10 2 -4 13 -5 -13 22 -4 -3 46
  -5 6 -7 -9 5 -5 -5 11 -6 -4 -4 15 -6 7 4 -16 7 -11 -1 -6 10 6 -4 -7 1 2 -5 12 -13 -5 -2 12 0 -13 3 -1 -17 -2 -3 1 2 -1 6 -2 1 7 34
  -2 12 -18 4 4 -3 11 11 -5 -1 3 -4 10 1 16 7 -6 -1 37 -3 -7 -1 -12 -8 -6 -1 4 21 2 8 0 -2 5 -1 -8 -4 -15 -11 1 5 -3 4 -8 10 13 -2 3 46
  3 9 2 5 -5 -13 1 -4 1 -14 -7 7 10 2 -14 3 5 -1 -7 -8 -6 -2 -9 -11 -7 -5 -3 1 -2 -1 -8 2 -3 7 -5 -4 1 2 -4 9 -2 -1 3 6 -12 8 1 -4 30
  -2 2 -13 -9 7 -13 -4 -9 -1 -2 2 8 3 3 9 -1 -9 -9 15 3 -6 -3 0 -7 -7 -7 0 5 -2 2 -6 4 -8 2 8 6 -12 -14 5 1 -6 -1 0 1 5 1 9 12 8 44
  -7 5 2 0 16 2 5 7 9 10 21 -8 -4 -2 1 -5 -3 13 8 2 -5 6 -8 3 2 -2 5 13 7 0 -6 4 5 2 -2 -6 4 -9 6 6 12 5 -1 6 13 2 0 8 -10 0 36
  4 -6 11 -1 3 -7 11 -7 -2 5 -2 -7 0 -10 -3 -11 -3 8 1 7 -11 1 5 -2 -8 6 -2 -1 9 2 7 -4 -2 1 0 4 2 -2 -2 -3 -8 2 7 -4 11 1 -4 -4 -4 1 5 36
  -11 5 -5 11 0 2 -5 3 -11 -2 -13 12 4 1 -3 1 8 -1 8 -11 -1 12 -15 -6 -2 5 9 5 -12 -2 -13 9 11 0 -17 -15 -6 7 4 13 16 -10 -1 9 -7 14 12 11 5 -2 2 -16 42
  -5 8 5 8 5 1 -7 -6 1 2 9 5 -3 -11 -16 -5 11 7 6 -8 -2 6 -11 3 -6 -1 8 5 -2 -1 -11 14 8 10 -8 -3 9 2 11 -1 5 -7 5 5 3 3 4 -4 4 2 5 -2 5 36
  7 9 -1 9 -6 5 1 7 -4 -3 2 -12 7 -6 -4 9 2 7 7 -12 -7 3 -14 -3 2 1 4 2 8 0 0 1 10 10 0 2 2 7 -1 -9 3 2 -3 10 -5 -9 -9 4 5 -12 -5 -6 -3 5 36
  -7 2 3 -5 2 0 -5 8 3 -2 -2 19 -6 6 -4 -8 10 -4 -17 -3 12 8 6 -6 -1 0 0 -7 -11 -7 -3 8 0 -6 -3 3 -4 5 -10 2 11 -5 2 -1 -7 3 15 -14 3 -4 -4 -10 8 6 -6 32
  -13 -6 -2 16 -2 23 -14 10 -10 -18 -19 2 -11 -2 0 2 13 7 3 -17 -1 1 -16 9 13 -6 1 -18 -8 3 -15 13 7 1 0 -5 15 10 6 0 13 -10 8 -4 -14 11 4 -4 2 -7 -7 -11 15 4 6 6 48
  -3 12 3 -6 -6 0 13 15 15 -3 0 9 5 0 0 1 4 -4 -19 8 6 9 5 -12 -2 5 -4 -5 4 -5 5 4 3 -13 -13 1 -5 9 -8 9 8 5 -14 6 -4 -6 6 -8 4 -15 1 -6 8 0 -2 18 -4 42
  -12 -8 -2 -5 2 -14 13 1 -15 -7 -2 10 -3 15 -4 -8 -8 -2 -2 -2 11 9 -3 -12 -7 -4 -4 3 -2 4 2 -1 -2 -3 10 -5 -7 -5 -2 3 -8 0 15 2 5 10 7 1 2 9 -3 6 4 -2 -9 5 -6 1 34
  -14 -20 21 12 19 20 1 -4 0 4 11 -26 -29 3 -7 -7 1 16 -8 -1 12 2 8 32 9 -23 -5 -9 -5 2 -22 -1 5 16 5 -10 10 9 10 4 9 -9 18 -10 18 4 -6 -7 -10 -4 15 11 -12 5 -5 -3 4 -8 2 64
  6 16 0 -12 -4 -16 4 -6 9 10 14 2 9 -12 -6 2 4 -8 -3 8 5 8 1 4 -11 3 7 17 11 -2 0 -2 14 -4 -12 -1 -2 -5 8 3 5 12 -2 10 7 -8 -1 -1 6 -2 11 3 2 11 3 2 -16 11 0 0 42
  15 -3 14 -8 -7 4 -11 -22 13 16 3 -24 -16 -18 3 4 3 -9 -11 0 0 0 12 26 9 -1 -4 -6 7 -7 1 -13 -3 4 2 5 5 7 10 -4 1 -1 -2 -6 -3 -7 -10 -18 -7 -9 1 4 -17 7 4 -6 -2 -4 -16 19 8 50
  -1 7 6 0 -9 -9 14 2 4 -15 -2 6 11 -2 -6 -5 3 8 -9 -1 -2 -2 -7 -11 -1 -1 -2 -17 6 6 2 4 3 7 -5 0 5 8 -6 -4 -6 1 -8 1 -6 -6 0 -13 16 -3 -13 1 -4 0 7 2 1 13 1 -7 1 -3 36
  0 -14 0 5 -10 -8 11 -11 -20 6 -13 -8 8 10 -1 24 -19 -10 12 3 4 -10 -8 -12 -12 -4 -9 -3 -8 -5 5 -13 -6 0 0 -11 -6 6 7 9 -9 -5 5 12 3 0 -7 3 0 4 -7 1 4 -3 -1 -8 -9 -5 8 -3 -3 0 -3 42
  2 -3 -5 -3 -14 -9 3 -15 -15 2 -9 5 5 -3 2 6 -10 4 9 0 0 5 2 -5 -9 5 8 1 -5 4 2 -6 -1 4 -1 9 0 1 4 -11 2 -6 0 4 -3 -5 -6 -1 -1 -2 -10 -4 6 1 7 -1 1 -8 5 -9 1 -3 0 8 34
  12 -3 9 2 -2 7 1 -9 13 14 4 -25 -7 -4 10 12 -6 -5 2 11 -3 -3 9 15 9 -6 -7 0 -1 -7 -2 -16 -9 3 -5 -1 -4 6 3 4 1 4 0 4 8 -7 -8 10 -11 -6 3 7 -11 -9 1 -10 -7 0 -10 17 0 14 -7 5 -4 36
  -14 1 -14 6 -2 2 2 3 -14 -14 -14 19 6 -5 5 -2 8 5 24 -7 -7 0 -9 -8 -6 1 8 -7 0 15 -3 12 10 2 -4 2 6 -4 1 -5 1 -12 -5 0 0 4 1 12 5 12 -9 -6 14 5 -1 -1 22 -9 6 -9 -8 -17 2 -1 12 -16 42
  2 11 -15 -7 -1 2 -11 14 -3 -12 -6 7 5 -2 4 -9 0 1 -3 -6 -10 9 -8 -10 3 11 2 5 9 4 2 14 2 -5 8 8 -6 -6 -3 -4 4 6 -5 -2 -13 1 8 2 3 3 -2 -5 9 -4 7 5 13 -1 3 -20 -5 -11 -1 -11 -1 -18 8 36
  8 6 -5 -12 7 12 -16 7 6 0 0 -5 -13 2 -7 -15 7 -6 -16 2 6 -3 11 8 14 6 -8 9 6 -5 4 12 -9 -4 4 8 -9 -6 4 0 8 1 6 -10 -8 2 9 -1 -7 -6 3 0 -4 0 -5 3 2 3 1 14 -2 7 -9 -16 -12 6 -8 5 46
  -13 -18 -2 6 -2 4 11 8 -29 -10 -12 3 5 1 -5 4 -4 9 12 -1 5 -9 -8 -5 -8 -4 4 -13 -1 10 7 1 11 5 9 -2 9 0 5 -11 -8 -1 11 1 6 -3 -13 -4 -3 2 -4 8 -6 -5 5 -6 9 -11 11 11 -2 -4 4 10 7 -8 13 1 -12 44
  -1 -1 14 7 15 -8 -6 1 -7 -3 10 -7 -19 7 -16 -20 12 -3 -8 -17 7 14 1 20 8 -8 -5 25 -3 -14 -8 2 3 8 12 -13 0 5 -2 4 -2 -1 30 3 12 21 9 -5 3 -5 6 6 -1 8 -2 1 -2 -6 10 17 7 5 -5 -10 -4 -1 -10 2 13 -6 48
  1 3 13 6 5 -1 1 4 2 -2 1 3 0 -1 -12 -18 7 7 -15 -1 -3 12 0 0 6 4 -2 1 3 -7 0 9 1 6 -8 -6 -2 11 -8 5 6 2 8 4 0 2 7 -7 1 -12 0 7 5 1 1 7 -2 11 6 3 -1 -6 9 -9 -3 2 -8 1 10 -6 13 32
  0 -2 8 -5 -12 -17 1 -21 7 5 8 4 -7 10 3 7 -4 -9 -3 1 11 6 -5 4 3 -12 -1 -3 -14 -6 -3 -14 -4 -3 -2 -10 0 3 5 0 -2 0 9 8 -1 -1 4 -6 4 3 -3 -10 6 6 -4 8 -2 -1 3 -7 8 4 1 6 11 3 -5 -8 -15 -9 3 0 44
  4 14 -15 1 -4 -12 8 12 -13 -9 -4 7 11 5 -6 8 0 -2 19 -13 -9 0 -11 -19 -9 14 -2 19 2 6 7 2 0 -2 4 6 -7 -5 -9 -3 -5 3 -5 9 -6 5 -1 13 8 0 -3 -3 4 2 9 -6 -1 -3 0 -23 -2 -8 -1 5 7 -12 7 11 -6 -2 0 -11 -8 38
  2 1 -1 6 -4 -1 0 -9 8 3 11 -13 3 -4 5 18 -5 3 11 -3 -7 3 -6 6 -2 -4 5 2 8 1 -10 -11 1 5 5 -1 14 -3 6 0 -2 6 -3 7 -2 0 -17 4 2 7 4 -6 -4 2 11 -11 3 -11 -7 -3 4 5 -6 6 6 3 5 1 -19 2 -6 -17 6 6 36
  2 7 -3 -10 6 -11 2 -1 -4 3 4 0 -1 -8 7 -17 -5 2 1 -2 -8 18 6 1 -1 6 3 14 11 -1 3 4 5 4 7 7 -12 -5 -4 -5 6 0 -3 2 9 1 7 3 -2 5 6 6 1 -2 3 -1 -9 0 8 -1 7 0 1 -9 7 -6 1 11 1 0 15 8 -6 2 -4 32
  1 -11 14 1 -8 -4 9 -2 -2 2 -11 0 -1 -1 -1 -1 3 -9 -12 4 8 0 5 0 -1 -1 -8 -2 -2 -11 9 -10 -1 -13 -2 -8 5 8 -1 7 -7 5 12 0 1 5 3 -10 -3 -13 -2 9 -1 -9 -8 4 0 11 4 3 1 5 0 5 -4 8 -7 -10 -2 4 3 12 7 -14 -9 -3 34
  8 -9 8 0 6 -2 -20 -8 -12 12 -13 9 -5 2 -13 -5 8 -18 -18 -2 4 1 15 7 6 6 -1 4 -13 -21 3 1 -3 0 2 -2 -11 12 -3 3 10 -9 16 0 -6 8 8 -17 1 -2 -7 -1 8 -2 -7 8 -3 1 1 -1 2 6 -4 5 3 -2 -9 -3 8 -1 12 11 4 -7 -15 6 11 42
  -8 8 -18 -9 -8 -15 2 -1 -14 -1 6 27 8 3 0 -6 9 -3 26 -3 12 4 -16 -13 -13 8 12 16 -11 14 6 5 6 -7 1 0 -2 -13 5 -14 -8 0 6 4 5 -5 12 12 -1 11 0 -9 13 15 -3 2 0 -11 12 -20 8 -15 -4 -1 16 -19 20 3 -11 4 -1 -9 15 11 2 0 -9 -6 56
  7 14 -6 2 -4 9 -13 8 6 -2 -9 7 5 -6 -6 6 14 -8 -8 2 -7 6 -1 -4 1 6 -1 2 -4 2 -6 11 1 2 -11 6 -9 8 -7 6 12 -3 -8 6 -12 -4 3 1 10 -6 -10 -11 10 5 8 11 9 13 -6 -8 4 -2 4 -7 -2 0 4 6 6 -8 -2 4 -4 4 -5 -3 -4 4 -1 30
  8 5 11 0 0 2 -12 -16 10 10 7 -17 -9 -6 -9 -8 1 8 -8 0 -7 16 -3 11 10 7 1 -1 5 -7 -8 3 -5 14 -8 -2 4 4 7 3 15 -2 9 7 -7 0 0 -4 -6 -9 11 4 7 3 2 1 0 -2 -3 11 5 11 -7 -2 0 8 -13 5 12 -12 8 11 3 -8 -1 4 -2 4 -8 1 36
  -3 -11 -8 4 -2 -1 2 5 -23 -8 -10 -2 -2 14 3 5 -4 -5 20 -11 11 -18 -11 -2 -3 -7 -5 3 -11 0 6 -5 -3 -5 10 -7 -4 -6 3 -6 -16 -3 15 -3 3 6 4 11 -3 6 -7 4 -4 -5 0 -8 7 -20 2 -1 -12 -6 -6 13 0 0 5 0 1 12 2 -9 -1 8 -2 -10 -3 -3 10 -9 -8 36
  0 14 -11 -11 -5 9 10 24 8 -6 -1 0 8 -14 8 -3 7 -3 -4 10 0 5 1 -8 -4 12 -2 9 21 8 10 10 11 -19 -2 9 -3 -5 3 2 1 15 -16 -3 1 -9 1 2 -4 -10 6 3 -2 -2 4 0 3 20 -1 -2 10 0 5 -14 -9 -2 2 9 10 1 0 2 -18 0 -2 7 6 -13 -5 9 -7 -11 44
  -9 -14 19 19 7 17 -1 7 -10 -1 -12 2 -11 -6 -13 -11 20 11 -8 -14 3 3 5 3 3 6 4 -17 -8 -10 0 4 5 7 -3 -6 12 14 -7 -4 4 -14 13 -6 -1 10 -1 -17 -3 -17 -6 5 0 6 5 8 14 5 -4 13 -15 4 5 -6 -2 -4 1 -4 3 9 6 11 -12 -7 -9 -4 9 9 -10 2 0 -1 3 40
  -2 -4 15 1 14 7 -1 6 9 4 -1 -9 -10 -1 -2 -6 -1 3 -23 -4 0 2 9 4 7 -8 -7 -13 6 -15 -8 0 -2 4 -1 -4 -3 7 -7 14 15 -5 1 -3 0 7 4 -13 1 -6 10 6 -4 -3 -2 14 2 15 0 23 1 8 3 -3 -16 6 -13 0 14 -1 5 13 -8 -17 -12 4 10 9 -33 2 10 -9 3 10 44
  -11 -8 10 10 6 -8 1 -6 -14 3 1 2 -14 16 -12 0 8 -4 2 -23 13 9 -8 3 0 -10 1 4 -17 -10 -8 -8 6 8 4 -19 -1 6 -6 5 5 -16 17 8 2 18 5 -3 6 -2 1 -11 14 9 4 10 1 -3 9 8 2 1 -7 11 4 -9 0 -4 -5 1 14 -5 12 6 2 0 -2 9 7 -1 3 3 -16 6 3 38
  8 1 -3 -6 8 14 -14 12 0 6 3 -14 -15 0 5 -4 8 -10 -8 -2 1 6 10 18 19 6 -8 10 2 -2 -2 1 2 -1 15 4 -12 7 -5 -6 7 2 1 -4 3 -2 2 -4 -11 -9 1 -4 -6 -9 7 -4 4 -3 -6 15 -4 16 -6 -8 -6 10 -9 8 17 -1 16 1 -11 0 0 10 -4 8 -13 6 6 -2 9 3 7 1 42
  -6 -6 13 16 2 15 1 5 3 -8 6 -14 -6 1 -7 2 -2 14 -11 -6 1 3 -6 11 8 -11 -3 -14 5 -2 -6 5 7 11 3 -8 16 14 2 1 2 -1 1 0 1 -4 -9 -11 0 -16 -3 -4 -7 7 15 -1 10 5 -7 12 -7 8 9 0 -4 0 -6 0 -5 2 5 6 -2 -4 5 -2 0 -6 -17 -1 3 -6 2 14 8 2 4 32
  

• 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
  240 -118 60 -55 153 121 64 -26 57 -161 -22 119 -94 42 -32 -7 -67 -89 80 103 52 69 -170 10 -62 65 67 34 61 -49 -39 -90 38 46 -36 19 0 -6 9 -32 6 -20 -49 -19 6 10 21 -5 38 -29 2 -30 -26 -17 -12 13 15 8 -10 5 -11 9 3 10 8 -10 4 -7 2 5 1 2 -7 -2 2 1 -5 7 -4 -9 0 -4 -4 4 3 -2 -1 -2
  -41 -7 2 -44 39 -36 -49 9 64 1 -70 -27 30 -45 -10 60 -9 66 35 -19 54 32 27 -20 31 -17 -11 20 -7 19 36 34 8 -6 14 -35 -2 33 -24 -19 -6 -4 8 -6 -9 10 -3 -9 2 6 -3 5 -5 5 -6 -7 -14 -7 1 -4 5 -6 0 0 2 5 -7 -4 6 -2 2 4 3 -5 2 -4 2 -2 -2 3 2 3 0 -6 -6 2 -3 -1
  366 93 316 -261 319 431 217 -225 -111 -316 63 224 -12 133 120 74 -51 -147 29 62 5 62 -258 87 -155 139 83 28 -12 -84 -28 -62 68 48 3 61 33 -37 6 13 -4 -9 -22 -20 19 25 -22 -3 33 -32 6 -32 -17 -29 -14 0 14 6 17 1 -24 5 -6 12 1 5 16 5 14 -5 -11 8 -2 -7 -1 -2 -1 16 5 -3 6 -9 -9 10 3 4 2 -2
  191 58 218 -84 117 208 118 -89 -81 -127 20 144 -44 101 76 11 -27 -92 -3 53 -11 -4 -136 61 -82 83 64 18 39 -30 -42 -33 17 31 17 38 11 -49 14 13 2 -5 -15 7 11 -6 -3 13 21 -11 0 -20 -4 -9 -3 6 14 10 3 2 -13 5 -1 6 2 1 4 4 3 -1 -9 -3 -6 0 0 -2 -3 9 5 -6 5 -5 -2 2 0 4 5 -1
  238 83 326 -334 317 373 142 -210 3 -293 -25 145 74 68 111 138 -69 -66 87 -5 68 64 -162 64 -114 119 52 44 -56 -38 16 20 90 40 45 13 30 -2 -49 5 5 -3 9 -7 0 38 -36 -23 32 -25 10 -13 -9 -9 -9 -24 -9 -8 30 -13 -17 -4 -10 11 2 23 5 11 23 -13 -5 14 8 -14 -6 -6 1 17 10 6 3 -5 -6 6 -4 2 -2 -1
  37 -51 -13 54 -30 -24 16 34 10 4 10 36 -60 8 -16 -35 -9 -27 0 40 -8 -16 -26 7 -6 10 15 5 38 -5 -28 -28 -1 14 -8 5 -5 -22 13 -3 9 -10 -14 4 -7 -11 17 10 9 0 -5 -5 1 -1 5 8 7 6 -8 1 -5 6 -1 6 7 -2 -6 0 -6 5 3 -5 -9 2 0 -1 -3 4 1 -4 3 -1 3 0 4 -3 3 2
  -147 -5 -130 105 -114 -199 -102 94 0 124 -7 -104 -20 -70 -26 -15 54 89 -41 -11 -9 -33 108 -44 54 -71 -51 -9 9 32 17 21 -25 -25 -5 -37 -1 15 1 -4 1 -3 9 0 -18 -15 5 2 -31 30 -8 10 10 8 8 -4 -7 8 -1 -8 7 -6 -2 -3 2 3 -5 -8 -10 2 -1 -2 1 6 -5 -2 3 -6 -4 3 2 5 0 -3 -6 0 0 1
  -372 -70 -243 233 -296 -449 -239 231 124 288 -87 -223 -10 -167 -85 -28 48 182 -20 -52 27 -85 271 -93 147 -131 -86 0 18 114 37 89 -47 -42 29 -96 -18 30 -30 -12 21 -6 33 27 -41 -33 19 -1 -43 53 -17 33 25 33 23 -12 -23 3 -4 -16 19 -10 -1 -8 3 7 -24 -7 -13 1 9 -6 2 6 -6 -5 1 -7 -5 4 2 13 9 -9 -7 -5 1 -1
  266 86 205 -315 300 385 180 -223 -67 -290 76 151 76 69 105 107 -33 -86 36 -12 -2 87 -183 74 -137 102 29 -2 -92 -90 15 -12 87 28 7 46 31 6 -19 16 -13 -1 4 -25 12 52 -40 -14 30 -38 14 -16 -13 -26 -16 -20 -3 -11 25 2 -22 0 -9 13 1 16 11 7 27 -11 -5 15 7 -17 -3 -7 4 15 7 3 3 -8 -7 9 -2 5 -5 3
  496 37 339 -394 467 546 259 -283 -57 -451 52 258 21 134 115 114 -92 -173 95 65 48 139 -339 98 -202 175 98 27 -51 -129 -16 -71 121 62 -8 74 40 -14 -18 0 -6 -10 -34 -30 25 58 -38 -23 63 -61 17 -42 -31 -35 -30 -13 8 -4 24 6 -32 7 -6 17 2 14 23 9 31 -15 -13 15 6 -18 -3 -4 1 21 11 -2 2 -11 -12 13 0 6 -3 -1
  134 82 91 -207 189 147 101 -110 -48 -174 55 31 52 -28 110 107 3 3 -7 -26 8 28 -71 62 -110 64 -11 -28 -115 -49 31 62 88 6 48 -5 44 6 -45 22 3 -14 22 1 -16 28 -51 -19 13 -8 8 -4 5 -8 -6 -38 -19 -7 38 -5 -29 -8 -20 7 2 34 4 12 35 -26 -12 16 14 -19 -9 -15 9 19 12 9 4 -5 -4 6 -11 8 -7 1
  -37 59 181 -136 82 147 -12 -92 58 -46 -99 9 113 28 19 82 -51 13 65 -70 68 29 15 -12 34 23 27 45 -7 42 37 54 -6 -1 41 -11 -14 29 -39 -11 -6 21 24 7 23 19 -10 -10 11 -9 6 9 -6 9 -15 -2 -11 -15 -1 -6 21 -5 9 -8 -8 0 2 5 5 0 1 5 9 -4 5 3 -2 -6 1 5 -3 4 0 -8 -4 3 -3 -6
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  204 -170 114 32 191 -36 -8 56 176 -58 -252 127 -174 68 -78 9 -72 -5 119 181 159 72 -179 5 22 49 102 108 186 7 -26 -89 -13 64 -33 -23 -18 -5 24 -82 -4 -12 -74 -24 -13 -24 56 7 40 -2 -16 -37 -30 8 -1 37 21 18 -36 -5 2 5 6 17 24 -19 -12 -16 -22 21 0 -10 -20 10 13 8 -12 -9 -6 -7 -2 0 2 -9 9 -6 8 3
  276 -257 -183 231 53 -194 43 144 51 -49 -31 63 -313 -33 -96 -99 -13 -50 4 249 69 37 -185 -22 -13 21 82 28 164 -71 -85 -129 4 51 -64 -11 -4 -28 41 -62 16 -47 -105 -27 -12 -34 55 17 39 -5 -21 -42 -29 -17 -15 41 30 29 -40 2 -18 19 -13 17 21 -20 1 -18 -16 14 -14 -7 -21 10 3 -7 -11 9 3 -9 6 -6 -2 -1 9 -3 0 -3
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  -327 41 -391 166 -283 -267 -95 105 -78 265 134 -204 109 -141 -54 -49 114 133 -126 -135 -120 -35 234 -41 109 -157 -142 -99 -123 -10 54 61 -44 -90 -26 -4 -4 42 15 38 -26 14 44 0 -8 5 -27 0 -46 25 -5 43 21 0 -4 -8 -29 -23 4 16 12 -6 5 -19 -17 3 4 7 1 -12 2 -1 11 2 -2 3 9 -15 2 10 4 5 2 0 -1 5 -2 5
  235 63 424 -373 362 415 105 -222 76 -306 -109 164 98 119 89 149 -109 -87 139 -7 108 77 -185 61 -109 144 79 81 -19 -3 21 -2 78 55 38 16 19 4 -47 -21 6 5 7 -2 15 37 -18 -27 36 -32 14 -20 -15 -4 -8 -18 -2 -6 24 -11 -11 -5 -1 9 3 16 8 9 18 -5 0 17 8 -8 -1 4 -4 10 2 4 -6 -3 -2 4 -2 -1 -5 0
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  23 110 -89 25 68 -82 38 -13 -111 10 51 -45 -15 -30 70 40 82 65 -98 16 -29 6 -4 28 -28 -20 -41 -36 -69 -30 44 30 4 -25 -3 -14 44 -3 17 18 -14 -12 1 -25 -19 -5 -24 0 -26 23 -16 2 6 -4 8 -7 -2 7 16 -1 -17 -3 -19 3 2 6 4 -5 7 -3 -12 3 4 -8 1 -10 7 4 3 4 8 -6 -4 3 0 1 6 -2
  287 -284 -15 102 175 -71 15 88 191 -89 -236 138 -281 4 -139 -25 -84 -11 141 252 193 111 -221 -44 61 23 136 117 241 -48 -60 -118 -16 76 -54 -34 -38 19 20 -96 -3 -14 -114 -50 -9 -16 67 11 61 -19 -21 -40 -55 -7 -22 54 19 16 -61 -8 11 8 6 22 20 -37 -9 -28 -30 25 -5 -14 -24 11 13 0 -14 -6 -2 -4 7 -2 -8 -10 9 0 7 -8
  -150 32 -18 16 -139 -1 -66 0 -20 58 52 -60 81 2 -14 -34 3 -10 -4 -78 -47 -32 109 -45 40 -40 -20 -20 -21 29 -2 15 -27 -22 2 8 -24 9 -4 19 5 22 26 10 22 10 -2 -1 -19 -8 11 21 7 4 3 -4 -2 -8 -5 3 17 1 9 -8 -14 -6 4 -2 -5 3 7 3 2 6 0 2 1 -6 -1 3 -3 -2 0 3 -3 -1 -2 -3
  -211 20 -332 204 -312 -233 -61 101 -132 153 237 -195 13 -130 0 -99 95 30 -143 -86 -136 -91 200 -62 25 -94 -88 -116 -102 -9 -9 53 -15 -59 0 -3 5 2 -7 53 19 -2 30 22 5 -1 -22 3 -36 12 7 32 26 -3 -2 -19 -12 -7 9 6 0 3 -12 -11 -16 7 8 3 10 -10 -6 4 11 -1 -12 -11 10 10 8 6 2 -2 1 3 -7 1 -6 -2
  -62 226 6 -52 -20 67 81 -84 -191 37 125 -37 107 9 120 33 78 23 -128 -98 -108 -42 61 60 -45 -9 -55 -78 -132 -22 38 85 -2 -46 40 20 34 -13 3 65 -21 15 47 5 3 6 -54 4 -28 12 5 15 21 -1 7 -20 -10 -8 29 7 -11 -8 -11 -10 -14 16 10 10 17 -15 -9 1 13 -11 0 -4 9 3 8 6 8 -5 -5 2 -7 7 1 -2
  

• PROPERTIES
  Unimodular=1
  

• REFERENCES
  I. Gauvin, Even unimodular ideal lattices of rank 88 over Q(zeta_115), preprint, 15 October 2026 (arXiv identifier pending),
  and its addendum: two further extremal lattices, X4969 and X4049 (October 2026).
  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 : 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.
  X4049 = (P, b), an ideal lattice over L = Q(zeta), zeta = zeta_115 (rank 88): P = (4049, zeta^2 + 105*zeta + 2084) 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 method of the paper (same certifying program, same execution model of section 5.4:
  M1, M2, H1-H3), as reported in the addendum: both sides of the criterion were swept completely, each side twice on two different
  machines, with byte-identical receipts 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 P; X4049_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 X4049_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 of the paper (Remark 2.6) and the stabiliser of the class (support {3,331}, see the
  addendum), it is the normaliser in Aut of the group of multiplications by roots of unity. The full automorphism group has not been
  determined: |Aut| >= 460.
  A vector of norm 8 (transported to this basis), as a row of coordinates in this basis:
  NORM8: 156 20 204 -136 153 209 87 -95 16 -128 -55 123 -7 88 33 30 -58 -72 61 33 35 27 -120 45 -56 76 61 36 29 -19 -20 -20 22 38 15 21 0 -13 -4 -4 -2 10 -11 -5 9 8 -2 0 26 -22 7 -17 -12 -4 -1 3 9 1 1 -3 -4 0 3 6 2 -2 2 1 1 2 -1 0 -2 -3 4 3 -4 3 2 -1 0 -3 -2 2 1 2 1 -1
  

• LAST_LINE
  

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  Haftungsausschluss/Disclaimer
  
  
  Gabriele Nebe