An entry from the Catalogue of Lattices, which is a joint project of
Gabriele Nebe, RWTH Aachen University (nebe@math.rwth-aachen.de) and
Neil J. A. Sloane (njasloane@gmail.com)

Last modified Fri Jul 18 13:26:56 CEST 2014

INDEX FILE | ABBREVIATIONS

Contents of this file

NAME
DIMENSION
DETERMINANT
MINIMAL_NORM
KISSING_NUMBER
PROPERTIES
GROUP_ORDER
NOTES
REFERENCES
BASIS (needs to be divided by 97)
GRAM
GROUP_GENERATORS
LAST_LINE

• NAME
  T27
  

• DIMENSION
  27
  

• DETERMINANT
  1
  

• MINIMAL_NORM
  3
  

• KISSING_NUMBER
  1640
  

• PROPERTIES
  INTEGRAL=1
  

• GROUP_ORDER
  1268047872
  

• NOTES
  Borcherds' unimodular 27-dim lattice of min norm 3 and group ___.
  

• REFERENCES
  SPLAG, 3rd edition.
  Borcherds' Thesis.
  Bacher and Venkov, Re'seaux entiers unimodulaires...
  

• BASIS (needs to be divided by 97)
  27 27
  1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 51 59 61
  0 97 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8633
  0 0 97 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1843
  0 0 0 97 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4462
  0 0 0 0 97 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7081
  0 0 0 0 0 97 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 291
  0 0 0 0 0 0 97 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2910
  0 0 0 0 0 0 0 97 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5529
  0 0 0 0 0 0 0 0 97 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8148
  0 0 0 0 0 0 0 0 0 97 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1358
  0 0 0 0 0 0 0 0 0 0 97 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3977
  0 0 0 0 0 0 0 0 0 0 0 97 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6596
  0 0 0 0 0 0 0 0 0 0 0 0 97 0 0 0 0 0 0 0 0 0 0 0 0 0 9215
  0 0 0 0 0 0 0 0 0 0 0 0 0 97 0 0 0 0 0 0 0 0 0 0 0 0 2425
  0 0 0 0 0 0 0 0 0 0 0 0 0 0 97 0 0 0 0 0 0 0 0 0 0 0 5044
  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 97 0 0 0 0 0 0 0 0 0 0 7663
  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 97 0 0 0 0 0 0 0 0 0 873
  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 97 0 0 0 0 0 0 0 0 3492
  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 97 0 0 0 0 0 0 0 6111
  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 97 0 0 0 0 0 0 8730
  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 97 0 0 0 0 0 1940
  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 97 0 0 0 0 4559
  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 97 0 0 0 7178
  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 97 0 0 388
  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 97 0 5626
  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 97 6693
  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9409
  

• GRAM
  27 27
  3 0 0 0 -1 0 -1 0 0 -1 -1 -1 0 0 -1 0 -1 0 -1 -1 1 -1 0 -1 -1 -1 -2
  0 3 0 0 1 0 0 0 0 0 0 1 -1 -1 1 1 0 -1 -1 0 0 -1 0 -1 -1 0 1
  0 0 3 0 1 1 0 -1 1 1 0 1 0 0 0 0 0 -1 -1 -1 0 0 0 -1 0 -1 0
  0 0 0 3 1 1 1 0 0 1 1 0 0 -1 0 -1 1 1 1 1 -1 1 1 1 1 -1 -1
  -1 1 1 1 3 1 1 -1 1 1 1 1 -1 -1 1 -1 1 0 0 0 0 0 0 0 0 -1 1
  0 0 1 1 1 3 1 0 1 0 1 1 -1 -1 1 -1 1 1 -1 1 1 -1 0 -1 1 -2 -1
  -1 0 0 1 1 1 3 0 0 0 1 1 0 -1 0 -1 1 1 1 1 0 0 0 0 0 -1 1
  0 0 -1 0 -1 0 0 3 -1 -1 0 -1 1 1 0 1 0 0 -1 1 -1 1 -1 0 0 1 -1
  0 0 1 0 1 1 0 -1 3 0 0 0 -1 -1 1 -1 1 0 0 0 2 -1 0 0 0 -1 1
  -1 0 1 1 1 0 0 -1 0 3 1 1 0 0 0 0 1 -1 1 -1 -1 1 1 1 1 1 0
  -1 0 0 1 1 1 1 0 0 1 3 1 -1 0 0 -1 1 1 1 0 0 1 0 0 2 -1 0
  -1 1 1 0 1 1 1 -1 0 1 1 3 -1 0 0 0 0 -1 0 0 0 -1 1 -1 1 0 1
  0 -1 0 0 -1 -1 0 1 -1 0 -1 -1 3 1 -1 1 -1 -1 0 -1 -1 1 -1 1 -1 1 -1
  0 -1 0 -1 -1 -1 -1 1 -1 0 0 0 1 3 -1 1 -1 -1 0 -1 -1 1 0 1 1 2 -1
  -1 1 0 0 1 1 0 0 1 0 0 0 -1 -1 3 0 1 1 -1 1 0 -1 0 0 0 0 1
  0 1 0 -1 -1 -1 -1 1 -1 0 -1 0 1 1 0 3 -1 -2 -1 -1 -1 0 0 0 -1 2 0
  -1 0 0 1 1 1 1 0 1 1 1 0 -1 -1 1 -1 3 1 1 1 0 1 0 1 1 0 1
  0 -1 -1 1 0 1 1 0 0 -1 1 -1 -1 -1 1 -2 1 4 1 2 0 0 0 0 1 -2 0
  -1 -1 -1 1 0 -1 1 -1 0 1 1 0 0 0 -1 -1 1 1 4 0 0 1 1 2 1 0 1
  -1 0 -1 1 0 1 1 1 0 -1 0 0 -1 -1 1 -1 1 2 0 4 -1 0 1 0 1 0 1
  1 0 0 -1 0 1 0 -1 2 -1 0 0 -1 -1 0 -1 0 0 0 -1 4 -2 -1 -1 -1 -2 0
  -1 -1 0 1 0 -1 0 1 -1 1 1 -1 1 1 -1 0 1 0 1 0 -2 4 -1 2 1 1 0
  0 0 0 1 0 0 0 -1 0 1 0 1 -1 0 0 0 0 0 1 1 -1 -1 4 0 1 1 0
  -1 -1 -1 1 0 -1 0 0 0 1 0 -1 1 1 0 0 1 0 2 0 -1 2 0 4 1 2 0
  -1 -1 0 1 0 1 0 0 0 1 2 1 -1 1 0 -1 1 1 1 1 -1 1 1 1 4 0 -1
  -1 0 -1 -1 -1 -2 -1 1 -1 1 -1 0 1 2 0 2 0 -2 0 0 -2 1 1 2 0 5 1
  -2 1 0 -1 1 -1 1 -1 1 0 0 1 -1 -1 1 0 1 0 1 1 0 0 0 0 -1 1 5
  

• GROUP_GENERATORS
  2
  27 27
  0 0 0 -1 1 0 0 2 0 1 -1 0 0 -1 -1 0 -1 1 -1 -1 0 0 1 1 1 -1 1
  0 0 0 -1 0 1 0 1 0 0 -1 1 0 -1 0 0 -1 1 0 -1 0 1 1 1 0 0 0
  1 0 1 -1 1 -1 -1 2 -1 0 0 1 1 -2 0 0 0 1 0 0 1 0 1 1 1 0 0
  0 1 -1 0 -1 1 1 -2 1 1 0 -2 0 2 0 0 0 -1 0 1 -1 0 -1 -1 0 0 0
  0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
  1 0 1 -2 0 0 -1 3 0 0 0 2 0 -3 0 -1 -1 1 0 -1 0 1 1 2 0 0 0
  -1 0 0 1 -2 0 0 -1 1 -1 1 0 -1 1 0 -1 1 -1 0 0 -2 -1 -1 0 -2 0 -1
  -1 0 0 1 -1 1 0 -1 0 0 1 -1 -1 1 0 0 0 -1 0 0 -1 -1 -1 0 -1 0 0
  1 0 1 -1 1 -1 -1 2 0 -1 0 2 1 -3 0 0 -1 2 0 -1 1 1 1 1 1 1 0
  0 1 -1 1 -1 1 1 -3 0 1 0 -3 0 3 1 0 0 -2 1 2 0 0 -1 -2 0 0 0
  0 0 -2 1 -2 1 1 -3 2 1 1 -1 -1 3 1 0 0 -2 1 2 -2 0 -2 -2 -2 0 -1
  1 0 0 -1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 -1 0 -1
  0 0 1 1 0 -1 0 -1 -1 0 1 -1 0 1 0 0 1 -1 0 1 0 -2 -1 0 0 0 0
  0 0 -1 1 0 0 0 -2 0 1 1 -1 0 3 0 1 1 -1 0 2 0 -1 -1 -1 -1 -1 0
  0 0 1 0 0 1 -1 2 -1 -1 -1 1 0 -2 0 -1 -1 1 0 -2 1 1 1 1 1 1 1
  0 0 1 -1 1 0 -1 1 -1 0 0 1 1 -1 0 0 0 1 0 0 1 0 1 1 0 0 0
  -2 1 -1 1 -2 1 0 -1 1 0 0 -1 -1 1 -1 -1 0 0 -1 -1 -1 0 0 0 -1 0 0
  -1 0 -1 1 -1 1 0 1 1 0 -1 0 -1 0 -1 -1 0 0 -1 -2 -1 0 0 0 0 0 1
  -1 1 -2 1 -1 0 1 -3 2 0 0 -1 0 3 0 0 1 -1 0 1 -1 0 -1 -2 -1 0 -1
  0 0 0 0 0 1 0 0 0 -1 0 0 0 -1 0 0 0 0 0 -1 0 1 0 0 0 1 0
  1 -1 1 -2 1 -2 -1 3 1 -1 0 4 0 -4 0 0 -1 2 0 -1 0 1 1 2 0 0 -1
  -2 1 -2 2 -2 2 1 -4 1 1 1 -4 -1 4 -1 0 1 -2 0 1 -2 -2 -2 -2 -1 0 0
  1 1 -1 0 1 0 1 -2 0 1 -1 -1 2 2 1 1 0 0 0 2 2 2 0 -2 1 0 0
  -1 1 -1 1 -1 1 0 -3 1 0 1 -2 0 3 -1 0 1 -1 0 1 -1 -1 -1 -1 -1 0 0
  1 1 -2 0 -1 2 1 -3 1 1 1 -2 0 3 1 0 0 -2 1 2 -1 1 -1 -2 -1 0 0
  -1 0 0 2 0 0 0 -3 -1 0 1 -2 0 3 0 1 1 -1 0 2 1 -1 -1 -1 -1 0 0
  -1 -1 0 1 0 -1 -1 0 0 -2 0 2 0 -1 0 0 1 1 0 -1 0 0 0 0 -1 1 -1
  27 27
  0 1 0 0 0 1 0 1 0 1 -1 -1 0 0 0 -1 -1 0 0 -1 0 0 0 0 1 0 1
  -1 1 0 0 0 1 0 2 0 1 -2 0 0 -1 -2 0 -1 2 -1 -2 1 0 1 1 1 0 1
  0 -1 0 2 -1 0 1 -1 -1 0 0 -2 -1 1 1 0 0 -2 0 0 0 -1 -1 -1 1 0 1
  0 0 0 -1 0 -1 -1 1 0 0 0 2 1 -1 0 0 0 1 -1 0 1 1 1 1 0 -1 0
  0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
  0 1 0 0 -1 2 1 -1 -1 0 0 -3 0 1 1 -1 -1 -2 1 0 0 0 0 -1 1 0 1
  -1 0 -1 0 -1 1 0 -1 0 0 0 -1 -1 1 0 0 0 -1 0 0 -1 0 0 0 -1 -1 0
  0 1 -1 0 1 1 1 -2 0 1 -1 -2 1 2 0 1 0 0 0 1 1 1 0 -2 1 0 0
  0 0 1 1 0 -1 0 1 -1 0 0 -1 0 0 0 0 0 0 0 0 1 -1 0 0 1 0 1
  0 -1 1 -1 0 -1 -1 2 0 -1 1 2 0 -3 -1 0 0 1 -1 -1 0 0 1 2 0 0 0
  -1 0 0 0 -1 0 0 -1 1 -1 1 0 0 0 0 0 0 0 0 0 -1 0 0 0 -1 0 -1
  0 -1 0 0 -1 0 0 0 0 0 1 0 -1 0 0 0 0 -1 0 0 -1 -1 0 1 -1 -1 0
  2 -1 0 -1 2 -1 0 0 -1 0 0 1 1 -1 1 1 0 0 0 1 1 1 0 0 1 0 0
  1 -1 -1 0 0 -1 1 -2 1 1 1 0 0 2 1 1 1 -1 0 2 -1 0 -1 -1 -1 -1 -1
  0 1 0 0 0 1 1 0 0 0 -1 -1 0 0 0 0 -1 0 1 0 1 1 0 -1 1 1 0
  0 0 0 0 1 -1 1 0 0 1 -1 0 0 0 -1 1 0 1 -1 0 1 0 0 0 1 0 0
  -1 0 1 -1 0 0 -1 2 0 -1 0 1 0 -2 -1 0 0 1 0 -1 0 0 1 1 0 0 0
  -1 1 -1 0 -1 1 0 -1 1 -1 0 0 0 1 1 -1 0 -1 1 0 -1 1 0 -1 -1 0 -1
  -1 -1 0 -1 0 -2 -2 2 1 -1 1 4 0 -2 -1 0 1 2 -1 -1 -1 0 1 2 -2 -1 -1
  -1 1 -1 1 -1 1 0 -2 0 0 0 -1 0 3 0 0 1 -1 0 1 0 0 0 -1 -1 -1 0
  0 1 1 0 0 1 0 1 -1 0 0 -2 0 -1 0 -1 -1 0 1 -1 0 -1 0 0 1 1 1
  0 -1 1 -1 1 -2 -1 1 0 -1 0 3 1 -2 0 1 1 2 -1 0 1 1 1 1 0 0 -1
  -1 0 -2 1 -2 0 0 -2 2 1 1 0 -1 3 -1 0 1 -1 -1 1 -2 -1 -1 0 -2 -2 0
  1 -1 1 -2 1 -3 -1 2 0 0 1 4 1 -2 0 1 1 2 -1 1 1 1 1 2 -1 -1 -1
  0 -1 0 0 -1 -1 0 -1 1 -1 2 1 0 1 1 0 1 -1 0 1 -1 0 0 0 -2 -1 -1
  1 -1 0 -1 1 -2 0 0 1 1 1 2 0 0 -1 2 1 1 -1 2 0 0 0 1 -1 -1 -1
  -1 -1 1 1 0 -2 -1 1 0 -1 0 2 -1 -1 -1 1 1 1 0 0 0 -1 0 1 -1 0 -1
  

• LAST_LINE
  

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  Haftungsausschluss/Disclaimer
  
  
  Gabriele Nebe