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 Di 9. Jun 19:06:42 CEST 2020

INDEX FILE | ABBREVIATIONS

Contents of this file

NAME
DIM
GRAM
DET
GROUP_ORDER
GROUP_GENERATORS
PROPERTIES
MINIMAL_NORM
KISSING_NUMBER
NOTES
REFERENCES
LAST_LINE

• NAME
  dim24modular21Group6912
  

• DIM
  24
  

• GRAM
  24 24
  12 4 4 4 2 4 4 0 4 0 -5 -4 3 -4 3 -4 -6 3 -2 -2 -4 -3 -2 -3
  4 12 0 -4 -4 2 4 0 4 0 -1 -1 1 -4 -2 4 3 1 2 2 -2 -4 -6 4
  4 0 12 2 -4 -4 1 0 1 0 -7 -4 1 -1 4 -2 0 1 -1 -1 4 2 0 1
  4 -4 2 12 6 0 -4 0 -4 0 1 -2 -2 -4 1 -4 -4 -2 -6 2 -1 2 2 -1
  2 -4 -4 6 14 0 -2 0 -2 0 0 -1 2 -1 -4 -3 -3 2 -3 0 -3 4 2 -7
  4 2 -4 0 0 12 2 0 2 0 -2 4 4 -1 1 -1 -7 4 0 -1 -4 -4 -1 -1
  4 4 1 -4 -2 2 12 -4 4 -4 -3 2 2 0 -3 0 3 0 -2 -6 0 1 -6 2
  0 0 0 0 0 0 -4 12 4 4 -3 -4 2 -4 1 -4 -1 6 2 -2 -4 -6 2 -1
  4 4 1 -4 -2 2 4 4 12 4 -7 -1 0 0 -2 0 0 2 2 -2 -3 -6 -2 -4
  0 0 0 0 0 0 -4 4 4 12 -3 -2 3 4 3 -4 -3 5 -2 2 -1 0 -2 0
  -5 -1 -7 1 0 -2 -3 -3 -7 -3 18 -1 -8 2 2 5 6 -4 -2 5 0 -2 3 6
  -4 -1 -4 -2 -1 4 2 -4 -1 -2 -1 12 2 1 -5 4 1 -2 0 0 0 4 -1 0
  3 1 1 -2 2 4 2 2 0 3 -8 2 14 0 0 -5 -7 9 0 -4 0 3 -3 0
  -4 -4 -1 -4 -1 -1 0 -4 0 4 2 1 0 12 1 0 0 -1 -2 2 3 4 2 -1
  3 -2 4 1 -4 1 -3 1 -2 3 2 -5 0 1 14 -3 -3 3 0 0 -1 -1 2 2
  -4 4 -2 -4 -3 -1 0 -4 0 -4 5 4 -5 0 -3 12 6 -6 2 6 2 -1 2 1
  -6 3 0 -4 -3 -7 3 -1 0 -3 6 1 -7 0 -3 6 16 -5 0 1 2 2 -2 6
  3 1 1 -2 2 4 0 6 2 5 -4 -2 9 -1 3 -6 -5 16 1 -4 -4 -2 -2 -1
  -2 2 -1 -6 -3 0 -2 2 2 -2 -2 0 0 -2 0 2 0 1 12 -2 0 -3 0 -2
  -2 2 -1 2 0 -1 -6 -2 -2 2 5 0 -4 2 0 6 1 -4 -2 12 2 -1 2 1
  -4 -2 4 -1 -3 -4 0 -4 -3 -1 0 0 0 3 -1 2 2 -4 0 2 12 5 -2 6
  -3 -4 2 2 4 -4 1 -6 -6 0 -2 4 3 4 -1 -1 2 -2 -3 -1 5 16 -2 2
  -2 -6 0 2 2 -1 -6 2 -2 -2 3 -1 -3 2 2 2 -2 -2 0 2 -2 -2 12 -6
  -3 4 1 -1 -7 -1 2 -1 -4 0 6 0 0 -1 2 1 6 -1 -2 1 6 2 -6 16
  

• DET
  3^12*7^12
  

• GROUP_ORDER
  2^8 * 3^3
  

• GROUP_GENERATORS
  [6 1 19 22 12 -21 5 -1 33 38 20 -35 0 0 0 0 0 0 14 12 0 -20 4 4]
  [1 -17 -6 -11 -9 14 -17 -7 -8 -54 -36 49 0 0 0 0 0 0 -42 -30 -24 56 4 -10]
  [6 -11 29 8 -3 -7 1 -29 54 36 8 -28 0 0 0 0 0 0 28 24 0 -38 8 6]
  [6 -65 -2 -21 -27 35 -50 -32 -8 -167 -120 154 0 0 0 0 0 0 -144 -102 -82 190
  16 -34]
  [7 -48 77 -26 -46 28 5 -131 155 91 3 -56 0 0 0 0 0 0 122 100 18 -162 22 24]
  [16 -112 71 -31 -60 49 -50 -141 133 -98 -124 112 0 0 0 0 0 0 -62 -32 -80 80
  38 -18]
  [1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
  [4 -4 2 9 5 -7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
  [1 0 -3 -8 -5 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
  [12 -15 6 28 15 -21 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
  [0 3 -8 -33 -21 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
  [14 -15 0 5 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
  [103 -85 457 230 40 -217 109 -286 826 776 304 -658 0 0 0 0 0 0 439 369 42
  -609 105 105]
  [93 -853 -100 -230 -306 427 -558 -418 -286 -2033 -1425 1897 0 0 0 0 0 0
  -1566 -1118 -861 2079 147 -378]
  [54 -318 181 -78 -160 133 -114 -360 319 -258 -319 295 0 0 0 0 0 0 -168 -90
  -200 216 93 -48]
  [182 -774 438 65 -208 105 -309 -665 695 -739 -791 767 0 0 0 0 0 0 -738 -465
  -623 950 231 -180]
  [57 -281 327 -79 -187 112 -20 -489 614 175 -109 -68 0 0 0 0 0 0 204 196 -69
  -282 110 36]
  [205 -774 559 258 -103 -70 -239 -617 804 -571 -695 599 0 0 0 0 0 0 -776 -490
  -627 984 237 -181]
  [42 -84 252 0 -84 0 58 -265 481 386 100 -301 1 0 0 0 0 0 292 243 42 -399 63
  63]
  [84 -672 -84 -126 -210 294 -387 -337 -250 -1499 -1059 1414 -93 -71 -42 126 0
  -21 -1125 -803 -609 1491 105 -273]
  [36 -240 114 -84 -138 126 -72 -282 202 -204 -247 238 -15 -12 -8 21 0 -3 -105
  -54 -128 135 63 -33]
  [114 -618 234 -78 -246 210 -227 -559 361 -712 -699 742 -67 -52 -31 92 0 -15
  -547 -351 -426 710 153 -141]
  [24 -210 192 -150 -198 168 -13 -388 379 54 -130 28 -1 -2 -1 3 -1 0 157 146
  -30 -213 71 24]
  [138 -606 336 102 -138 42 -176 -499 461 -557 -600 581 -62 -48 -28 84 0 -13
  -582 -374 -437 744 159 -142]
  
  [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 0 1 -2 -5 -4 -2 4 0 0 0 0 0 0 0 0 0 0 0 0]
  [0 0 0 0 0 0 5 -3 2 10 6 -8 0 0 0 0 0 0 0 0 0 0 0 0]
  [0 0 0 0 0 0 3 -9 -15 -11 -6 12 0 0 0 0 0 0 0 0 0 0 0 0]
  [0 0 0 0 0 0 17 -9 6 37 23 -30 0 0 0 0 0 0 0 0 0 0 0 0]
  [0 0 0 0 0 0 18 -18 -12 18 12 -11 0 0 0 0 0 0 0 0 0 0 0 0]
  [-6 -1 -19 -22 -12 21 -5 1 -33 -38 -20 35 0 0 0 0 0 0 -14 -12 0 20 -4 -4]
  [1 3 11 17 9 -14 16 -1 20 51 29 -42 0 0 0 0 0 0 32 24 14 -44 0 8]
  [-21 -4 -53 -82 -49 77 -10 -19 -85 -92 -53 91 0 0 0 0 0 0 -2 -6 16 8 -12 -4]
  [0 23 17 39 27 -35 47 8 44 158 99 -133 0 0 0 0 0 0 114 84 52 -154 -4 28]
  [-77 -39 -182 -316 -198 301 -42 -106 -288 -355 -219 357 0 0 0 0 0 0 -8 -20
  50 30 -38 -16]
  [-65 -2 -137 -219 -132 210 19 -72 -199 -119 -68 147 0 0 0 0 0 0 120 76 102
  -146 -38 18]
  [-197 5 -518 -775 -440 707 -73 -161 -772 -779 -424 763 1 0 0 0 0 0 -29 -63
  132 93 -111 -39]
  [24 158 309 457 275 -406 432 -110 339 1259 763 -1022 30 28 -11 -48 18 12 939
  700 400 -1269 -15 228]
  [-123 9 -226 -393 -236 371 81 -189 -374 -153 -82 223 -6 -3 -10 6 6 0 237 153
  188 -288 -69 36]
  [-205 134 -348 -523 -281 483 259 -266 -612 109 125 57 18 17 -11 -29 14 8 699
  474 463 -884 -139 134]
  [-212 -44 -456 -791 -481 742 10 -300 -712 -627 -369 678 -12 -8 -11 15 5 -2
  131 59 199 -123 -109 -5]
  [-196 222 -379 -471 -203 406 266 -154 -643 173 219 -38 12 12 -11 -21 12 7
  698 469 474 -882 -153 139]
  [-252 -18 -705 -999 -564 924 -138 -162 -1113 -1179 -636 1134 0 0 0 0 0 0
  -176 -189 132 303 -153 -81]
  [6 159 291 444 264 -378 558 -111 525 1704 1014 -1386 0 0 0 0 0 0 1275 952
  547 -1731 -15 312]
  [-162 -3 -330 -537 -321 513 84 -213 -498 -237 -129 321 0 0 0 0 0 0 309 198
  254 -378 -93 48]
  [-282 114 -573 -792 -432 753 284 -270 -847 14 88 183 0 0 0 0 0 0 823 549 588
  -1043 -187 158]
  [-274 -74 -634 -1040 -628 985 -28 -346 -986 -930 -542 979 0 0 0 0 0 0 121 37
  252 -96 -148 -17]
  [-275 224 -623 -754 -350 680 281 -128 -907 64 194 88 0 0 0 0 0 0 834 551 605
  -1053 -207 166]
  
  [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 0 -2 0 4 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0]
  [0 0 0 0 0 0 -6 4 -2 -9 -5 7 0 0 0 0 0 0 0 0 0 0 0 0]
  [0 0 0 0 0 0 -9 0 10 -8 -8 7 0 0 0 0 0 0 0 0 0 0 0 0]
  [0 0 0 0 0 0 -21 16 -5 -28 -15 21 0 0 0 0 0 0 0 0 0 0 0 0]
  [0 0 0 0 0 0 -28 13 7 -33 -22 26 0 0 0 0 0 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 0]
  [-2 0 4 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
  [-6 4 -2 -9 -5 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
  [-9 0 10 -8 -8 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
  [-21 16 -5 -28 -15 21 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
  [-28 13 7 -33 -22 26 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
  [-57 24 -27 -102 -57 84 -57 24 -27 -102 -57 84 0 0 0 0 0 0 -1 0 0 0 0 0]
  [-90 39 27 -108 -69 84 -90 39 27 -108 -69 84 0 0 0 0 0 0 4 -4 28 0 -21 0]
  [-60 30 3 -81 -48 63 -60 30 3 -81 -48 63 0 0 0 0 0 0 14 9 14 -18 -6 3]
  [-134 59 -3 -193 -115 154 -134 59 -3 -193 -115 154 0 0 0 0 0 0 10 3 25 -11
  -16 2]
  [-75 38 -5 -107 -62 84 -75 38 -5 -107 -62 84 0 0 0 0 0 0 14 10 12 -19 -4 3]
  [-131 57 4 -185 -112 147 -131 57 4 -185 -112 147 0 0 0 0 0 0 24 14 29 -31
  -14 6]
  [-57 24 -27 -102 -57 84 -57 24 -27 -102 -57 84 -1 0 0 0 0 0 0 0 0 0 0 0]
  [-90 39 27 -108 -69 84 -90 39 27 -108 -69 84 4 -4 28 0 -21 0 0 0 0 0 0 0]
  [-60 30 3 -81 -48 63 -60 30 3 -81 -48 63 14 9 14 -18 -6 3 0 0 0 0 0 0]
  [-134 59 -3 -193 -115 154 -134 59 -3 -193 -115 154 10 3 25 -11 -16 2 0 0 0 0
  0 0]
  [-75 38 -5 -107 -62 84 -75 38 -5 -107 -62 84 14 10 12 -19 -4 3 0 0 0 0 0 0]
  [-131 57 4 -185 -112 147 -131 57 4 -185 -112 147 24 14 29 -31 -14 6 0 0 0 0
  0 0]
  
  [-6 0 -26 -30 -16 28 -5 0 -26 -30 -16 28 -7 -6 0 10 -2 -2 -7 -6 0 10 -2 -2]
  [-12 -10 -36 -62 -34 56 -12 -9 -36 -62 -34 56 -14 -11 -4 20 -2 -4 -14 -11 -4
  20 -2 -4]
  [-22 -10 -76 -102 -60 98 -22 -10 -75 -102 -60 98 -19 -16 -3 28 -4 -6 -19 -16
  -3 28 -4 -6]
  [-38 -16 -124 -190 -96 168 -38 -16 -124 -189 -96 168 -33 -27 -7 49 -8 -10
  -33 -27 -7 49 -8 -10]
  [-72 -62 -232 -340 -214 336 -72 -62 -232 -340 -213 336 -52 -44 -8 78 -11 -18
  -52 -44 -8 78 -11 -18]
  [-104 -64 -342 -502 -286 472 -104 -64 -342 -502 -286 473 -84 -70 -15 125 -19
  -27 -84 -70 -15 125 -19 -27]
  [5 0 26 30 16 -28 6 0 26 30 16 -28 7 6 0 -10 2 2 7 6 0 -10 2 2]
  [12 9 36 62 34 -56 12 10 36 62 34 -56 14 11 4 -20 2 4 14 11 4 -20 2 4]
  [22 10 75 102 60 -98 22 10 76 102 60 -98 19 16 3 -28 4 6 19 16 3 -28 4 6]
  [38 16 124 189 96 -168 38 16 124 190 96 -168 33 27 7 -49 8 10 33 27 7 -49 8
  10]
  [72 62 232 340 213 -336 72 62 232 340 214 -336 52 44 8 -78 11 18 52 44 8 -78
  11 18]
  [104 64 342 502 286 -473 104 64 342 502 286 -472 84 70 15 -125 19 27 84 70
  15 -125 19 27]
  [-56 11 -275 -310 -164 287 -60 -12 -264 -312 -168 294 -74 -63 0 105 -21 -21
  -73 -63 0 105 -21 -21]
  [-123 -97 -358 -644 -351 574 -126 -102 -390 -642 -354 588 -147 -116 -42 210
  -21 -42 -147 -115 -42 210 -21 -42]
  [-75 -39 -251 -375 -211 346 -78 -48 -264 -384 -216 360 -81 -66 -17 117 -15
  -24 -81 -66 -16 117 -15 -24]
  [-158 -60 -578 -833 -447 754 -166 -90 -596 -844 -460 782 -187 -153 -31 268
  -39 -54 -187 -153 -31 269 -39 -54]
  [-84 -44 -307 -430 -248 405 -90 -56 -322 -448 -256 426 -89 -74 -12 129 -20
  -27 -89 -74 -12 129 -19 -27]
  [-138 -9 -539 -713 -358 635 -152 -52 -566 -760 -398 698 -148 -124 -17 216
  -39 -44 -148 -124 -17 216 -39 -43]
  [49 11 271 320 172 -301 66 -12 282 318 168 -294 74 63 0 -105 21 21 73 63 0
  -105 21 21]
  [129 92 398 658 363 -602 126 108 366 660 360 -588 147 116 42 -210 21 42 147
  115 42 -210 21 42]
  [78 45 268 393 221 -368 78 42 258 384 216 -354 81 66 17 -117 15 24 81 66 16
  -117 15 24]
  [158 83 609 864 471 -800 170 66 592 854 458 -772 187 153 31 -268 39 54 187
  153 31 -269 39 54]
  [86 54 329 458 261 -435 90 46 314 440 254 -414 89 74 12 -129 20 27 89 74 12
  -129 19 27]
  [145 52 579 778 409 -716 148 8 550 728 364 -646 148 124 17 -216 39 44 148
  124 17 -216 39 43]
  
  [-6 0 -26 -30 -16 28 -6 0 -26 -30 -16 28 -7 -6 0 10 -2 -2 -7 -6 0 10 -2 -2]
  [-12 -10 -36 -62 -34 56 -12 -10 -36 -62 -34 56 -14 -11 -4 20 -2 -4 -14 -11
  -4 20 -2 -4]
  [-22 -10 -76 -102 -60 98 -22 -10 -76 -102 -60 98 -19 -16 -3 28 -4 -6 -19 -16
  -3 28 -4 -6]
  [-38 -16 -124 -190 -96 168 -38 -16 -124 -190 -96 168 -33 -27 -7 49 -8 -10
  -33 -27 -7 49 -8 -10]
  [-72 -62 -232 -340 -214 336 -72 -62 -232 -340 -214 336 -52 -44 -8 78 -11 -18
  -52 -44 -8 78 -11 -18]
  [-104 -64 -342 -502 -286 472 -104 -64 -342 -502 -286 472 -84 -70 -15 125 -19
  -27 -84 -70 -15 125 -19 -27]
  [5 0 26 30 16 -28 5 0 26 30 16 -28 7 6 0 -10 2 2 7 6 0 -10 2 2]
  [12 9 36 62 34 -56 12 9 36 62 34 -56 14 11 4 -20 2 4 14 11 4 -20 2 4]
  [22 10 75 102 60 -98 22 10 75 102 60 -98 19 16 3 -28 4 6 19 16 3 -28 4 6]
  [38 16 124 189 96 -168 38 16 124 189 96 -168 33 27 7 -49 8 10 33 27 7 -49 8
  10]
  [72 62 232 340 213 -336 72 62 232 340 213 -336 52 44 8 -78 11 18 52 44 8 -78
  11 18]
  [104 64 342 502 286 -473 104 64 342 502 286 -473 84 70 15 -125 19 27 84 70
  15 -125 19 27]
  [-56 11 -275 -310 -164 287 -66 12 -282 -318 -168 294 -74 -63 0 105 -21 -21
  -73 -63 0 105 -21 -21]
  [-123 -97 -358 -644 -351 574 -126 -108 -366 -660 -360 588 -147 -116 -42 210
  -21 -42 -147 -115 -42 210 -21 -42]
  [-75 -39 -251 -375 -211 346 -78 -42 -258 -384 -216 354 -81 -66 -17 117 -15
  -24 -81 -66 -16 117 -15 -24]
  [-158 -60 -578 -833 -447 754 -170 -66 -592 -854 -458 772 -187 -153 -31 268
  -39 -54 -187 -153 -31 269 -39 -54]
  [-84 -44 -307 -430 -248 405 -90 -46 -314 -440 -254 414 -89 -74 -12 129 -20
  -27 -89 -74 -12 129 -19 -27]
  [-138 -9 -539 -713 -358 635 -148 -8 -550 -728 -364 646 -148 -124 -17 216 -39
  -44 -148 -124 -17 216 -39 -43]
  [49 11 271 320 172 -301 60 12 264 312 168 -294 74 63 0 -105 21 21 73 63 0
  -105 21 21]
  [129 92 398 658 363 -602 126 102 390 642 354 -588 147 116 42 -210 21 42 147
  115 42 -210 21 42]
  [78 45 268 393 221 -368 78 48 264 384 216 -360 81 66 17 -117 15 24 81 66 16
  -117 15 24]
  [158 83 609 864 471 -800 166 90 596 844 460 -782 187 153 31 -268 39 54 187
  153 31 -269 39 54]
  [86 54 329 458 261 -435 90 56 322 448 256 -426 89 74 12 -129 20 27 89 74 12
  -129 19 27]
  [145 52 579 778 409 -716 152 52 566 760 398 -698 148 124 17 -216 39 44 148
  124 17 -216 39 43]
  

• PROPERTIES
  INTEGRAL=1
  MODULAR=21
  

• MINIMAL_NORM
  12
  

• KISSING_NUMBER
  744
  

• NOTES
  Even 21-modular lattice constructed from totally definite quaternion algebra over maximal real subfield of cyclotomic field.
  

• REFERENCES
  Xiaolu Hou, "Algebraic Constructions of Modular Lattices (Doctoral Thesis)", in preparation, Nanyang Technological University, Singapore.
  

• LAST_LINE
  

  ---

  Haftungsausschluss/Disclaimer
  
  
  Gabriele Nebe