[Introduction] - Groups of Genus: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
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The table below gives the number of congruence subgroups of genus
0 to 24,
giving the total number, the number of congruence subgroups up
to conjugacy in PSL(2,Z) and PGL(2,Z),
and the maximal index and maximal level for each genus.
The same is data is also given for the torsion-free subgroups
(the groups with c2=c3=0 in the main tables)
and the cycloidal subgroups (the groups with one equivalence class
of cusps).
All Groups | Torsion Free Groups | Cycloidal Groups | |||||||||||||
Genus | #All | #PSL | #PGL | Max. Level | Max. Index | #All | #PSL | #PGL | Max. Level | Max. Index | #All | #PSL | #PGL | Max. Level | Max. Index |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0 | 1116 | 132 | 121 | 48 | 72 | 254 | 33 | 33 | 32 | 60 | 151 | 26 | 18 | 21 | 21 |
1 | 2801 | 187 | 163 | 52 | 108 | 459 | 48 | 48 | 36 | 108 | 311 | 25 | 16 | 42 | 42 |
2 | 4107 | 177 | 145 | 78 | 108 | 672 | 49 | 49 | 64 | 108 | 410 | 14 | 8 | 63 | 63 |
3 | 6513 | 284 | 241 | 96 | 168 | 1809 | 108 | 105 | 72 | 168 | 638 | 24 | 13 | 84 | 84 |
4 | 7257 | 261 | 215 | 108 | 180 | 1665 | 87 | 86 | 81 | 180 | 824 | 21 | 11 | 80 | 80 |
5 | 9386 | 303 | 256 | 126 | 192 | 3028 | 133 | 125 | 75 | 192 | 696 | 12 | 6 | 126 | 126 |
6 | 10416 | 230 | 175 | 126 | 192 | 1780 | 55 | 45 | 121 | 180 | 378 | 10 | 5 | 126 | 126 |
7 | 18191 | 480 | 388 | 156 | 216 | 6216 | 213 | 191 | 128 | 216 | 1057 | 22 | 12 | 132 | 132 |
8 | 13726 | 277 | 212 | 169 | 220 | 2671 | 83 | 76 | 96 | 156 | 675 | 9 | 5 | 126 | 126 |
9 | 21014 | 469 | 403 | 154 | 288 | 6711 | 208 | 203 | 128 | 288 | 468 | 8 | 4 | 154 | 154 |
10 | 15622 | 304 | 235 | 168 | 324 | 4483 | 133 | 120 | 118 | 324 | 1972 | 30 | 15 | 168 | 168 |
11 | 27466 | 489 | 381 | 198 | 288 | 8450 | 195 | 179 | 147 | 240 | 874 | 10 | 5 | 198 | 198 |
12 | 18095 | 269 | 198 | 210 | 330 | 4978 | 93 | 70 | 142 | 300 | 418 | 6 | 3 | 165 | 165 |
13 | 33241 | 664 | 549 | 231 | 384 | 12447 | 343 | 303 | 162 | 384 | 628 | 9 | 5 | 231 | 231 |
14 | 22871 | 268 | 178 | 252 | 300 | 4581 | 72 | 53 | 167 | 192 | 1098 | 6 | 3 | 252 | 252 |
15 | 40880 | 596 | 485 | 240 | 384 | 16743 | 289 | 263 | 179 | 288 | 440 | 8 | 4 | 240 | 240 |
16 | 30809 | 410 | 294 | 243 | 364 | 8607 | 143 | 123 | 243 | 360 | 930 | 10 | 5 | 220 | 220 |
17 | 54794 | 819 | 667 | 289 | 480 | 17453 | 351 | 317 | 242 | 480 | 478 | 6 | 3 | 210 | 210 |
18 | 24935 | 273 | 191 | 264 | 384 | 4819 | 71 | 60 | 214 | 288 | 774 | 10 | 5 | 264 | 264 |
19 | 60648 | 812 | 647 | 273 | 504 | 24287 | 411 | 375 | 256 | 504 | 320 | 4 | 2 | 240 | 240 |
20 | 31137 | 308 | 203 | 286 | 408 | 9396 | 122 | 85 | 239 | 300 | 0 | 0 | 0 | 0 | 0 |
21 | 66841 | 888 | 729 | 308 | 480 | 27542 | 504 | 450 | 256 | 480 | 2352 | 12 | 6 | 308 | 308 |
22 | 36135 | 365 | 284 | 361 | 486 | 11206 | 152 | 132 | 263 | 432 | 0 | 0 | 0 | 0 | 0 |
23 | 59450 | 686 | 537 | 338 | 504 | 22798 | 312 | 271 | 274 | 384 | 630 | 2 | 1 | 315 | 315 |
24 | 42289 | 336 | 212 | 336 | 546 | 6903 | 78 | 51 | 284 | 336 | 2232 | 22 | 11 | 336 | 336 |
All | 659740 | 10287 | 8109 | 361 | 546 | 209958 | 4286 | 3813 | 284 | 504 | 18754 | 306 | 166 | 336 | 336 |
Chris Cummins and Sebastian Pauli, computed with MAGMA