[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

Congruence Subgroups of PSL(2,Z)

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^ Statistics

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#PGLMax. LevelMax. Index #All#PSL#PGLMax. LevelMax. Index #All#PSL#PGLMax. LevelMax. 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