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Author: William A. Stein
Primes of ordinary reduction of X0(p)

Primes of ordinary reduction of X0(p)

William A. Stein

September, 1998

Let p and $\ell$ be distinct primes. The modular curve X0(p) is ordinary at $\ell$ iff the Hecke operator T is invertible modulo $\ell$. The following is a table which enumerates, for each prime p between 23 and 997, the set of primes $\ell\leq 101$ for which the modular curve X0(p) is not ordinary.1



p non-ordinary $\ell\leq 101$
23 43
29 -
31 -
37 2, 3, 5, 17, 19
41 -
43 2, 7, 17, 37
47 -
53 3, 5, 11, 17
59 2,
61 31, 101
67 2, 41, 71, 97
71 3, 5, 37
73 3, 43, 59, 71, 79
79 -
83 2, 5, 47, 73, 89
89 7, 29, 41, 101
97 7, 23
101 2, 7
103 -
107 2
109 3, 13, 79
113 7, 11
127 3, 37
131 2, 11, 29
137 7, 29
139 2, 7, 13, 19, 53
149 3, 13
151 5, 13, 37, 41, 83
157 5
163 2, 3, 17
167 11
173 -
179 2, 3, 17, 53, 71
181 29
191 13, 17, 71
193 5
197 2, 3, 5, 17, 59
199 7, 11, 53, 83
211 2, 29, 67
223 -
227 2, 7, 19, 89
229 7, 17, 37
233 23
239 29, 97
241 101
251 2
257 23
263 13, 19
269 2, 3, 73
271 5, 7
277 5, 23
281 13, 19, 59
283 2
293 3
307 2, 3, 5, 7, 13, 23, 29
311 -
313 19, 31
317 5, 11
331 2, 11, 41
337 5, 61
347 2, 5, 41, 61, 101
349 2
353 2, 5, 13, 19, 37
359 3, 5, 13, 23, 43, 47, 53, 97
367 5, 13
373 2, 7
379 2
383 13
389 2, 5, 7, 31, 79
397 3, 5, 7
401 2
409 2, 83
419 2
421 13, 41
431 3, 11
433 89
439 3, 31
443 2, 5, 13, 29
449 7
457 7, 61
461 5, 17, 31
463 5
467 2, 3
479 17
487 2, 3, 43, 67
491 2
499 2, 5, 19
503 3, 5, 7, 17, 29, 37, 71, 89, 101
509 5, 13
521 3, 5
523 2, 5, 7, 11, 41, 43
541 -
547 2, 7, 17, 23
557 2, 5, 23, 31, 43, 89
563 2, 97
569 7, 71
571 2, 17, 19, 23, 37, 41, 43, 47, 79, 83
577 2, 3, 5, 23, 29, 47
587 2, 11, 19, 43
593 3, 23, 31, 59, 83
599 19
601 -
607 31, 59
613 79
617 23
619 2, 5, 7, 41
631 5, 101
641 17, 59
643 2, 7, 13, 43, 67
647 3, 29, 61, 79
653 83
659 2, 3, 7, 11, 19, 23, 29, 79, 83
661 3
673 29, 43
677 2, 5, 43, 59, 101
683 2
691 2, 5, 47, 73
701 2, 11, 79
709 2, 41, 61, 67
719 7, 11
727 -
733 5, 7, 11, 29, 89, 101
739 2, 3, 5
743 5, 13
751 2, 29
757 2, 3
761 2
769 7
773 2, 3, 5, 19, 37, 53
787 2, 79
797 5, 7, 47, 53, 61
809 2, 47
811 3, 7, 11
821 3, 11, 79
823 43
827 2, 3, 5, 7, 13, 23, 41, 59
829 2, 3, 11, 19, 31
839 3, 5
853 29, 43
857 17, 23
859 2, 5, 43
863 3, 7, 11, 19, 47
877 2
881 7
883 2, 3, 59
887 2
907 2
911 11
919 2, 5, 13
929 5, 11, 13
937 -
941 5, 79
947 2
953 3, 5
967 -
971 2
977 7
983 7, 11
991 83
997 2, 5, 11, 13, 29, 31, 41, 79, 83, 97