Non-ordinary reduction of X_{0}(p)

William A. Stein

September 28, 1998

Let p and ℓ be distinct primes. The modular curve X_{0}(p) is ordinary at ℓ iff the Hecke operator T_{ℓ} is invertible modulo ℓ. The following is a table which enumerates, for each prime p between 23 and 997, the set of primes ℓ ≤ 101 for which the modular curve X_{0}(p) is not ordinary.^{1}

p non-ordinary ℓ ≤ 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

p non-ordinary ℓ ≤ 101

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 -

1Constructing this table took 26 hours (CPU time) on a 350 Mhz Pentium II.

p non-ordinary ℓ ≤ 101
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

p non-ordinary ℓ ≤ 101
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