Non-ordinary reduction of X₀(p)

William A. Stein

Let p and $\ell$ be distinct primes. The modular curve X₀(p) is ordinary at $\ell$iff the Hecke operator $T_{\ell}$ 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

p	non-ordinary $\ell\leq 101$
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	-

p	non-ordinary $\ell\leq 101$
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 $\ell\leq 101$
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