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

 p non-ordinary 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