Generator points of E(Q)/2E(Q) on y^2=(x+p)(x^2+p^2) (1 < p < 1000, prime) (using Cremona's mwrank.exe) Ranks of some of the curve are not completely determined yet. ("*" in front of "p") p |rank| (xp, yp, zp), height -----+---+---------------------------------------------------------------------- 2 | 0 | 3 | 0 | 5 | 0 | 7 | 1 | (58, 245, 8), 2.07029 11 | 0 | 13 | 0 | 17 | 1 | (-150, 1295, 27), 5.3747 19 | 0 | 23 | 1 | (114939198, -1016523755, 7762392), 11.0954 29 | 0 | 31 | 1 | (839832, 7433335, 13824), 7.36225 37 | 0 | * 41 | 1 | (522, 5043, 8), 2.2577 43 | 0 | 47 | 1 | (-1909196146536566729160, -335263671052024189265, 61600297392921964032), 30.7247 53 | 0 | 59 | 0 | 61 | 0 | 67 | 0 | 71 | 1 | (-69418888049010, -813232264308227, 2253243231000), 19.3051 73 | 1 | (18480, -274415, 343), 8.73999 79 | 1 | (21572, 405665, 64), 4.3702 83 | 0 | 89 | 1 | (-42701723190, -665725746673, 1228480911), 18.7846 97 | 1 | (20562432, -497854945, 35937), 13.4474 101 | 0 | 103 | 1 | (44773463028904610, -744118556611750475, 195697921531112), 23.0266 107 | 0 | 109 | 0 | * 113 | 1 | (60414, 1085365, 216), 4.74698 127 | 1 | (46668, 2338705, 1728), 5.37542 131 | 0 | * 137 | 1 | (46466, 844605, 343), 4.53906 139 | 0 | 149 | 0 | 151 | 1 | (1488627602, -31971448195, 3652264), 11.3031 157 | 0 | 163 | 0 | * 167 | 0 | 173 | 0 | 179 | 0 | 181 | 0 | 191 | 1 | (4793387604314147880, -404010957615296813783, 674987715648000), 26.3957 193 | 1 | (-91440, 372385, 729), 10.0031 197 | 0 | 199 | 1 | (26088543224006379270, -736808906216681732699, 34732130875883000), 26.8854 211 | 0 | * 223 | 1 | (-9003843295845292542013006572275148, 5539676294795420571246122402521985615, 1683348635263490053962220858612032), 51.4305 227 | 0 | 229 | 0 | 233 | 1 | (-281227572848334762, -6301490607105836975, 2862864576873657), 29.5211 239 | 1 | (125402706042080148, -2961390524696286335, 489378389062464), 23.2287 241 | 1 | (609280, 20319887, 4913), 11.634 251 | 0 | * 257 | 1 | (-7951081654950, -58058888074039, 49633171875), 16.7639 263 | 1 | (-5841161298, -75806111395, 40001688), 12.0133 269 | 0 | 271 | 1 | (779761713192, -21485222201185, 1204550144), 15.0786 277 | 0 | 281 | 1 | (330072438, -8318454913, 804357), 15.5295 283 | 0 | 293 | 0 | 307 | 0 | * 311 | 1 | (65353269980927167369164068949666185862887530795793437612859703798, -3689807256247525781124321902366747452551950730909311091275551343885, 528925212286464559060277796967404172043852941089841910706608376), 96.8374 * 313 | 1 | (23397374, -1105580165, 10648), 8.21831 317 | 0 | 331 | 0 | 337 | 3 | (19200, 567845, 27), 3.24942 | | (-5671960, 171829897, 42875), 7.45404 | | (225688364, 11690225015, 85184), 9.70824 347 | 0 | 349 | 0 | * 353 | 1 | (25368439409225196, -1264781295871719535, 10365935926464), 22.0043 359 | 1 | (721662, 41886325, 216), 5.87956 * 367 | 0 | 373 | 0 | 379 | 0 | * 383 | 0 | 389 | 0 | 397 | 0 | 401 | 1 | (356019583817508906, -18174182659808252399, 2178748078144119), 29.9903 * 409 | 1 | (1293360, -41318407, 3375), 6.04506 419 | 0 | 421 | 0 | 431 | 1 | (952135638482444024657525028, -202071975851299630004855464015, 25262082902831570884160064), 39.3764 433 | 1 | (-608785968965400, -4432610857077935, 2191427717669), 25.3602 439 | 1 | (532128032626040725482707422, -44271492253797231708685333045, 5028762338676117137987864), 38.3293 443 | 0 | 449 | 1 | (6759273473916543030, -215678903847154570943, 10131117927634341), 31.5121 * 457 | 1 | (20410, 1044245, 8), 3.21655 461 | 0 | * 463 | 0 | 467 | 0 | * 479 | 0 | * 487 | 0 | 491 | 0 | 499 | 0 | * 503 | 0 | 509 | 0 | 521 | 1 | (-35094111483920, -278563708744437, 105154048000), 17.2649 523 | 0 | 541 | 0 | 547 | 0 | 557 | 0 | 563 | 0 | * 569 | 1 | (-814960, -37880037, 4096), 5.88934 571 | 0 | * 577 | 1 | (14040, 1664645, 1), 3.21656 587 | 0 | * 593 | 0 | 599 | 1 | (-8970, 340502149, 27000), 7.17205 601 | 1 | (1324400, -83083057, 343), 12.2462 * 607 | 0 | 613 | 0 | 617 | 1 | (-14384464950, -212399012735, 38958219), 18.4214 619 | 0 | 631 | 1 | (5036710245798, -200131850092915, 8564793336), 15.8743 641 | 1 | (190422474, -7897278241, 132651), 15.4194 643 | 0 | * 647 | 0 | 653 | 0 | 659 | 0 | 661 | 0 | * 673 | 0 | 677 | 0 | 683 | 0 | 691 | 0 | 701 | 0 | 709 | 0 | * 719 | 0 | * 727 | 0 | 733 | 0 | 739 | 0 | * 743 | 0 | 751 | 1 | (7467717252660, -534786930910097, 1481544000), 16.0729 757 | 0 | * 761 | 1 | (35724210, -6753129981, 1000), 8.46688 769 | 1 | (-2623976, 78302497, 6859), 12.8769 773 | 0 | 787 | 0 | 797 | 0 | * 809 | 1 | (3081610, -178673313, 6859), 6.38373 811 | 0 | 821 | 0 | * 823 | 0 | 827 | 0 | 829 | 0 | * 839 | 0 | 853 | 0 | * 857 | 0 | 859 | 0 | * 863 | 0 | 877 | 0 | * 881 | 1 | (-33730, 2328483, 125), 3.56434 883 | 0 | * 887 | 0 | 907 | 0 | * 911 | 0 | 919 | 1 | (778221841290, -40513210272989, 328509000), 14.2676 * 929 | 0 | * 937 | 0 | 941 | 0 | 947 | 0 | * 953 | 0 | * 967 | 0 | 971 | 0 | * 977 | 0 | * 983 | 0 | * 991 | 0 | 997 | 0 | ------+---+---------------------------------------------------------------------