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Generator points of E(Q)/2E(Q) on y^2=x^3+ax+b (1 <=a <= 20, 1 <=b <= 20) (using Cremona's mwrank.exe) Ranks of some of the curve are not completely determined yet. (leftmost "x") a, b |rank| (xp, yp, zp), height --------+---+------------------------------------------------------------------- 1, 1 | 1 | (0, 1, 1), 0.4762 1, 2 | 0 | 1, 3 | 1 | (-1, 1, 1), 0.4878 1, 4 | 1 | (0, 2, 1), 0.9251 1, 5 | 0 | 1, 6 | 1 | (-1, 2, 1), 0.1457 1, 7 | 1 | (1, 3, 1), 0.8357 1, 8 | 1 | (2, 23, 8), 2.5493 1, 9 | 2 | (0, 3, 1), 1.1947 | | (-14, 11, 8), 2.6228 1, 10 | 0 | 1, 11 | 2 | (-1, 3, 1), 0.9255 | | (-2, 1, 1), 1.3126 1, 12 | 0 | 1, 13 | 1 | (4, 9, 1), 1.7748 1, 14 | 2 | (1, 4, 1), 0.6911 | | (-2, 2, 1), 1.3889 1, 15 | 1 | (2, 5, 1), 1.5074 1, 16 | 1 | (0, 4, 1), 1.3863 1, 17 | 0 | 1, 18 | 1 | (-1, 4, 1), 0.7420 1, 19 | 1 | (3, 7, 1), 0.3714 1, 20 | 0 | 2, 1 | 1 | (0, 1, 1), 0.2035 2, 2 | 0 | 2, 3 | 1 | (3, -6, 1), 1.4494 2, 4 | 2 | (2, 4, 1), 0.9197 | | (7, 19, 1), 2.1020 2, 5 | 0 | 2, 6 | 1 | (1, 3, 1), 1.1602 2, 7 | 1 | (-1, 2, 1), 1.1039 2, 8 | 0 | 2, 9 | 2 | (0, 3, 1), 0.8493 | | (-14, 3, 8), 2.5991 2, 10 | 0 | 2, 11 | 1 | (57, 134, 27), 3.6657 2, 12 | 1 | (-1, 3, 1), 1.2896 2, 13 | 2 | (2, 5, 1), 1.1417 | | (18, 43, 8), 2.9098 2, 14 | 0 | 2, 15 | 1 | (-15, 100, 27), 3.5580 2, 16 | 2 | (-2, 2, 1), 1.0685 | | (11, 37, 1), 2.5465 2, 17 | 0 | 2, 18 | 0 | 2, 19 | 1 | (-1, 4, 1), 1.4455 2, 20 | 0 | 3, 1 | 1 | (0, 1, 1), 0.6566 3, 2 | 1 | (2, 4, 1), 1.2175 3, 3 | 0 | 3, 4 | 1 | (0, 2, 1), 0.9474 3, 5 | 1 | (4, 9, 1), 1.1499 3, 6 | 0 | 3, 7 | 0 | 3, 8 | 1 | (-1, 2, 1), 0.7917 3, 9 | 1 | (0, 3, 1), 0.6500 3, 10 | 0 | 3, 11 | 1 | (2, 5, 1), 0.3941 3, 12 | 1 | (1, 4, 1), 1.0147 3, 13 | 1 | (-1, 3, 1), 0.2965 3, 14 | 0 | 3, 15 | 1 | (-2, 1, 1), 1.3778 3, 16 | 1 | (0, 4, 1), 1.3878 3, 17 | 0 | 3, 18 | 1 | (-2, 2, 1), 1.4402 3, 19 | 0 | 3, 20 | 1 | (-1, 4, 1), 1.1097 4, 1 | 1 | (0, 1, 1), 0.2977 4, 2 | 0 | 4, 3 | 0 | 4, 4 | 1 | (0, 2, 1), 0.2815 4, 5 | 0 | 4, 6 | 1 | (-1, 1, 1), 1.0334 4, 7 | 0 | 4, 8 | 0 | 4, 9 | 2 | (0, 3, 1), 0.7436 | | (2, 5, 1), 0.9823 4, 10 | 1 | (3, 7, 1), 1.6218 4, 11 | 1 | (1, 4, 1), 1.3503 4, 12 | 0 | 4, 13 | 0 | 4, 14 | 2 | (2, 31, 8), 2.7448 | | (155, 571, 125), 4.6578 4, 15 | 1 | (-14, 13, 8), 2.7447 4, 16 | 1 | (0, 4, 1), 0.6967 4, 17 | 1 | (-2, 1, 1), 0.9455 4, 18 | 1 | (-14, 19, 8), 2.8088 4, 19 | 1 | (9, 28, 1), 2.3977 4, 20 | 2 | (2, 6, 1), 0.7434 | | (4, 10, 1), 1.1677 5, 1 | 1 | (0, 1, 1), 0.8509 5, 2 | 0 | 5, 3 | 1 | (1, 3, 1), 0.7816 5, 4 | 1 | (0, 2, 1), 1.0106 5, 5 | 1 | (12, 73, 27), 3.3111 5, 6 | 0 | 5, 7 | 2 | (3, 7, 1), 1.2470 | | (2, 23, 8), 2.5547 5, 8 | 0 | 5, 9 | 2 | (0, 3, 1), 1.2159 | | (18, 45, 8), 2.8882 5, 10 | 1 | (9, 28, 1), 1.1554 5, 11 | 0 | 5, 12 | 0 | 5, 13 | 0 | 5, 14 | 0 | 5, 15 | 1 | (-1, 3, 1), 0.9998 5, 16 | 2 | (0, 4, 1), 1.3933 | | (-14, 11, 8), 2.7574 5, 17 | 0 | 5, 18 | 1 | (2, -6, 1), 1.5966 5, 19 | 2 | (-2, 1, 1), 1.4358 | | (5, 13, 1), 1.6409 x 5, 20 | 0-2 | 6, 1 | 1 | (0, 1, 1), 0.5836 6, 2 | 1 | (1, 3, 1), 0.4159 6, 3 | 1 | (2, 17, 8), 2.4333 6, 4 | 2 | (0, 2, 1), 0.7051 | | (6, 16, 1), 1.6743 6, 5 | 1 | (2, 5, 1), 1.0604 6, 6 | 0 | 6, 7 | 0 | 6, 8 | 1 | (-1, 1, 1), 1.1385 6, 9 | 1 | (0, 3, 1), 0.3340 6, 10 | 0 | 6, 11 | 1 | (-1, 2, 1), 1.2400 6, 12 | 1 | (4, 10, 1), 1.4705 6, 13 | 0 | 6, 14 | 1 | (5, 13, 1), 1.9646 6, 15 | 1 | (7, 20, 1), 2.1998 6, 16 | 2 | (8, 24, 1), 1.2249 | | (15, 59, 1), 2.8245 6, 17 | 0 | 6, 18 | 1 | (1, 5, 1), 1.5026 6, 19 | 2 | (18, 53, 8), 3.0348 | | (-51, -26, 27), 3.6229 6, 20 | 0 | 7, 1 | 2 | (0, 1, 1), 0.9997 | | (15, -59, 1), 2.3323 7, 2 | 0 | 7, 3 | 1 | (2, 5, 1), 1.4044 7, 4 | 1 | (0, 2, 1), 1.0946 7, 5 | 0 | 7, 6 | 0 | 7, 7 | 0 | 7, 8 | 2 | (1, 4, 1), 0.9891 | | (8, -24, 1), 2.2761 7, 9 | 2 | (-1, 1, 1), 0.7226 | | (5, 13, 1), 1.4777 7, 10 | 0 | 7, 11 | 0 | 7, 12 | 1 | (-1, 2, 1), 0.9259 7, 13 | 0 | 7, 14 | 2 | (2, 6, 1), 1.5709 | | (50, 139, 8), 3.5066 7, 15 | 1 | (30, -165, 1), 3.4584 7, 16 | 2 | (3, 8, 1), 1.3796 | | (13, 48, 1), 2.3561 7, 17 | 1 | (16, 65, 1), 2.0806 7, 18 | 1 | (-14, 5, 8), 2.7862 7, 19 | 1 | (10, 33, 1), 2.4921 7, 20 | 0 | 8, 1 | 1 | (7, 20, 1), 1.3482 8, 2 | 0 | 8, 3 | 1 | (13, 48, 1), 2.6757 8, 4 | 1 | (0, 2, 1), 0.2133 8, 5 | 0 | 8, 6 | 0 | 8, 7 | 1 | (1, 4, 1), 1.3414 8, 8 | 0 | 8, 9 | 1 | (0, 3, 1), 0.8058 8, 10 | 1 | (-1, 1, 1), 1.2273 8, 11 | 1 | (-855, 46, 729), 5.6423 8, 12 | 1 | (2, 6, 1), 0.8674 8, 13 | 2 | (3, 8, 1), 1.7106 | | (2, 31, 8), 2.7624 8, 14 | 1 | (2115, 5723, 729), 6.1009 8, 15 | 1 | (2, 33, 8), 2.7994 8, 16 | 1 | (0, 4, 1), 0.3738 8, 17 | 0 | 8, 18 | 2 | (-1, 3, 1), 1.4074 | | (-28, 243, 64), 4.1983 8, 19 | 0 | 8, 20 | 0 | 9, 1 | 1 | (0, 1, 1), 1.1165 9, 2 | 0 | 9, 3 | 0 | 9, 4 | 1 | (0, 2, 1), 1.1791 9, 5 | 0 | 9, 6 | 1 | (1, 4, 1), 0.6598 9, 7 | 0 | 9, 8 | 0 | 9, 9 | 1 | (0, 3, 1), 0.5581 9, 10 | 1 | (3, -8, 1), 1.0046 9, 11 | 1 | (-1, 1, 1), 0.9203 9, 12 | 1 | (88, -826, 1), 4.4963 9, 13 | 1 | (12, 43, 1), 2.6343 9, 14 | 1 | (-1, 2, 1), 0.4100 9, 15 | 1 | (1, 5, 1), 1.1480 9, 16 | 1 | (0, 4, 1), 1.4238 9, 17 | 0 | 9, 18 | 0 | 9, 19 | 1 | (-1, 3, 1), 0.5322 9, 20 | 0 | 10, 1 | 2 | (0, 1, 1), 0.8199 | | (2, 15, 8), 2.5885 10, 2 | 1 | (2, 17, 8), 2.6035 10, 3 | 1 | (11, 38, 1), 2.5444 10, 4 | 1 | (0, 2, 1), 0.8723 10, 5 | 1 | (1, 4, 1), 1.3694 10, 6 | 0 | 10, 7 | 2 | (3, 8, 1), 1.6903 | | (36, 229, 64), 4.1135 10, 8 | 1 | (2, 6, 1), 1.2057 10, 9 | 1 | (0, 3, 1), 0.9679 10, 10 | 1 | (18, 53, 8), 2.9950 10, 11 | 1 | (-15, -62, 27), 3.5059 10, 12 | 1 | (-1, 1, 1), 1.3038 10, 13 | 1 | (6, 17, 1), 1.7642 10, 14 | 1 | (1, 5, 1), 1.4967 10, 15 | 1 | (-1, 2, 1), 1.3633 10, 16 | 1 | (0, 4, 1), 1.0891 10, 17 | 1 | (4, 11, 1), 1.5484 10, 18 | 0 | 10, 19 | 1 | (57, 190, 27), 3.8745 10, 20 | 1 | (-1, 3, 1), 1.4487 11, 1 | 1 | (0, 1, 1), 1.2120 x 11, 2 | 0-2 | 11, 3 | 0 | 11, 4 | 2 | (1, 4, 1), 1.0427 | | (3, 8, 1), 1.3438 11, 5 | 0 | 11, 6 | 1 | (2, 6, 1), 1.5563 11, 7 | 1 | (6, 17, 1), 0.6799 11, 8 | 1 | (3, 82, 27), 3.1847 11, 9 | 1 | (0, 3, 1), 1.3393 11, 10 | 0 | 11, 11 | 1 | (294, 1153, 216), 5.1040 11, 12 | 0 | 11, 13 | 2 | (-1, 1, 1), 0.8762 | | (1, 5, 1), 1.0394 11, 14 | 0 | 11, 15 | 1 | (430, 1209, 125), 5.0425 11, 16 | 2 | (0, 4, 1), 1.4490 | | (5, 14, 1), 1.6727 x 11, 17 | 0-2 | 11, 18 | 1 | (14, 54, 1), 2.7809 11, 19 | 1 | (2, 7, 1), 1.6769 11, 20 | 1 | (-15, 100, 27), 3.3235 12, 1 | 2 | (0, 1, 1), 0.7918 | | (3, 8, 1), 1.6979 12, 2 | 0 | 12, 3 | 2 | (1, 4, 1), 1.4113 | | (146, 635, 8), 4.3843 12, 4 | 1 | (14, 54, 1), 1.3484 12, 5 | 0 | 12, 6 | 0 | 12, 7 | 0 | 12, 8 | 0 | 12, 9 | 1 | (0, 3, 1), 0.3531 12, 10 | 1 | (1065, 6715, 27), 5.9173 12, 11 | 1 | (5, 14, 1), 2.0072 12, 12 | 1 | (1, 5, 1), 0.7043 12, 13 | 0 | 12, 14 | 2 | (-1, 1, 1), 1.3707 | | (2, 33, 8), 2.2944 12, 15 | 1 | (-55, 388, 125), 4.6375 12, 16 | 1 | (0, 4, 1), 0.7704 12, 17 | 2 | (-1, 2, 1), 1.4183 | | (48, -179, 27), 3.3828 12, 18 | 1 | (3, 9, 1), 1.2513 12, 19 | 0 | 12, 20 | 0 | 13, 1 | 1 | (0, 1, 1), 1.2926 13, 2 | 1 | (1, 4, 1), 0.3947 13, 3 | 0 | 13, 4 | 1 | (0, 2, 1), 1.3263 13, 5 | 2 | (2, 23, 8), 2.7530 | | (12, 89, 27), 3.5869 13, 6 | 1 | (5, 14, 1), 1.3075 13, 7 | 1 | (7, 21, 1), 1.8772 13, 8 | 0 | 13, 9 | 2 | (8, 25, 1), 1.5203 | | (-28, -115, 64), 3.3180 13, 10 | 0 | 13, 11 | 1 | (1, 5, 1), 1.1723 13, 12 | 0 | 13, 13 | 1 | (108, 1123, 1), 4.6993 13, 14 | 0 | 13, 15 | 2 | (-1, 1, 1), 1.0546 | | (2, 7, 1), 1.6724 13, 16 | 1 | (0, 4, 1), 1.4790 13, 17 | 0 | 13, 18 | 1 | (-1, 2, 1), 0.7510 13, 19 | 0 | 13, 20 | 0 | 14, 1 | 3 | (8, 25, 1), 1.9730 | | (2, 17, 8), 2.7453 | | (18, 53, 8), 3.0213 14, 2 | 1 | (86445, -366031, 91125), 9.0711 14, 3 | 0 | x 14, 4 | 1 | (0, 2, 1), 1.0118 14, 5 | 1 | (-6, 37, 27), 3.1965 14, 6 | 1 | (50, 147, 8), 3.5399 14, 7 | 0 | 14, 8 | 1 | (84, 244, 27), 3.6360 14, 9 | 1 | (0, 3, 1), 1.0677 14, 10 | 1 | (1, 5, 1), 1.5309 x 14, 11 | 0-2 | 14, 12 | 1 | (3, 9, 1), 1.7919 14, 13 | 1 | (2, 7, 1), 1.3274 14, 14 | 1 | (3030, -11773, 216), 6.3701 14, 15 | 1 | (15, -60, 1), 2.8449 14, 16 | 2 | (-1, 1, 1), 1.4300 | | (10, 34, 1), 2.1680 14, 17 | 1 | (-24, 53, 27), 3.2996 x 14, 18 | 0-2 | 14, 19 | 1 | (-1, 2, 1), 1.4689 14, 20 | 1 | (66, 224, 27), 3.6235 15, 1 | 1 | (0, 1, 1), 1.3621 15, 2 | 0 | 15, 3 | 1 | (4, -127, 64), 4.1590 15, 4 | 1 | (0, 2, 1), 1.3887 15, 5 | 1 | (44, 293, 1), 3.8266 15, 6 | 2 | (10, 34, 1), 2.5043 | | (2, 25, 8), 2.8207 15, 7 | 0 | 15, 8 | 0 | 15, 9 | 2 | (3, 9, 1), 0.7812 | | (0, 3, 1), 0.8892 15, 10 | 0 | 15, 11 | 1 | (2, 7, 1), 1.6783 15, 12 | 0 | 15, 13 | 1 | (18, 61, 8), 3.1073 15, 14 | 0 | 15, 15 | 0 | 15, 16 | 1 | (0, -4, 1), 1.5117 15, 17 | 1 | (-1, 1, 1), 0.9953 15, 18 | 1 | (6, 18, 1), 1.6197 15, 19 | 0 | 15, 20 | 2 | (-1, 2, 1), 1.1462 | | (16, 66, 1), 2.1763 16, 1 | 1 | (0, 1, 1), 0.9315 16, 2 | 1 | (98, 361, 8), 4.0563 16, 3 | 1 | (705, -2064, 125), 5.3227 16, 4 | 1 | (0, 2, 1), 0.4934 16, 5 | 0 | 16, 6 | 1 | (3, 9, 1), 1.7968 16, 7 | 2 | (18, 59, 8), 3.0922 | | (3, 80, 27), 3.6521 16, 8 | 1 | (1, 5, 1), 1.5592 16, 9 | 1 | (2, 7, 1), 0.2500 16, 10 | 0 | 16, 11 | 1 | (2, 31, 8), 2.8905 16, 12 | 1 | (6, 18, 1), 1.4683 16, 13 | 1 | (2, 33, 8), 2.9085 16, 14 | 1 | (11, 39, 1), 2.5957 16, 15 | 1 | (904, 3575, 512), 5.8523 16, 16 | 1 | (0, 4, 1), 0.1424 16, 17 | 0 | 16, 18 | 2 | (311, -5485, 1), 5.7464 | | (-1428, 3547, 1728), 6.4626 16, 19 | 1 | (1, 6, 1), 1.6418 16, 20 | 1 | (5, 15, 1), 2.0795 17, 1 | 1 | (0, 1, 1), 1.4233 17, 2 | 0 | 17, 3 | 2 | (3, 9, 1), 1.4573 | | (4, 129, 64), 4.2167 17, 4 | 2 | (0, 2, 1), 1.4450 | | (2, 23, 8), 2.8576 17, 5 | 1 | (-1964820, 5119343, 7414875), 11.9727 17, 6 | 1 | (6, 18, 1), 2.1583 17, 7 | 1 | (1, 5, 1), 0.4235 17, 8 | 1 | (-1278, 12043, 5832), 7.2374 17, 9 | 2 | (0, 3, 1), 1.4853 | | (2440, 7209, 512), 6.1905 17, 10 | 0 | 17, 11 | 0 | 17, 12 | 1 | (4, 12, 1), 1.9536 17, 13 | 0 | 17, 14 | 0 | 17, 15 | 1 | (5, 15, 1), 1.7272 17, 16 | 1 | (0, 4, 1), 1.5458 x 17, 17 | 0-2 | 17, 18 | 1 | (1, 6, 1), 0.7832 17, 19 | 1 | (-1, 1, 1), 1.1614 17, 20 | 0 | 18, 1 | 1 | (0, 1, 1), 1.1047 18, 2 | 0 | 18, 3 | 1 | (1878, -6173, 216), 5.9987 18, 4 | 2 | (0, 2, 1), 1.1245 | | (30, 136, 27), 3.4437 18, 5 | 1 | (2, 7, 1), 1.3579 18, 6 | 1 | (1, 5, 1), 1.5909 18, 7 | 1 | (-45, 86, 125), 4.6786 18, 8 | 1 | (4, 12, 1), 1.0585 18, 9 | 1 | (0, 3, 1), 0.4287 18, 10 | 1 | (5, 15, 1), 1.5218 18, 11 | 0 | 18, 12 | 0 | 18, 13 | 0 | 18, 14 | 0 | 18, 15 | 1 | (7, 22, 1), 1.0832 18, 16 | 2 | (0, 4, 1), 1.2163 | | (30, 166, 1), 3.1275 18, 17 | 1 | (1, 6, 1), 1.1093 18, 18 | 0 | 18, 19 | 1 | (3, -10, 1), 1.0746 18, 20 | 2 | (-1, 1, 1), 1.5316 | | (8, 26, 1), 2.0337 19, 1 | 1 | (0, 1, 1), 1.4778 19, 2 | 1 | (1130, -3776, 125), 5.6689 19, 3 | 2 | (-286, -1591, 2197), 6.6068 | | (6810, -21171, 1000), 6.8589 19, 4 | 2 | (0, 2, 1), 1.4959 | | (18, 61, 8), 3.1361 19, 5 | 2 | (5, 15, 1), 1.6098 | | (2, 25, 8), 2.9134 19, 6 | 0 | 19, 7 | 0 | 19, 8 | 1 | (7, 22, 1), 1.9316 19, 9 | 1 | (0, 3, 1), 1.5294 19, 10 | 0 | 19, 11 | 0 | 19, 12 | 1 | (8, 26, 1), 2.3737 19, 13 | 0 | 19, 14 | 0 | 19, 15 | 1 | (14, 55, 1), 2.8041 19, 16 | 3 | (3, 10, 1), 1.5334 | | (18, 67, 8), 3.1841 | | (45, -598, 125), 4.4822 19, 17 | 1 | (128, -1449, 1), 4.8690 19, 18 | 2 | (2, 8, 1), 1.7804 | | (3908, 30617, 64), 6.9207 19, 19 | 1 | (2, 39, 8), 3.0078 19, 20 | 0 | 20, 1 | 2 | (0, 1, 1), 1.0410 | | (2, 7, 1), 1.2667 20, 2 | 1 | (366996, -2266361, 592704), 10.4346 20, 3 | 1 | (327, -1214, 27), 4.8789 20, 4 | 2 | (0, 2, 1), 0.8266 | | (34, 103, 8), 3.3857 20, 5 | 1 | (18, -13121, 5832), 7.3066 20, 6 | 1 | (497, -2115, 343), 5.5742 20, 7 | 1 | (3, 82, 27), 3.7453 20, 8 | 0 | 20, 9 | 1 | (0, 3, 1), 1.0883 20, 10 | 1 | (2, 31, 8), 2.9656 20, 11 | 1 | (-62373, -162028, 132651), 9.3886 20, 12 | 1 | (2, 33, 8), 2.9780 20, 13 | 1 | (3, 10, 1), 1.8826 20, 14 | 1 | (-59619, -294245, 132651), 9.4121 20, 15 | 1 | (1, 6, 1), 1.6792 20, 16 | 1 | (2, 8, 1), 0.2259 20, 17 | 0 | 20, 18 | 1 | (18, 69, 8), 3.2066 20, 19 | 0 | 20, 20 | 0 |