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Generator points of E(Q)/2E(Q) on y^2=x^3+ax+b (1 <=a <= 20, -20 <=b <= -1) (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, -20 | 0 | 1, -19 | 1 | (4, 7, 1), 0.7997 1, -18 | 0 | 1, -17 | 0 | 1, -16 | 1 | (16, 64, 1), 2.8681 1, -15 | 1 | (34, 65, 8), 3.1732 1, -14 | 1 | (3, 4, 1), 0.8624 1, -13 | 0 | 1, -12 | 0 | 1, -11 | 1 | (18, 13, 8), 2.7450 1, -10 | 0 | 1, -9 | 2 | (2, 1, 1), 1.2658 | | (5, 11, 1), 1.5111 1, -8 | 1 | (18, 19, 8), 2.6982 1, -7 | 1 | (48, -17, 27), 3.3632 1, -6 | 1 | (2, 2, 1), 1.2052 1, -5 | 1 | (3, 5, 1), 1.0880 1, -4 | 1 | (4, 8, 1), 1.6241 1, -3 | 0 | 1, -2 | 0 | 1, -1 | 1 | (1, 1, 1), 0.2517 2, -20 | 0 | 2, -19 | 1 | (66, -19, 27), 3.3425 2, -18 | 1 | (9, 27, 1), 2.3725 2, -17 | 1 | (3, 4, 1), 1.5851 2, -16 | 0 | 2, -15 | 0 | 2, -14 | 1 | (13, 47, 1), 2.6768 x 2, -13 | 0-2 | 2, -12 | 0 | 2, -11 | 1 | (2, 1, 1), 0.9591 2, -10 | 1 | (16215, -35479, 3375), 7.2538 2, -9 | 1 | (18, 21, 8), 2.7226 2, -8 | 2 | (4, 8, 1), 1.3386 | | (3, 5, 1), 1.4910 2, -7 | 0 | 2, -6 | 0 | 2, -5 | 0 | 2, -4 | 0 | 2, -3 | 1 | (2, -3, 1), 0.7898 2, -2 | 1 | (1, 1, 1), 0.7564 2, -1 | 0 | 3, -20 | 1 | (3, 4, 1), 1.2656 3, -19 | 1 | (5, 11, 1), 1.4641 3, -18 | 1 | (18, 3, 8), 2.2861 3, -17 | 1 | (54, 397, 1), 4.0175 3, -16 | 0 | 3, -15 | 0 | 3, -14 | 0 | 3, -13 | 1 | (2, 1, 1), 1.3423 3, -12 | 1 | (4, 8, 1), 1.7280 3, -11 | 0 | 3, -10 | 1 | (2, 2, 1), 1.3028 3, -9 | 1 | (6, 15, 1), 1.4537 3, -8 | 0 | 3, -7 | 1 | (8, 23, 1), 2.2261 3, -6 | 1 | (10, 32, 1), 2.4143 3, -5 | 1 | (2, 3, 1), 0.6667 3, -4 | 0 | 3, -3 | 1 | (1, 1, 1), 0.4049 3, -2 | 0 | 3, -1 | 0 | 4, -20 | 1 | (18, 5, 8), 2.8603 4, -19 | 0 | 4, -18 | 1 | (5690, -13747, 1000), 6.6213 4, -17 | 0 | 4, -16 | 1 | (4, 8, 1), 1.0691 4, -15 | 2 | (6, 15, 1), 1.5791 | | (34, 71, 8), 3.1819 4, -14 | 2 | (3, 5, 1), 1.5720 | | (420, -1099, 64), 4.8776 4, -13 | 1 | (133, 1534, 1), 4.9008 4, -12 | 1 | (2, 2, 1), 0.4770 4, -11 | 0 | 4, -10 | 1 | (7, 19, 1), 2.1360 4, -9 | 1 | (18, 27, 8), 2.7498 4, -8 | 0 | 4, -7 | 1 | (2, 3, 1), 0.8147 4, -6 | 1 | (11, 37, 1), 2.5025 4, -5 | 0 | 4, -4 | 1 | (1, 1, 1), 0.9542 4, -3 | 1 | (3, 6, 1), 1.4524 4, -2 | 0 | 4, -1 | 2 | (1, 2, 1), 0.9302 | | (2, 1, 8), 2.1027 5, -20 | 2 | (18, 13, 8), 2.8698 | | (24, 118, 1), 3.2461 x 5, -19 | 0-2 | 5, -18 | 0 | 5, -17 | 2 | (7, 19, 1), 0.3520 | | (18, 19, 8), 1.3738 x 5, -16 | 0-2 | 5, -15 | 0 | 5, -14 | 1 | (2, 2, 1), 1.3802 5, -13 | 0 | 5, -12 | 0 | 5, -11 | 0 | 5, -10 | 0 | 5, -9 | 1 | (2, 3, 1), 1.3270 5, -8 | 0 | 5, -7 | 1 | (726, 1493, 216), 5.1728 5, -6 | 1 | (5, -12, 1), 0.8166 5, -5 | 1 | (1, 1, 1), 0.6804 5, -4 | 1 | (120, 278, 27), 3.9535 5, -3 | 1 | (4, 9, 1), 1.6753 5, -2 | 1 | (1, 2, 1), 0.3150 5, -1 | 0 | 6, -20 | 1 | (3, 5, 1), 1.6329 6, -19 | 2 | (2, 1, 1), 1.0911 | | (34, 73, 8), 3.2163 6, -18 | 1 | (18, 21, 8), 2.3170 6, -17 | 1 | (39, 244, 1), 3.7034 6, -16 | 1 | (2, 2, 1), 1.0674 6, -15 | 0 | 6, -14 | 0 | 6, -13 | 0 | 6, -12 | 0 | 6, -11 | 1 | (2, 3, 1), 0.2913 6, -10 | 1 | (39, 35, 27), 3.4333 6, -9 | 1 | (3, 6, 1), 1.0065 6, -8 | 0 | 6, -7 | 0 | 6, -6 | 1 | (1, 1, 1), 1.0896 6, -5 | 1 | (21, -10, 27), 3.2244 6, -4 | 1 | (2, 4, 1), 0.9663 6, -3 | 1 | (1, 2, 1), 1.0759 6, -2 | 0 | 6, -1 | 0 | 7, -20 | 0 | 7, -19 | 0 | 7, -18 | 2 | (2, 2, 1), 1.4452 | | (82, -269, 8), 3.8736 7, -17 | 0 | 7, -16 | 1 | (5, 12, 1), 1.5835 7, -15 | 0 | 7, -14 | 1 | (18, 29, 8), 2.8524 7, -13 | 1 | (2, 3, 1), 1.4086 7, -12 | 1 | (3, 6, 1), 1.2476 7, -11 | 1 | (4, 9, 1), 1.7579 7, -10 | 1 | (770, 2042, 125), 5.2741 7, -9 | 1 | (34, 79, 8), 3.1732 7, -8 | 1 | (9, 28, 1), 2.0055 7, -7 | 1 | (1, 1, 1), 0.6827 7, -6 | 1 | (2, 4, 1), 1.3606 7, -5 | 1 | (30, 55, 27), 3.3585 7, -4 | 1 | (1, 2, 1), 0.7883 7, -3 | 0 | 7, -2 | 1 | (6, 16, 1), 2.0137 7, -1 | 1 | (2, 7, 8), 2.3842 8, -20 | 1 | (2, 2, 1), 0.7812 8, -19 | 1 | (7665, 11134, 3375), 6.9353 8, -18 | 2 | (18, 27, 8), 2.8956 | | (27, 141, 1), 3.3561 8, -17 | 1 | (9, 28, 1), 2.3809 8, -16 | 0 | 8, -15 | 2 | (2, 3, 1), 0.9811 | | (4, 9, 1), 1.3280 8, -14 | 1 | (15, 59, 1), 2.8120 8, -13 | 1 | (27942, -317879, 216), 8.4575 8, -12 | 0 | 8, -11 | 0 | 8, -10 | 1 | (57, 109, 27), 3.6321 8, -9 | 1 | (260, 609, 64), 4.5458 8, -8 | 2 | (2, 4, 1), 0.7096 | | (1, 1, 1), 1.1942 8, -7 | 0 | 8, -6 | 0 | 8, -5 | 1 | (1, 2, 1), 1.1871 8, -4 | 0 | 8, -3 | 0 | 8, -2 | 1 | (3, 7, 1), 0.6094 8, -1 | 0 | 9, -20 | 0 | 9, -19 | 1 | (4, 9, 1), 1.2686 9, -18 | 0 | 9, -17 | 1 | (2, 3, 1), 0.9255 9, -16 | 1 | (48, 64, 27), 3.6246 9, -15 | 1 | (16, 65, 1), 2.8737 9, -14 | 1 | (6, 16, 1), 2.0669 9, -13 | 0 | 9, -12 | 1 | (380, 824, 125), 4.8552 9, -11 | 0 | 9, -10 | 1 | (2, -4, 1), 1.4404 9, -9 | 1 | (1, 1, 1), 0.8924 9, -8 | 1 | (8, 24, 1), 1.7221 9, -7 | 0 | 9, -6 | 1 | (1, 2, 1), 0.3676 9, -5 | 1 | (3, 7, 1), 1.2627 9, -4 | 1 | (12, 8, 27), 3.3282 9, -3 | 1 | (28, -149, 1), 3.3825 9, -2 | 0 | 9, -1 | 2 | (1, 3, 1), 0.3597 | | (2, 9, 8), 1.9606 10, -20 | 1 | (6, 16, 1), 1.7451 10, -19 | 1 | (2, 3, 1), 1.1574 10, -18 | 1 | (291, 989, 27), 4.7363 10, -17 | 0 | 10, -16 | 1 | (8, 24, 1), 1.9484 10, -15 | 0 | 10, -14 | 0 | 10, -13 | 1 | (7, 20, 1), 2.1863 10, -12 | 1 | (2, 4, 1), 1.1279 10, -11 | 1 | (10, -33, 1), 2.1188 10, -10 | 2 | (1, 1, 1), 1.2799 | | (338, -2203, 8), 5.1661 10, -9 | 0 | 10, -8 | 1 | (3, 7, 1), 1.6380 10, -7 | 1 | (1, 2, 1), 1.2765 10, -6 | 1 | (5, 13, 1), 1.9339 10, -5 | 1 | (18, 43, 8), 2.8931 10, -4 | 1 | (4, 10, 1), 1.4455 10, -3 | 1 | (2, 5, 1), 1.1193 10, -2 | 2 | (57, 431, 1), 4.0687 | | (132, 335, 64), 4.2528 10, -1 | 1 | (34, 87, 8), 3.2185 11, -20 | 1 | (7, 20, 1), 1.8621 11, -19 | 0 | 11, -18 | 0 | 11, -17 | 1 | (18, 35, 8), 2.9452 11, -16 | 1 | (39, 46, 27), 3.2637 11, -15 | 0 | 11, -14 | 1 | (2, 4, 1), 1.5055 11, -13 | 1 | (705684, 791783, 592704), 10.2186 11, -12 | 1 | (19, 84, 1), 2.6849 11, -11 | 2 | (3, 7, 1), 1.2036 | | (12, 43, 1), 2.6245 11, -10 | 1 | (36162, -255250, 729), 8.3304 11, -9 | 0 | 11, -8 | 2 | (4, 10, 1), 1.8134 | | (11, 38, 1), 2.2014 11, -7 | 0 | 11, -6 | 0 | 11, -5 | 1 | (2, 5, 1), 1.4940 11, -4 | 0 | 11, -3 | 1 | (1, 3, 1), 0.8665 11, -2 | 1 | (2, 7, 8), 2.6002 11, -1 | 0 | 12, -20 | 0 | 12, -19 | 1 | (11, 38, 1), 2.5655 12, -18 | 0 | 12, -17 | 1 | (18, 37, 8), 2.9651 12, -16 | 1 | (2, 4, 1), 0.4945 12, -15 | 0 | 12, -14 | 1 | (3, 7, 1), 1.6916 12, -13 | 0 | 12, -12 | 1 | (4, 10, 1), 0.3382 12, -11 | 0 | 12, -10 | 1 | (665, 1555, 343), 5.4040 12, -9 | 1 | (1, 2, 1), 1.3513 12, -8 | 0 | 12, -7 | 1 | (2, 5, 1), 1.0596 12, -6 | 0 | 12, -5 | 0 | 12, -4 | 1 | (1, 3, 1), 0.8130 12, -3 | 1 | (2, 1, 8), 2.6366 12, -2 | 0 | 12, -1 | 1 | (82, 277, 8), 3.8822 13, -20 | 0 | 13, -19 | 0 | 13, -18 | 1 | (2, 4, 1), 1.5608 13, -17 | 1 | (3, 7, 1), 1.3692 13, -16 | 1 | (4, 10, 1), 1.8580 13, -15 | 1 | (68, 7, 64), 4.1708 13, -14 | 0 | 13, -13 | 1 | (1, 1, 1), 1.0387 13, -12 | 0 | 13, -11 | 1 | (180, 409, 125), 4.6762 13, -10 | 1 | (1, 2, 1), 0.6915 13, -9 | 1 | (2, 5, 1), 0.7436 13, -8 | 0 | 13, -7 | 0 | 13, -6 | 0 | 13, -5 | 2 | (1, 3, 1), 1.0476 | | (36, -101, 64), 4.0990 13, -4 | 1 | (40, 254, 1), 3.7309 13, -3 | 0 | 13, -2 | 1 | (3, 8, 1), 0.6712 13, -1 | 0 | 14, -20 | 2 | (2, 4, 1), 1.2392 | | (4, 10, 1), 1.5322 14, -19 | 1 | (102, 253, 27), 3.6974 14, -18 | 1 | (34, 87, 8), 3.2946 14, -17 | 0 | 14, -16 | 1 | (30, 26, 27), 3.2863 14, -15 | 0 | 14, -14 | 2 | (9, 29, 1), 2.4097 | | (18, 43, 8), 3.0010 14, -13 | 1 | (570147, 1335446, 328509), 10.0013 14, -12 | 0 | 14, -11 | 2 | (2, 5, 1), 1.2272 | | (1, 2, 1), 1.4158 14, -10 | 0 | 14, -9 | 0 | 14, -8 | 0 | 14, -7 | 1 | (264, 881, 27), 4.3219 14, -6 | 2 | (1, 3, 1), 1.4243 | | (242, -1341, 8), 4.8541 14, -5 | 1 | (3, 8, 1), 1.7271 x 14, -4 | 0 | 14, -3 | 0 | 14, -2 | 0 | x 14, -1 | 0-2 | 15, -20 | 0 | 15, -19 | 1 | (20, 91, 1), 3.0926 15, -18 | 1 | (18, 78, 1), 2.4487 15, -17 | 2 | (132, -305, 64), 4.3832 | | (546, 1133, 343), 5.4301 15, -16 | 0 | 15, -15 | 1 | (1, 1, 1), 0.9826 15, -14 | 1 | (234, -3580, 1), 5.4630 15, -13 | 2 | (1734, 5399, 216), 5.9126 | | (5054, -20021, 343), 6.7120 15, -12 | 2 | (1, 2, 1), 1.0985 | | (112, 1186, 1), 4.7345 15, -11 | 0 | 15, -10 | 1 | (26, 134, 1), 3.3292 15, -9 | 0 | 15, -8 | 1 | (3, 8, 1), 1.3983 15, -7 | 1 | (8, 25, 1), 2.3243 15, -6 | 1 | (674, -6193, 8), 5.8412 15, -5 | 0 | 15, -4 | 1 | (5, 14, 1), 1.6620 15, -3 | 2 | (2, 7, 8), 2.7508 | | (380, 1051, 125), 4.9753 15, -2 | 1 | (2, 6, 1), 0.8800 15, -1 | 0 | 16, -20 | 0 | 16, -19 | 0 | x 16, -18 | 0-2 | 16, -17 | 0 | 16, -16 | 1 | (1, 1, 1), 1.4719 16, -15 | 2 | (8, 25, 1), 1.8733 | | (15, 60, 1), 2.8403 16, -14 | 1 | (7, 21, 1), 2.2392 16, -13 | 2 | (5202, -8209, 5832), 7.2375 | | (9667, -51820, 343), 7.2985 16, -12 | 0 | 16, -11 | 2 | (3, 8, 1), 1.7630 | | (66, 209, 8), 3.7424 16, -10 | 0 | 16, -9 | 1 | (5, 14, 1), 1.0470 16, -8 | 1 | (1, 3, 1), 1.4797 16, -7 | 1 | (4, 11, 1), 1.4353 16, -6 | 1 | (21, 71, 27), 3.6513 16, -5 | 1 | (2415, -7330, 343), 6.1336 16, -4 | 2 | (10, 34, 1), 1.8101 | | (2, 1, 8), 2.7785 16, -3 | 0 | 16, -2 | 1 | (162, 743, 8), 4.4899 16, -1 | 2 | (1, 4, 1), 1.5044 | | (4, 1, 64), 4.1593 17, -20 | 0 | 17, -19 | 0 | 17, -18 | 1 | (17, -72, 1), 2.2604 17, -17 | 2 | (1, 1, 1), 1.1512 | | (2, 5, 1), 1.6425 17, -16 | 1 | (1257120, 3252212, 531441), 10.4829 17, -15 | 1 | (100320, 872235, 1331), 9.1438 17, -14 | 2 | (1, 2, 1), 0.6322 | | (3, 8, 1), 1.0880 17, -13 | 1 | (93, 251, 27), 3.6909 17, -12 | 1 | (16, 66, 1), 2.8989 17, -11 | 1 | (4, 11, 1), 1.9115 17, -10 | 0 | 17, -9 | 2 | (18, 51, 8), 3.0658 | | (505, -1401, 125), 4.7892 17, -8 | 1 | (34, 95, 8), 3.3296 17, -7 | 1 | (48, 145, 27), 3.0094 17, -6 | 1 | (2, 6, 1), 1.6493 17, -5 | 0 | x 17, -4 | 0-2 | 17, -3 | 1 | (2, 9, 8), 2.8145 17, -2 | 1 | (1, 4, 1), 0.8337 17, -1 | 1 | (183, 557, 27), 4.0785 18, -20 | 0 | 18, -19 | 1 | (2, -5, 1), 0.5119 18, -18 | 1 | (1, 1, 1), 1.5223 18, -17 | 1 | (3, 8, 1), 1.7988 18, -16 | 0 | 18, -15 | 1 | (4, 11, 1), 0.3809 18, -14 | 1 | (2745, -13093, 125), 6.4015 18, -13 | 0 | 18, -12 | 0 | 18, -11 | 0 | 18, -10 | 1 | (1, 3, 1), 0.9803 18, -9 | 0 | 18, -8 | 2 | (2, 6, 1), 0.7719 | | (12, 44, 1), 2.3141 18, -7 | 0 | 18, -6 | 0 | 18, -5 | 1 | (27, 142, 1), 3.3707 18, -4 | 0 | 18, -3 | 1 | (1, 4, 1), 1.5488 18, -2 | 1 | (3, 1, 27), 3.6434 18, -1 | 1 | (2, 15, 8), 2.3042 19, -20 | 1 | (3, 8, 1), 1.4694 19, -19 | 2 | (4, 11, 1), 1.9413 | | (165, -487, 27), 3.9134 19, -18 | 1 | (20636, -18257, 21952), 8.2022 19, -17 | 0 | 19, -16 | 2 | (1, 2, 1), 1.2003 | | (50, 149, 8), 3.5774 19, -15 | 0 | 19, -14 | 1 | (30, 166, 1), 3.4705 19, -13 | 1 | (130, 351, 125), 4.7740 19, -12 | 1 | (39, 116, 27), 3.4607 19, -11 | 1 | (1, 3, 1), 1.0907 19, -10 | 1 | (2, 6, 1), 1.6860 19, -9 | 0 | 19, -8 | 0 | 19, -7 | 1 | (264, 901, 27), 4.7021 19, -6 | 1 | (6, 18, 1), 2.1657 19, -5 | 1 | (36, 155, 64), 4.2858 19, -4 | 2 | (2, 7, 8), 2.8666 | | (41, 264, 1), 3.4170 19, -3 | 1 | (3, 9, 1), 1.3623 19, -2 | 0 | 19, -1 | 0 | 20, -20 | 1 | (1, 1, 1), 1.5679 20, -19 | 1 | (122145, 58834, 132651), 9.4215 20, -18 | 1 | (68, 135, 64), 4.3495 20, -17 | 1 | (1, 2, 1), 1.5694 20, -16 | 0 | 20, -15 | 0 | 20, -14 | 1 | (19, 85, 1), 3.0590 20, -13 | 0 | 20, -12 | 2 | (2, 6, 1), 0.8373 | | (1, 3, 1), 1.5750 20, -11 | 0 | 20, -10 | 1 | (1071456, 3538621, 884736), 10.7338 20, -9 | 2 | (9, 30, 1), 2.4490 | | (130, 543, 8), 4.3092 20, -8 | 0 | 20, -7 | 0 | 20, -6 | 1 | (3, 9, 1), 1.8366 20, -5 | 2 | (2, 1, 8), 2.8889 | | (29, 158, 1), 3.4408 x 20, -4 | 0-2 | 20, -3 | 0 | 20, -2 | 1 | (18, 59, 8), 3.1376 20, -1 | 1 | (303810, -2612683, 1481544), 10.9868