Deprecated: The each() function is deprecated. This message will be suppressed on further calls in /home/zhenxiangba/zhenxiangba.com/public_html/phproxy-improved-master/index.php on line 456
Generator points of E(Q)/2E(Q) on y^2=x^3+ax+b (-20 <=a <= -1, -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 ----------+---+----------------------------------------------------------------- -20, -20 | 1 | (9, -23, 1), 2.4287 -20, -19 | 0 | -20, -18 | 2 | (-405, 447, 125), 4.8103 | | (356, 419, 64), 4.8330 -20, -17 | 2 | (-3, 4, 1), 1.5902 | | (324, 217, 64), 4.7638 -20, -16 | 1 | (-2, 4, 1), 0.4949 -20, -15 | 2 | (-4, 1, 1), 1.2168 | | (6, 9, 1), 1.6458 -20, -14 | 1 | (-30, 23, 8), 3.0621 -20, -13 | 0 | -20, -12 | 1 | (-4, 2, 1), 1.0204 -20, -11 | 0 | -20, -10 | 1 | (-1, 3, 1), 1.5210 -20, -9 | 2 | (5, 4, 1), 1.9585 | | (82, -235, 8), 3.9033 -20, -8 | 1 | (-3, 5, 1), 1.6886 -20, -7 | 2 | (-4, 3, 1), 1.2996 | | (22, 101, 1), 2.7216 -20, -6 | 0 | -20, -5 | 1 | (21, 94, 1), 3.1396 -20, -4 | 0 | -20, -3 | 1 | (-1, 4, 1), 1.5744 -20, -2 | 0 | -20, -1 | 1 | (13, 44, 1), 2.7070 -19, -20 | 0 | -19, -19 | 0 | -19, -18 | 0 | -19, -17 | 1 | (-1, 1, 1), 1.0560 -19, -16 | 1 | (624, 2944, 27), 5.4310 -19, -15 | 1 | (-30, 15, 8), 3.0149 -19, -14 | 2 | (13, 44, 1), 1.5091 | | (-2, 4, 1), 1.5237 -19, -13 | 1 | (147, -181, 27), 3.8782 -19, -12 | 1 | (-156, -285, 64), 4.3529 -19, -11 | 1 | (-4, 1, 1), 1.6955 -19, -10 | 0 | -19, -9 | 1 | (-1, 3, 1), 1.1534 -19, -8 | 1 | (-4, 2, 1), 1.7277 -19, -7 | 1 | (-17688, 53695, 35937), 8.4656 -19, -6 | 0 | -19, -5 | 2 | (5, 5, 1), 1.5930 | | (-2, 5, 1), 1.6167 -19, -4 | 1 | (1734, -4103, 216), 5.8863 -19, -3 | 1 | (-4, 3, 1), 1.7729 -19, -2 | 1 | (-1, 4, 1), 0.5166 -19, -1 | 0 | -18, -20 | 0 | -18, -19 | 1 | (-2, 3, 1), 0.4951 -18, -18 | 1 | (-3, 3, 1), 0.7511 -18, -17 | 0 | -18, -16 | 1 | (-1, 1, 1), 1.3722 -18, -15 | 0 | -18, -14 | 1 | (-30, 7, 8), 2.9916 -18, -13 | 2 | (-14, 29, 8), 2.8665 | | (201, 440, 27), 4.4568 -18, -12 | 1 | (-2, 4, 1), 1.1667 -18, -11 | 1 | (-3, 4, 1), 1.5987 -18, -10 | 1 | (5, 5, 1), 1.9520 -18, -9 | 0 | -18, -8 | 1 | (-1, 3, 1), 0.9291 -18, -7 | 2 | (8, 19, 1), 1.9564 | | (-14, 35, 8), 2.9378 -18, -6 | 0 | -18, -5 | 1 | (-30, 25, 8), 3.1021 -18, -4 | 1 | (-4, 2, 1), 1.3950 -18, -3 | 1 | (-2, 5, 1), 1.2612 -18, -2 | 1 | (-3, 5, 1), 1.6914 -18, -1 | 1 | (-1, 4, 1), 1.5378 -17, -20 | 1 | (-3, 2, 1), 1.0152 -17, -19 | 0 | -17, -18 | 0 | -17, -17 | 1 | (-2, 3, 1), 1.3775 -17, -16 | 0 | -17, -15 | 2 | (-1, 1, 1), 0.8779 | | (8, 19, 1), 2.3171 -17, -14 | 1 | (6, 10, 1), 2.0934 -17, -13 | 1 | (-14, 27, 8), 2.8246 -17, -12 | 1 | (-1, 2, 1), 1.0451 -17, -11 | 1 | (-30, 1, 8), 2.9997 -17, -10 | 1 | (-2, 4, 1), 1.5034 -17, -9 | 0 | -17, -8 | 1 | (-3, 4, 1), 1.2572 -17, -7 | 1 | (-1, 3, 1), 0.9940 -17, -6 | 1 | (14, 50, 1), 2.7706 -17, -5 | 0 | -17, -4 | 1 | (5, -6, 1), 1.5775 -17, -3 | 2 | (12, 39, 1), 2.6310 | | (-14, 37, 8), 2.9472 -17, -2 | 0 | -17, -1 | 1 | (-2, 5, 1), 0.5260 -16, -20 | 2 | (6, 10, 1), 1.4127 | | (-14, 13, 8), 2.5540 -16, -19 | 0 | -16, -18 | 0 | -16, -17 | 2 | (-14, 19, 8), 2.6758 | | (605, -544, 125), 5.1623 -16, -16 | 0 | -16, -15 | 1 | (-2, -3, 1), 0.9032 -16, -14 | 1 | (-1, 1, 1), 1.3057 x -16, -13 | 0-2 | -16, -12 | 1 | (-3, 3, 1), 1.5045 -16, -11 | 1 | (-1, 2, 1), 1.3634 x -16, -10 | 0 | -16, -9 | 2 | (9, 24, 1), 2.3958 | | (50, -93, 8), 3.4916 -16, -8 | 1 | (-2, 4, 1), 0.8010 -16, -7 | 0 | -16, -6 | 2 | (7, 15, 1), 2.1847 | | (-30, 9, 8), 3.0358 -16, -5 | 1 | (-3, 4, 1), 1.6093 -16, -4 | 0 | -16, -3 | 0 | -16, -2 | 0 | -16, -1 | 1 | (2886, -10063, 216), 6.3016 -15, -20 | 0 | -15, -19 | 1 | (-14, 11, 8), 2.4711 -15, -18 | 1 | (-2, 2, 1), 1.1686 -15, -17 | 1 | (-3, 1, 1), 0.9468 -15, -16 | 0 | -15, -15 | 0 | -15, -14 | 1 | (-3, -2, 1), 0.5360 -15, -13 | 2 | (-2, 3, 1), 0.8051 | | (7, 15, 1), 1.3044 -15, -12 | 1 | (16, 62, 1), 2.8918 -15, -11 | 1 | (-60, 91, 27), 3.6155 -15, -10 | 1 | (-1, 2, 1), 0.4096 -15, -9 | 1 | (-3, 3, 1), 0.6200 -15, -8 | 1 | (24, 116, 1), 3.2559 -15, -7 | 0 | -15, -6 | 1 | (-2, 4, 1), 1.4856 -15, -5 | 1 | (-1, 3, 1), 0.5138 -15, -4 | 0 | -15, -3 | 1 | (4, 1, 1), 1.7425 -15, -2 | 1 | (-3, 4, 1), 0.9220 -15, -1 | 1 | (5, 7, 1), 1.5505 -14, -20 | 1 | (7, -15, 1), 2.2145 -14, -19 | 2 | (-14, 3, 8), 2.2040 | | (34, 197, 1), 3.2408 -14, -18 | 0 | -14, -17 | 0 | -14, -16 | 1 | (-2, 2, 1), 0.8070 -14, -15 | 0 | -14, -14 | 1 | (-3, 1, 1), 1.3260 -14, -13 | 0 | -14, -12 | 1 | (-1, 1, 1), 1.2299 -14, -11 | 2 | (-3, 2, 1), 1.4229 | | (6, 11, 1), 1.7248 -14, -10 | 0 | -14, -9 | 2 | (34, -23, 8), 3.1969 | | (-87, 44, 27), 3.6842 x -14, -8 | 1 | -14, -7 | 1 | (4, 1, 1), 1.4113 -14, -6 | 2 | (5, 7, 1), 1.9144 | | (-14, 29, 8), 2.8122 -14, -5 | 0 | -14, -4 | 2 | (-2, 4, 1), 1.1311 | | (4, 2, 1), 1.3942 -14, -3 | 0 | -14, -2 | 0 | -14, -1 | 0 | -13, -20 | 0 | -13, -19 | 0 | -13, -18 | 1 | (34, 15, 8), 3.2317 -13, -17 | 2 | (6, -11, 1), 2.0864 | | (-14, 5, 8), 2.1726 -13, -16 | 0 | -13, -15 | 0 | -13, -14 | 1 | (-2, 2, 1), 1.1426 -13, -13 | 0 | -13, -12 | 0 | -13, -11 | 2 | (-1, 1, 1), 0.7257 | | (5, 7, 1), 1.4685 x -13, -10 | 0-2 | -13, -9 | 1 | (-2, 3, 1), 1.3376 -13, -8 | 2 | (-1, 2, 1), 0.9241 | | (8, 20, 1), 2.2848 -13, -7 | 1 | (-24, 53, 27), 3.4784 -13, -6 | 1 | (-14, 27, 8), 2.7692 -13, -5 | 0 | -13, -4 | 0 | -13, -3 | 2 | (-1, 3, 1), 0.8984 | | (13, 45, 1), 2.2252 -13, -2 | 1 | (-2, 4, 1), 1.4704 -13, -1 | 1 | (-6, 37, 27), 3.4967 -12, -20 | 0 | -12, -19 | 1 | (35, -206, 1), 3.6126 -12, -18 | 0 | -12, -17 | 0 | -12, -15 | 1 | (-2, 1, 1), 0.3312 -12, -14 | 0 | -12, -13 | 1 | (-14, 13, 8), 2.4303 -12, -12 | 1 | (-2, 2, 1), 0.2698 -12, -11 | 0 | -12, -10 | 2 | (-14, 19, 8), 2.5992 | | (23, 109, 1), 3.2131 -12, -9 | 0 | -12, -8 | 1 | (-3, 1, 1), 1.3865 -12, -7 | 1 | (-1, 2, 1), 1.2371 -12, -6 | 1 | (-15, 19, 27), 3.4108 -12, -5 | 1 | (-3, 2, 1), 1.4634 -12, -4 | 0 | -12, -3 | 1 | (7, 16, 1), 2.1530 -12, -2 | 1 | (-1, 3, 1), 0.6025 -12, -1 | 1 | (5, 8, 1), 1.8736 -11, -20 | 0 | -11, -19 | 2 | (44, -291, 1), 3.8292 | | (292, -325, 64), 4.6620 -11, -18 | 0 | -11, -17 | 0 | -11, -16 | 2 | (50, 101, 8), 3.4927 | | (32, 180, 1), 3.5252 -11, -15 | 1 | (34, 31, 8), 3.2097 -11, -14 | 0 | -11, -13 | 1 | (-2, 1, 1), 0.8000 -11, -12 | 1 | (-14, 11, 8), 2.3310 -11, -11 | 1 | (4, 3, 1), 1.7613 -11, -10 | 1 | (-2, -2, 1), 1.1347 -11, -9 | 1 | (-1, 1, 1), 0.7462 -11, -8 | 1 | (-24, 28, 27), 3.3212 -11, -7 | 1 | (-14, 21, 8), 2.6322 -11, -6 | 1 | (-1, 2, 1), 0.5091 -11, -5 | 2 | (-2, 3, 1), 1.3281 | | (9, 25, 1), 2.0182 -11, -4 | 1 | (4, 4, 1), 1.7169 -11, -3 | 0 | -11, -2 | 1 | (-3, 2, 1), 0.7899 -11, -1 | 1 | (-1, 3, 1), 0.9620 -10, -20 | 1 | (4, 2, 1), 1.4550 -10, -19 | 1 | (138, 215, 27), 3.8157 -10, -18 | 1 | (836, -2919, 64), 5.4833 -10, -17 | 1 | (7, 16, 1), 2.1931 -10, -16 | 1 | (102, 10, 27), 3.5954 -10, -15 | 1 | (4, 3, 1), 1.4310 -10, -14 | 1 | (9, 25, 1), 2.3867 x -10, -13 | 0-2 | -10, -12 | 1 | (-14, 3, 8), 1.7102 -10, -11 | 1 | (-2, 1, 1), 0.4788 -10, -10 | 1 | (-33, 17, 27), 3.1008 -10, -9 | 0 | -10, -8 | 2 | (-2, 2, 1), 0.7916 | | (4, 4, 1), 1.3904 -10, -7 | 0 | -10, -6 | 0 | -10, -5 | 1 | (-1, 2, 1), 1.1662 -10, -4 | 0 | -10, -3 | 1 | (-2, 3, 1), 0.9795 -10, -2 | 1 | (-3, 1, 1), 1.4413 -10, -1 | 0 | -9, -20 | 1 | (12, 40, 1), 2.6387 -9, -19 | 1 | (4, 3, 1), 1.2439 -9, -18 | 1 | (6, 12, 1), 1.3426 -9, -17 | 1 | (14, 51, 1), 2.2181 -9, -16 | 1 | (5, 8, 1), 1.5868 -9, -15 | 0 | -9, -14 | 0 | -9, -13 | 1 | (102, -71, 27), 3.9202 -9, -12 | 1 | (4, 4, 1), 1.7559 -9, -11 | 0 | -9, -10 | 1 | (-14, -5, 8), 1.9119 -9, -9 | 1 | (-2, 1, 1), 0.8644 -9, -8 | 0 | -9, -7 | 1 | (-1, 1, 1), 0.5183 -9, -6 | 1 | (-2, 2, 1), 1.1463 -9, -5 | 0 | -9, -4 | 1 | (-1, 2, 1), 0.7828 -9, -3 | 1 | (4, 5, 1), 1.6910 -9, -2 | 0 | -9, -1 | 1 | (-2, 3, 1), 0.7762 -8, -20 | 1 | (10, 30, 1), 1.7894 -8, -19 | 1 | (34, 39, 8), 3.2161 -8, -18 | 1 | (11, 35, 1), 1.7537 -8, -17 | 1 | (66, 175, 8), 3.7035 -8, -16 | 1 | (4, -4, 1), 0.7342 -8, -15 | 0 | -8, -14 | 0 | -8, -13 | 0 | -8, -12 | 0 | -8, -11 | 0 | -8, -10 | 0 | -8, -9 | 0 | -8, -8 | 0 | -8, -7 | 1 | (-2, -1, 1), 0.4491 -8, -6 | 2 | (-14, 13, 8), 2.3853 | | (93, 73, 27), 3.7985 -8, -5 | 0 | -8, -4 | 1 | (-2, 2, 1), 0.4653 -8, -3 | 1 | (-1, -2, 1), 1.0920 -8, -2 | 1 | (3, 1, 1), 1.4591 -8, -1 | 0 | -7, -20 | 1 | (4, 4, 1), 1.7910 -7, -19 | 0 | -7, -18 | 0 | -7, -17 | 0 | -7, -16 | 1 | (461040, 170648, 132651), 9.5412 -7, -15 | 2 | (16, 63, 1), 2.8786 | | (228, -147, 64), 4.4597 -7, -14 | 0 | -7, -13 | 0 | -7, -12 | 0 | -7, -11 | 1 | (4, 5, 1), 1.7388 -7, -10 | 0 | -7, -9 | 1 | (5, 9, 1), 1.5415 -7, -8 | 0 | -7, -7 | 0 | -7, -6 | 0 | -7, -5 | 2 | (-2, 1, 1), 0.9612 | | (3, 1, 1), 1.1396 -7, -4 | 0 | -7, -3 | 0 | -7, -2 | 1 | (-2, 2, 1), 1.1736 -7, -1 | 0 | -6, -20 | 0 | -6, -19 | 0 | -6, -18 | 1 | (33, -189, 1), 3.0010 -6, -17 | 1 | (207, -2978, 1), 5.3413 -6, -16 | 0 | -6, -15 | 1 | (4, 5, 1), 1.4133 -6, -14 | 1 | (5, 9, 1), 1.3624 -6, -13 | 0 | -6, -12 | 1 | (7, 17, 1), 2.1609 -6, -11 | 1 | (6, 13, 1), 1.6877 -6, -10 | 0 | -6, -9 | 0 | -6, -8 | 1 | (3, 1, 1), 1.5119 -6, -7 | 0 | -6, -6 | 0 | -6, -5 | 1 | (3, -2, 1), 1.4764 -6, -4 | 1 | (-1, 1, 1), 0.7761 -6, -3 | 1 | (-2, 1, 1), 0.6647 -6, -2 | 0 | -6, -1 | 1 | (-1, 2, 1), 1.0179 -5, -20 | 0 | -5, -19 | 2 | (5, 9, 1), 1.4706 | | (7, 17, 1), 1.7220 -5, -18 | 1 | (34, -49, 8), 3.2011 -5, -17 | 1 | (6, 13, 1), 2.0580 -5, -16 | 1 | (13, 46, 1), 2.3452 -5, -15 | 1 | (24, -117, 1), 3.2468 -5, -14 | 0 | -5, -13 | 1 | (196, -41, 64), 4.3398 -5, -12 | 0 | -5, -11 | 1 | (3, 1, 1), 1.0741 -5, -10 | 0 | -5, -9 | 0 | -5, -8 | 2 | (3, 2, 1), 1.1583 | | (9, 26, 1), 2.0024 -5, -7 | 0 | -5, -6 | 0 | -5, -5 | 0 | -5, -4 | 0 | -5, -3 | 1 | (-14, 5, 8), 2.1901 -5, -2 | 0 | -5, -1 | 1 | (-2, 1, 1), 1.0597 -4, -20 | 0 | -4, -19 | 1 | (2310, -7361, 216), 6.1112 -4, -18 | 1 | (578, 4911, 8), 5.6903 -4, -17 | 1 | (9, 26, 1), 2.3776 -4, -16 | 0 | -4, -15 | 0 | -4, -14 | 1 | (3, 1, 1), 1.5593 -4, -13 | 0 | -4, -12 | 1 | (4, 6, 1), 1.0384 -4, -11 | 1 | (3, 2, 1), 1.5315 -4, -10 | 1 | (4269, -53675, 27), 7.2699 -4, -9 | 1 | (34, -57, 8), 3.1371 -4, -8 | 0 | -4, -7 | 0 | -4, -6 | 1 | (3, 3, 1), 1.4722 -4, -5 | 1 | (5, 10, 1), 1.8381 -4, -4 | 0 | -4, -3 | 0 | -4, -2 | 1 | (-1, 1, 1), 0.6457 -4, -1 | 0 | -3, -20 | 0 | -3, -19 | 1 | (20, 89, 1), 3.0806 -3, -18 | 0 | -3, -17 | 1 | (3, 1, 1), 1.2348 -3, -16 | 1 | (4, 6, 1), 1.0240 -3, -15 | 1 | (920, 2377, 125), 5.4228 -3, -14 | 1 | (3, 2, 1), 0.8633 -3, -13 | 0 | -3, -12 | 1 | (84, 80, 27), 3.7578 -3, -11 | 1 | (12, 41, 1), 2.6041 -3, -10 | 1 | (5, 10, 1), 1.1817 -3, -9 | 1 | (3, 3, 1), 0.6092 -3, -8 | 0 | -3, -7 | 0 | -3, -6 | 0 | -3, -5 | 0 | -3, -4 | 1 | (8, -22, 1), 2.2149 -3, -3 | 1 | (4, 7, 1), 1.6278 -3, -1 | 1 | (-1, 1, 1), 0.2643 -2, -20 | 2 | (4, 6, 1), 1.4325 | | (18, 76, 1), 2.6381 -2, -19 | 1 | (10, 31, 1), 2.1207 -2, -18 | 0 | -2, -17 | 2 | (612, -1853, 64), 5.1966 | | (1477, 2526, 343), 5.7025 -2, -16 | 0 | -2, -15 | 1 | (5, -10, 1), 1.9044 -2, -14 | 1 | (75, -37, 27), 3.7056 -2, -13 | 1 | (11, 36, 1), 2.5316 -2, -12 | 2 | (8, 22, 1), 1.9103 | | (14, 52, 1), 2.3957 -2, -11 | 0 | -2, -10 | 0 | -2, -9 | 0 | -2, -8 | 1 | (6, 14, 1), 1.6521 -2, -7 | 1 | (4, 7, 1), 1.3298 -2, -6 | 0 | -2, -5 | 2 | (3, 4, 1), 1.4410 | | (18, -11, 8), 2.6367 -2, -4 | 0 | -2, -3 | 1 | (2, 1, 1), 0.7729 -2, -2 | 0 | -2, -1 | 0 | -1, -20 | 2 | (5, 10, 1), 1.5829 | | (8, 22, 1), 2.2851 -1, -19 | 0 | -1, -18 | 1 | (8118, 18954, 1331), 6.8615 -1, -17 | 1 | (22, 103, 1), 3.1636 -1, -16 | 1 | (75, 44, 27), 3.3772 -1, -15 | 1 | (3, 3, 1), 1.1005 -1, -14 | 1 | (6, 14, 1), 2.0337 -1, -13 | 1 | (73494, -116983, 19683), 8.2758 -1, -12 | 1 | (7, 18, 1), 1.7985 -1, -11 | 1 | (4, 7, 1), 1.7140 -1, -10 | 1 | (1014, 2035, 216), 5.4045 -1, -9 | 1 | (18, 3, 8), 2.7102 -1, -8 | 1 | (3, 4, 1), 1.1385 -1, -7 | 0 | -1, -6 | 0 | -1, -5 | 1 | (2, 1, 1), 1.1738 -1, -4 | 0 | -1, -3 | 0 | -1, -2 | 1 | (2, 2, 1), 1.0660 -1, -1 | 0 |