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, 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 ---------+---+------------------------------------------------------------------ -20, 1 | 1 | (0, 1, 1), 1.0407 -20, 2 | 1 | (-14, 45, 8), 3.0534 -20, 3 | 1 | (-3, 6, 1), 1.7672 -20, 4 | 2 | (0, 2, 1), 0.8231 | | (18, 74, 1), 2.1237 -20, 5 | 1 | (2, 1, 8), 2.8795 -20, 6 | 1 | (-1, 5, 1), 1.6224 -20, 7 | 0 | -20, 8 | 0 | -20, 9 | 2 | (0, 3, 1), 1.0717 | | (34, 7, 8), 3.1478 -20, 10 | 0 | -20, 11 | 2 | (5, 6, 1), 1.8771 | | (-14, 51, 8), 3.0950 -20, 12 | 0 | -20, 13 | 0 | -20, 14 | 0 | -20, 15 | 0 | -20, 16 | 1 | (0, -4, 1), 0.8551 -20, 17 | 2 | (4, 1, 1), 1.1957 | | (-2, 7, 1), 1.2879 -20, 18 | 1 | (34, 25, 8), 3.0855 -20, 19 | 0 | -20, 20 | 2 | (4, 2, 1), 0.9371 | | (-4, 6, 1), 1.2230 -19, 1 | 2 | (8, 19, 1), 2.2873 | | (-14, 43, 8), 3.0262 -19, 2 | 0 | -19, 3 | 0 | -19, 4 | 2 | (-4, 4, 1), 1.8234 | | (50, 91, 8), 3.4641 -19, 5 | 1 | (18180, -89405, 729), 7.6815 -19, 6 | 2 | (5, 6, 1), 1.2005 | | (-2, 6, 1), 1.6887 -19, 7 | 1 | (-1, 5, 1), 1.2590 -19, 8 | 0 | -19, 9 | 1 | (0, 3, 1), 1.5101 -19, 10 | 0 | -19, 11 | 0 | -19, 12 | 0 | -19, 13 | 2 | (4, 1, 1), 1.6760 | | (2, 23, 8), 2.8761 -19, 14 | 0 | -19, 15 | 1 | (7, 15, 1), 1.7790 -19, 16 | 2 | (4, 2, 1), 1.6507 | | (-14, 53, 8), 3.0934 -19, 17 | 1 | (-8040, -23563, 3375), 7.1820 -19, 18 | 1 | (9, 24, 1), 0.9452 -19, 19 | 2 | (5, 7, 1), 0.4924 | | (-2, 7, 1), 0.7648 -19, 20 | 1 | (1740, 2375, 1728), 6.3497 -18, 1 | 2 | (0, 1, 1), 1.1044 | | (34, 9, 8), 2.6249 -18, 2 | 1 | (3, 1, 27), 3.6414 -18, 3 | 0 | -18, 4 | 1 | (0, 2, 1), 1.1197 -18, 5 | 0 | -18, 6 | 0 | -18, 7 | 0 | -18, 8 | 2 | (-4, -4, 1), 1.4849 | | (7, 15, 1), 1.5988 -18, 9 | 2 | (12, 39, 1), 1.5309 | | (2, 17, 8), 2.8418 -18, 10 | 0 | -18, 11 | 0 | -18, 12 | 1 | (4, 2, 1), 1.3237 -18, 13 | 1 | (6, 11, 1), 1.6487 -18, 14 | 1 | (5, 7, 1), 1.8364 -18, 15 | 0 | -18, 16 | 1 | (0, 4, 1), 1.1510 -18, 17 | 1 | (4, 3, 1), 0.7271 -18, 18 | 1 | (1, 1, 1), 1.3511 -18, 19 | 1 | (-1, 6, 1), 1.0678 -18, 20 | 0 | -17, 1 | 1 | (0, 1, 1), 1.4229 -17, 2 | 0 | -17, 3 | 0 | -17, 4 | 1 | (0, -2, 1), 1.4394 -17, 5 | 2 | (2, 7, 8), 2.8007 | | (4, 1, 1), 1.7104 -17, 6 | 1 | (-30, 33, 8), 3.1705 -17, 7 | 1 | (6, 11, 1), 2.0175 -17, 8 | 1 | (4, 2, 1), 1.6890 -17, 9 | 2 | (-1, 5, 1), 1.1075 | | (5, 7, 1), 1.3963 -17, 10 | 1 | (-2, 6, 1), 1.6691 -17, 11 | 1 | (10, 29, 1), 2.4366 -17, 12 | 2 | (-4, 4, 1), 1.8392 | | (11, 34, 1), 2.1715 -17, 13 | 1 | (4, 3, 1), 1.6462 -17, 14 | 1 | (2, 25, 8), 2.8219 -17, 15 | 0 | -17, 16 | 1 | (0, -4, 1), 1.4691 -17, 17 | 1 | (1, 1, 1), 0.8538 -17, 18 | 0 | -17, 19 | 0 | -17, 20 | 2 | (1, 2, 1), 0.9511 | | (16, 62, 1), 2.8352 -16, 1 | 3 | (4, 1, 1), 1.2646 | | (-4, 1, 1), 1.2802 | | (15, 56, 1), 2.8178 -16, 2 | 1 | (-4596, -8767, 1728), 6.6224 -16, 3 | 0 | -16, 4 | 2 | (4, 2, 1), 0.7828 | | (-4, 2, 1), 0.8440 -16, 5 | 1 | (-15, 100, 27), 3.6764 -16, 6 | 1 | (19, 81, 1), 3.0252 -16, 7 | 0 | -16, 8 | 0 | -16, 9 | 2 | (4, 3, 1), 1.2059 | | (-4, 3, 1), 1.3430 x -16, 10 | 1 | (-1, 5, 1), 1.5500 -16, 11 | 1 | (3, 82, 27), 3.6186 -16, 12 | 1 | (-2, 6, 1), 0.9648 -16, 13 | 0 | -16, 14 | 0 | -16, 15 | 1 | (-3, -6, 1), 1.7603 -16, 16 | 1 | (0, 4, 1), 0.0511 -16, 17 | 0 | -16, 18 | 1 | (-14, 51, 8), 3.0363 -16, 19 | 2 | (5, 8, 1), 1.7466 | | (2, 31, 8), 2.7746 -16, 20 | 0 | -15, 1 | 2 | (-30, 17, 8), 3.0885 | | (480, 19, 125), 4.9064 -15, 2 | 1 | (-1, 4, 1), 0.7857 -15, 3 | 1 | (-2, 5, 1), 1.5814 -15, 4 | 1 | (0, 2, 1), 1.3807 -15, 5 | 2 | (4, 3, 1), 1.1392 | | (2, 9, 8), 2.1910 -15, 6 | 0 | -15, 7 | 1 | (-3, 5, 1), 1.3499 -15, 8 | 2 | (8, 20, 1), 2.2384 | | (-14, 43, 8), 2.9789 -15, 9 | 1 | (0, 3, 1), 0.8520 -15, 10 | 0 | -15, 11 | 1 | (-1, 5, 1), 1.1819 -15, 12 | 2 | (4, 4, 1), 1.6207 | | (2, 23, 8), 2.7571 -15, 13 | 1 | (-4, 3, 1), 1.2657 -15, 14 | 1 | (5, -8, 1), 1.0951 -15, 15 | 1 | (1, 1, 1), 0.8898 -15, 16 | 1 | (0, 4, 1), 1.4012 -15, 17 | 0 | -15, 18 | 1 | (1, -2, 1), 0.3379 -15, 19 | 1 | (3, 1, 1), 0.8569 -15, 20 | 1 | (-4, 4, 1), 1.8512 -14, 1 | 2 | (4, 3, 1), 1.3615 | | (-30, 7, 8), 3.0644 -14, 2 | 0 | -14, 3 | 1 | (-1, 4, 1), 1.4577 -14, 4 | 1 | (0, 2, 1), 1.0020 -14, 5 | 1 | (-2, 5, 1), 1.2257 -14, 6 | 0 | -14, 7 | 1 | (2, 15, 8), 2.7150 -14, 8 | 1 | (4, 4, 1), 1.3012 -14, 9 | 2 | (-4, 1, 1), 1.4246 | | (34, 41, 8), 3.0729 -14, 10 | 2 | (13, 45, 1), 2.6566 | | (-30, -25, 8), 3.1456 -14, 11 | 1 | (7, 16, 1), 2.0942 -14, 12 | 2 | (-1, 5, 1), 1.5042 | | (6, 12, 1), 1.6038 -14, 13 | 0 | -14, 14 | 1 | (1, 1, 1), 1.1911 -14, 15 | 0 | -14, 16 | 2 | (3, 1, 1), 1.2504 | | (-2, 6, 1), 1.2787 -14, 17 | 2 | (1, 2, 1), 1.1464 | | (4, 5, 1), 1.1505 -14, 18 | 0 | -14, 19 | 1 | (3, 2, 1), 1.0801 -14, 20 | 0 | -13, 1 | 1 | (0, 1, 1), 1.2917 -13, 2 | 1 | (-42, 116, 27), 3.6731 -13, 3 | 0 | -13, 4 | 3 | (4, 4, 1), 1.6726 | | (7, 16, 1), 1.7807 | | (2, 7, 8), 2.6661 -13, 5 | 0 | -13, 6 | 1 | (6, 12, 1), 1.9852 -13, 7 | 1 | (-2, 5, 1), 1.5624 -13, 8 | 0 | -13, 9 | 1 | (0, 3, 1), 1.3340 -13, 10 | 0 | -13, 11 | 0 | -13, 12 | 0 | -13, 13 | 2 | (9, 25, 1), 1.0242 | | (2, 25, 8), 1.8688 -13, 14 | 1 | (30, 26, 27), 3.2972 -13, 15 | 0 | -13, 16 | 2 | (3, 2, 1), 0.8224 | | (0, 4, 1), 1.2968 -13, 17 | 0 | -13, 18 | 1 | (18, 3, 8), 1.5483 -13, 19 | 1 | (2, 1, 1), 0.7960 -13, 20 | 0 | -12, 1 | 1 | (0, 1, 1), 0.7907 -12, 2 | 0 | -12, 3 | 1 | (2, 1, 8), 2.6210 -12, 4 | 1 | (0, 2, 1), 0.5840 -12, 5 | 1 | (-1, 4, 1), 1.4121 -12, 6 | 0 | -12, 7 | 1 | (-3, 4, 1), 1.6324 -12, 8 | 0 | -12, 9 | 1 | (0, -3, 1), 0.2836 -12, 10 | 1 | (3, 1, 1), 1.3312 -12, 11 | 0 | -12, 12 | 1 | (1, 1, 1), 1.0844 -12, 13 | 1 | (3, 2, 1), 1.2352 -12, 14 | 1 | (-1, 5, 1), 1.4300 -12, 15 | 1 | (1, 2, 1), 0.9750 -12, 17 | 1 | (2, 1, 1), 0.3340 -12, 18 | 1 | (3, 3, 1), 0.6634 -12, 19 | 0 | -12, 20 | 1 | (10, 30, 1), 0.7811 -11, 1 | 1 | (0, 1, 1), 1.2107 -11, 2 | 1 | (-2, 4, 1), 1.4570 -11, 3 | 1 | (-3, 3, 1), 1.2169 -11, 4 | 2 | (2, 9, 8), 2.5853 | | (12, 40, 1), 2.5864 -11, 5 | 1 | (4, 5, 1), 1.6324 -11, 6 | 0 | -11, 7 | 1 | (3, 1, 1), 1.0201 -11, 8 | 0 | -11, 9 | 1 | (0, 3, 1), 1.2504 -11, 10 | 1 | (-3, -4, 1), 0.5960 -11, 11 | 2 | (5, 9, 1), 1.3813 | | (2, 23, 8), 2.5768 -11, 12 | 0 | -11, 13 | 0 | -11, 14 | 0 | -11, 15 | 2 | (2, 1, 1), 0.7834 | | (-1, 5, 1), 1.0549 -11, 16 | 2 | (4, 6, 1), 1.4884 | | (18, 13, 8), 2.3917 -11, 17 | 1 | (8, 21, 1), 2.1617 -11, 18 | 1 | (2, 2, 1), 1.1398 -11, 19 | 2 | (1, 3, 1), 0.9019 | | (18, 19, 8), 2.6094 -11, 20 | 0 | -10, 1 | 2 | (4, 5, 1), 1.3169 | | (-14, 29, 8), 2.7678 -10, 2 | 1 | (-69, 89, 27), 3.6752 -10, 3 | 0 | -10, 4 | 2 | (0, 2, 1), 0.8467 | | (11, 35, 1), 2.5005 -10, 5 | 0 | -10, 6 | 2 | (5, 9, 1), 1.7878 | | (2, 15, 8), 2.5459 -10, 7 | 2 | (3, 2, 1), 1.3353 | | (2, 17, 8), 2.5456 -10, 8 | 0 | -10, 9 | 1 | (0, -3, 1), 0.8496 -10, 10 | 1 | (1, 1, 1), 0.9437 -10, 11 | 0 | -10, 12 | 1 | (3, -3, 1), 0.9898 -10, 13 | 2 | (2, 1, 1), 0.4091 | | (18, 11, 8), 2.2300 -10, 14 | 0 | -10, 15 | 0 | -10, 16 | 2 | (0, 4, 1), 0.9721 | | (7, 17, 1), 2.0538 -10, 17 | 0 | -10, 18 | 2 | (18, 21, 8), 2.6502 | | (17, 69, 1), 2.8917 -10, 19 | 1 | (3, 4, 1), 1.4303 -10, 20 | 0 | -9, 1 | 2 | (8, 21, 1), 1.6762 | | (-14, 27, 8), 2.1784 -9, 2 | 0 | -9, 3 | 1 | (2, 7, 8), 2.4811 -9, 4 | 2 | (3, 2, 1), 1.0294 | | (0, 2, 1), 1.1448 -9, 5 | 0 | -9, 6 | 1 | (-2, 4, 1), 1.4403 -9, 7 | 1 | (6, 13, 1), 1.9122 -9, 8 | 1 | (-1, -4, 1), 0.9696 -9, 9 | 1 | (1, 1, 1), 0.3876 -9, 10 | 0 | -9, 11 | 1 | (2, 1, 1), 0.7049 -9, 12 | 1 | (1, 2, 1), 0.6312 -9, 13 | 0 | -9, 14 | 1 | (2, 2, 1), 1.1135 -9, 15 | 1 | (-2, 5, 1), 1.5526 -9, 16 | 2 | (3, 4, 1), 1.0565 | | (-3, 4, 1), 1.3157 -9, 17 | 1 | (16, 63, 1), 2.2890 -9, 18 | 0 | -9, 19 | 1 | (2, 3, 1), 0.7844 -9, 20 | 1 | (5, 10, 1), 1.4918 -8, 1 | 2 | (0, 1, 1), 0.5958 | | (6, 13, 1), 1.5067 -8, 2 | 2 | (7, 17, 1), 2.0924 | | (2, 1, 8), 2.4143 -8, 3 | 0 | -8, 4 | 1 | (0, 2, 1), 0.1657 -8, 5 | 0 | -8, 6 | 1 | (3, 3, 1), 1.2820 -8, 7 | 1 | (-3, -2, 1), 1.5370 -8, 8 | 1 | (1, -1, 1), 0.7222 -8, 9 | 1 | (2, 1, 1), 0.1382 -8, 10 | 0 | -8, 11 | 1 | (1, 2, 1), 1.0077 -8, 12 | 1 | (2, 2, 1), 0.4005 -8, 13 | 1 | (3, 4, 1), 1.3690 -8, 14 | 0 | -8, 15 | 1 | (5, 10, 1), 1.8032 -8, 16 | 1 | (0, 4, 1), 0.3173 -8, 17 | 2 | (4, 7, 1), 1.1988 | | (19, 82, 1), 3.0056 -8, 18 | 1 | (-1, 5, 1), 0.6914 -8, 19 | 2 | (-3, 4, 1), 1.6798 | | (2, 33, 8), 2.7894 -8, 20 | 0 | -7, 1 | 1 | (0, 1, 1), 0.9946 -7, 2 | 0 | -7, 3 | 2 | (3, 3, 1), 0.9974 | | (2, 9, 8), 2.3572 -7, 4 | 1 | (0, 2, 1), 1.0255 -7, 5 | 0 | -7, 6 | 0 | -7, 7 | 1 | (1, 1, 1), 0.1296 -7, 8 | 0 | -7, 9 | 1 | (0, 3, 1), 1.1183 -7, 10 | 2 | (2, 2, 1), 1.0678 | | (9, 26, 1), 1.4137 -7, 11 | 0 | -7, 12 | 0 | -7, 13 | 1 | (4, 7, 1), 1.6262 -7, 14 | 0 | -7, 15 | 2 | (1, 3, 1), 0.9229 | | (2, 3, 1), 1.3077 -7, 16 | 1 | (0, 4, 1), 1.3674 -7, 17 | 0 | -7, 18 | 0 | -7, 19 | 1 | (3, 5, 1), 0.1291 -7, 20 | 1 | (48, 98, 27), 3.6151 -6, 1 | 1 | (0, 1, 1), 0.5757 -6, 2 | 0 | -6, 3 | 0 | -6, 4 | 1 | (0, -2, 1), 0.5971 -6, 5 | 1 | (2, -1, 1), 0.4189 -6, 6 | 1 | (1, 1, 1), 0.5550 -6, 7 | 1 | (3, 4, 1), 1.2493 -6, 8 | 1 | (2, 2, 1), 0.6867 -6, 9 | 1 | (0, 3, 1), 0.2576 -6, 10 | 1 | (-3, 1, 1), 1.5376 -6, 11 | 1 | (-1, 4, 1), 1.3495 -6, 12 | 1 | (-2, 4, 1), 1.1282 -6, 13 | 1 | (2, 3, 1), 0.3950 -6, 14 | 1 | (1, 3, 1), 0.7181 -6, 15 | 0 | -6, 16 | 2 | (3, 5, 1), 1.5275 | | (6, 14, 1), 1.6514 -6, 17 | 0 | -6, 18 | 1 | (-3, 3, 1), 1.0912 -6, 19 | 0 | -6, 20 | 2 | (-1, 5, 1), 1.5191 | | (8, 22, 1), 1.9083 -5, 1 | 1 | (0, 1, 1), 0.8376 -5, 2 | 1 | (-2, -2, 1), 1.2074 -5, 3 | 1 | (2, 1, 1), 0.8917 -5, 4 | 1 | (0, -2, 1), 0.7983 -5, 5 | 1 | (4, 7, 1), 1.4749 -5, 6 | 1 | (2, 2, 1), 0.9847 -5, 7 | 1 | (-2, 3, 1), 1.3499 -5, 8 | 1 | (1, 2, 1), 0.6834 -5, 9 | 1 | (0, 3, 1), 1.1725 -5, 10 | 1 | (6, 14, 1), 1.9581 -5, 11 | 2 | (10, 31, 1), 2.4105 | | (2, 25, 8), 2.6123 -5, 12 | 1 | (-1, 4, 1), 1.0215 -5, 13 | 2 | (12, 41, 1), 2.5843 | | (18, 29, 8), 2.7572 -5, 14 | 1 | (-2, 4, 1), 1.4945 -5, 15 | 0 | -5, 16 | 2 | (0, 4, 1), 1.3797 | | (7, 18, 1), 1.7871 -5, 17 | 0 | -5, 18 | 2 | (2, 4, 1), 1.4498 | | (-14, 37, 8), 2.9225 -5, 19 | 1 | (18, 35, 8), 2.8821 -5, 20 | 2 | (1, 4, 1), 1.0871 | | (11, 36, 1), 2.1849 -4, 1 | 2 | (0, 1, 1), 0.2727 | | (3, 4, 1), 1.3102 -4, 2 | 0 | -4, 3 | 0 | -4, 4 | 1 | (6, 14, 1), 1.0066 -4, 5 | 0 | -4, 6 | 1 | (-1, 3, 1), 1.1723 -4, 7 | 1 | (1, 2, 1), 1.0234 -4, 8 | 0 | -4, 9 | 2 | (0, 3, 1), 0.7216 | | (11, 36, 1), 2.4932 -4, 10 | 1 | (3, 5, 1), 1.4718 -4, 11 | 0 | -4, 12 | 1 | (1, 3, 1), 1.2560 -4, 13 | 1 | (-1, 4, 1), 1.3823 -4, 14 | 1 | (21, 91, 27), 3.5106 -4, 15 | 0 | -4, 16 | 2 | (4, 8, 1), 1.0447 | | (14, 52, 1), 1.6984 -4, 17 | 0 | -4, 18 | 1 | (2, 33, 8), 2.8004 -4, 19 | 2 | (-3, 2, 1), 1.6191 | | (34, 71, 8), 3.1906 -4, 20 | 0 | -3, 1 | 1 | (0, 1, 1), 0.6009 -3, 3 | 1 | (1, 1, 1), 0.2874 -3, 4 | 1 | (0, 2, 1), 0.9000 -3, 5 | 0 | -3, 6 | 1 | (1, 2, 1), 0.3174 -3, 7 | 1 | (3, 5, 1), 1.0857 -3, 8 | 0 | -3, 9 | 1 | (0, 3, 1), 0.6407 -3, 10 | 0 | -3, 11 | 1 | (5, 11, 1), 1.5090 -3, 12 | 1 | (4, 8, 1), 1.7093 -3, 13 | 0 | -3, 14 | 1 | (-1, 4, 1), 0.3545 -3, 15 | 0 | -3, 16 | 1 | (0, 4, 1), 1.3849 -3, 17 | 0 | -3, 18 | 1 | (1, 4, 1), 0.3818 -3, 19 | 1 | (-3, 1, 1), 1.2552 -3, 20 | 0 | -2, 1 | 0 | -2, 2 | 1 | (1, 1, 1), 0.5958 -2, 3 | 1 | (-1, 2, 1), 0.9609 -2, 4 | 1 | (0, 2, 1), 0.5709 -2, 5 | 2 | (-2, 1, 1), 0.8499 | | (2, 17, 8), 2.3650 -2, 6 | 1 | (5, 11, 1), 1.8099 -2, 7 | 0 | -2, 8 | 2 | (4, 8, 1), 1.3258 | | (-2, 2, 1), 0.9560 -2, 9 | 1 | (0, 3, 1), 0.8466 -2, 10 | 1 | (1, 3, 1), 1.2347 -2, 11 | 0 | -2, 12 | 1 | (2, 4, 1), 1.0517 -2, 13 | 1 | (-2, 3, 1), 1.0785 -2, 14 | 0 | -2, 15 | 2 | (-1, 4, 1), 1.4050 | | (-14, 29, 8), 2.8343 -2, 16 | 1 | (0, 4, 1), 1.0393 -2, 17 | 1 | (1, 4, 1), 1.4138 -2, 18 | 1 | (9, 27, 1), 2.3698 -2, 19 | 1 | (15, 58, 1), 2.8133 -2, 20 | 1 | (-2, 4, 1), 1.1958 -1, 1 | 1 | (1, 1, 1), 0.0498 -1, 2 | 0 | -1, 3 | 1 | (2, 3, 1), 1.1166 -1, 4 | 2 | (1, 2, 1), 0.6155 | | (4, 8, 1), 1.6175 -1, 5 | 0 | -1, 6 | 0 | -1, 7 | 1 | (-2, 1, 1), 1.2383 -1, 8 | 0 | -1, 9 | 2 | (1, 3, 1), 0.7580 | | (-1, 3, 1), 0.7758 -1, 10 | 2 | (2, 4, 1), 1.3748 | | (18, 35, 8), 2.8019 -1, 11 | 0 | -1, 12 | 1 | (3, 6, 1), 1.2314 -1, 13 | 0 | -1, 14 | 0 | -1, 15 | 2 | (6, 15, 1), 2.0435 | | (-14, 27, 8), 2.8185 -1, 16 | 2 | (-1, 4, 1), 1.0686 | | (16, 64, 1), 2.8677 -1, 17 | 0 | -1, 18 | 0 | -1, 19 | 1 | (2, 5, 1), 1.5399 -1, 20 | 0 |