- | En: Y2Z=X3+(n2-6n-3)X2Z+16nXZ2 y2=x3+(n2-6n-3)x2+16nx |
(x+y+z)(1/x+1/y+1/z)=n | - | |||
n | [a1,a2,a3,a4,a6] j(En) CM:Complex Multiplication. conductor of En |
rank of En(Q) |
En(Q)/En(Q)tors の生成元 [X:Y:Z] En(Q)tors En(Q)torsの生成元 |
En(Q)/En(Q)tors の生成元の高さ |
[x:y:z] (最小解と次の解) |
n |
-1 | [0, 4, 0, -16, 0] 16384/5 20 |
0 | - Z/6Z [-4, 8] |
- | - | -1 |
-2 | [0, 13, 0, -32, 0] 18609625/1188 66 |
0 | - Z/6Z [-8, 24] |
- | - | -2 |
-3 | [0, 24, 0, -48, 0] 54000 CM 36 |
0 | - Z/6Z [-12, 48] |
- | - | -3 |
-4 | [0, 37, 0, -64, 0] 3803721481/26000 130 |
1 | [64 : -640 : 1] Z/6Z [-16, 80] |
1.17046415359472 | [1 : 2 : -6] [1 : -3 : -6] |
-4 |
-5 | [0, 52, 0, -80, 0] 1594753024/4725 420 |
0 | - Z/6Z [-20, 120] |
- | - | -5 |
-6 | [0, 69, 0, -96, 0] 4767078987/6860 630 |
0 | - Z/6Z [-24, 168] |
- | - | -6 |
-7 | [0, 88, 0, -112, 0] 128787625/98 14 |
0 | - Z/6Z [-28, 224] |
- | - | -7 |
-8 | [0, 109, 0, -128, 0] 1845026709625/793152 102 |
0 | - Z/6Z [-32, 288] |
- | - | -8 |
-9 | [0, 132, 0, -144, 0] 488095744/125 180 |
0 | - Z/6Z [-36, 360] |
- | - | -9 |
-10 | [0, 157, 0, -160, 0] 15868125221689/2528900 2090 |
1 | [-8 : -104 : 1] Z/6Z [-40, 440] |
1.58004991707949 | [1 : 3 : -12] [1 : -4 : -12] |
-10 |
-11 | [0, 184, 0, -176, 0] 158792223184/16335 660 |
1 | [100 : -1680 : 1] Z/6Z [-44, 528] |
1.86793279055472 | [1 : -6 : -10] [3 : 5 : -30] |
-11 |
-12 | [0, 213, 0, -192, 0] 3592121380875/246064 1638 |
1 | [48 : -496 : 27] Z/6Z [-48, 624] |
2.9354641578894 | [4 : 35 : -84] [-5 : 12 : 105] |
-12 |
-13 | [0, 244, 0, -208, 0] 13608288256000/637637 4004 |
0 | - Z/6Z [-52, 728] |
- | - | -13 |
-14 | [0, 277, 0, -224, 0] 463702796512201/15214500 4830 |
0 | - Z/6Z [-56, 840] |
- | - | -14 |
-15 | [0, 312, 0, -240, 0] 8527173507/200 90 |
0 | - Z/6Z [-60, 960] |
- | - | -15 |
-16 | [0, 349, 0, -256, 0] 1841373668746009/31443200 170 |
0 | - Z/6Z [-64, 1088] |
- | - | -16 |
-17 | [0, 388, 0, -272, 0] 216727177216000/2738853 2652 |
0 | - Z/6Z [-68, 1224] |
- | - | -17 |
-18 | [0, 429, 0, -288, 0] 8671983378625/82308 342 |
1 | [-6 : -171 : 8] Z/6Z [-72, 1368] |
1.0648513558535 | [1 : 4 : -20] [1 : -5 : -20] |
-18 |
-19 | [0, 472, 0, -304, 0] 43725490482256/315875 2660 |
1 | [484 : -14960 : 1] Z/6Z [-76, 1520] |
2.96077212296937 | [3 : -14 : -66] [7 : 33 : -154] |
-19 |
-20 | [0, 517, 0, -320, 0] 19302534392242249/107427600 6090 |
1 | [144 : -3696 : 1] Z/6Z [-80, 1680] |
2.0018566081256 | [1 : -10 : -15] [2 : 3 : -30] |
-20 |
-21 | [0, 564, 0, -336, 0] 75216478666752/326095 13860 |
0 | - Z/6Z [-84, 1848] |
- | - | -21 |
-22 | [0, 613, 0, -352, 0] 53508049906515625/182553668 15686 |
1 | [896 : 19648 : 343] Z/6Z [-88, 2024] |
5.48934516696152 | [60 : 2516 : -3885] [-68 : 105 : 4403] |
-22 |
-23 | [0, 664, 0, -368, 0] 21081759765625/57132 138 |
0 | - Z/6Z [-92, 2208] |
- | - | -23 |
-24 | [0, 717, 0, -384, 0] 5066026756449723/11000000 990 |
0 | - Z/6Z [-96, 2400] |
- | - | -24 |
-25 | [0, 772, 0, -400, 0] 13310810713145344/23343125 4420 |
1 | [2500 : -143000 : 1] Z/6Z [-100, 2600] |
2.71203065247207 | [1 : 15 : -40] [3 : -8 : -120] |
-25 |
-26 | [0, 829, 0, -416, 0] 326355561310674169/465699780 2730 |
0 | - Z/6Z [-104, 2808] |
- | - | -26 |
-27 | [0, 888, 0, -432, 0] 2640279346000/3087 252 |
0 | - Z/6Z [-108, 3024] |
- | - | -27 |
-28 | [0, 949, 0, -448, 0] 733736571525483625/707476112 15022 |
2 | [-16 : -496 : 1], [12288 : -460288 : 27] Z/6Z [-112, 3248] |
2.13676074496033 4.25392692429032 |
[1 : 5 : -30] [1 : -6 : -30] [21 : -165 : -560] [33 : 112 : -880] |
-28 |
-29 | [0, 1012, 0, -464, 0] 67411307099324416/53929125 33060 |
1 | [242 : -7755 : 8] Z/6Z [-116, 3480] |
4.13817729242365 | [12 : -77 : -364] [33 : 156 : -1001] |
-29 |
-30 | [0, 1077, 0, -480, 0] 58015885327629867/38728300 36270 |
1 | [-756 : -27180 : 343] Z/6Z [-120, 3720] |
5.49123017151537 | [156 : -2109 : -3367] [228 : 364 : -4921] |
-30 |
-31 | [0, 1144, 0, -496, 0] 549131937598369/307520 310 |
1 | [196 : -7168 : 1] Z/6Z [-124, 3968] |
1.95523821419308 | [1 : -21 : -15] [5 : 7 : -1057] |
-31 |
-32 | [0, 1213, 0, -512, 0] 3195392484115617625/1508779008 2706 |
1 | [5780 : -204204 : 125] Z/6Z [-128, 4224] |
4.34954503066335 | [15 : -136 : -440] [51 : 165 : -1496] |
-32 |
-33 | [0, 1284, 0, -528, 0] 10402975660032000/4161311 47124 |
0 | - Z/6Z [-132, 4488] |
- | - | -33 |
-34 | [0, 1357, 0, -544, 0] 6260849528983300441/2131230500 51170 |
0 | - Z/6Z [-136, 4760] |
- | - | -34 |
-35 | [0, 1432, 0, -560, 0] 33766427105425744/9823275 4620 |
0 | - Z/6Z [-140, 5040] |
- | - | -35 |
-36 | [0, 1509, 0, -576, 0] 16232905099479601/4052240 3330 |
1 | [18928 : 746096 : 343] Z/6Z [-144, 5328] |
6.82571727616647 | [770 : -8151 : -24339] [1430 : 4270 : -45201] |
-36 |
-37 | [0, 1588, 0, -592, 0] 1004385121189888000/215969333 64676 |
0 | - Z/6Z [-148, 5624] |
- | - | -37 |
-38 | [0, 1669, 0, -608, 0] 21656629953561291625/4025861892 69654 |
0 | - Z/6Z [-152, 5928] |
- | - | -38 |
-39 | [0, 1752, 0, -624, 0] 261984288445803/42250 1170 |
1 | [64 : -2720 : 1] Z/6Z [-156, 6240] |
2.68362485347205 | [1 : -12 : -33] [4 : 11 : -132] |
-39 |
-40 | [0, 1837, 0, -640, 0] 38494263748526418169/5403406400 2870 |
1 | [32 : 1376 : 1] Z/6Z [-160, 6560] |
2.34388546902757 | [1 : 6 : -42] [1 : -7 : -42] |
-40 |
-41 | [0, 1924, 0, -656, 0] 3175432607945703424/389193525 17220 |
0 | - Z/6Z [-164, 6888] |
- | - | -41 |
-42 | [0, 2013, 0, -672, 0] 2468010046615954875/264917324 92106 |
1 | [-32856 : -927368 : 27] Z/6Z [-168, 7224] |
7.66387594743436 | [2460 : 11859 : -101639] [-3540 : 30340: 146261] |
-42 |
-43 | [0, 2104, 0, -688, 0] 339345250021762000/31993247 24596 |
1 | [1887876 : 129335536 : 729] Z/6Z [-172, 7568] |
8.45826732910586 | [5166 : -23715 : -272510] [11439 : 131446 : -603415] |
-43 |
-44 | [0, 2197, 0, -704, 0] 112603088219295873961/9350154000 17490 |
1 | [256 : -12672 : 1] Z/6Z [-176, 7920] |
2.55634880034839 | [1 : -21 : -28] [3 : 4 : -84] |
-44 |
-45 | [0, 2292, 0, -720, 0] 12444451776495616/912525 4140 |
0 | - Z/6Z [-180, 8280] |
- | - | -45 |
-46 | [0, 2389, 0, -736, 0] 186123345206731842889/12082920740 118910 |
0 | - Z/6Z [-184, 8648] |
- | - | -46 |
-47 | [0, 2488, 0, -752, 0] 57971836400640625/3340008 1974 |
0 | - Z/6Z [-188, 9024] |
- | - | -47 |
-48 | [0, 2589, 0, -768, 0] 11165451838341046875/572244736 2394 |
1 | [704 : -23744 : 1331] Z/6Z [-192, 9408] |
3.83101824860721 | [12 : 65 : -572] [-15 : 132 : 715] |
-48 |
-49 | [0, 2692, 0, -784, 0] 23809656960517881856/1087953125 4060 |
0 | - Z/6Z [-196, 9800] |
- | - | -49 |
-50 | [0, 2797, 0, -800, 0] 479241517683532341529/19566022500 30090 |
1 | [89100 : -4758300 : 1331] Z/6Z [-200, 10200] |
5.21430844250903 | [209, -2736, 63] [-1008, -3344, 77] |
-50 |
-51 | [0, 2904, 0, -816, 0] 86846853774358512/3174665 39780 |
1 | [7396 : -750608 : 1] Z/6Z [-204, 10608] |
6.65642974042261 | [266 : -825 : -19866] [475 : 11438 : -35475] |
-51 |
-52 | [0, 3013, 0, -832, 0] 748777804614503079625/24556367888 84058 |
2 | [-1 : -62 : 1], [67280 : -4008496 : 125] Z/6Z [-208, 11024] |
3.68629321398363 5.2029233790362 |
[5 : 18 : -230] [9 : -115 : -414] [45 : -770 : -1827] [110 : 261 : -4466] |
-52 |
-53 | [0, 3124, 0, -848, 0] 58141435482554368000/1713975957 19716 |
0 | - Z/6Z [-212, 11448] |
- | - | -53 |
-54 | [0, 3237, 0, -864, 0] 1579250141304807889/41926500 6930 |
1 | [36 : -2052 : 1] Z/6Z [-216, 11880] |
1.97239038546929 | [1 : 7 : -56] [1 : -8 : -56] |
-54 |
-55 | [0, 3352, 0, -880, 0] 346553430870203929/8300600 770 |
0 | - Z/6Z [-220, 12320] |
- | - | -55 |
-56 | [0, 3469, 0, -896, 0] 1743880770596311067449/37749741120 51870 |
1 | [2268 : -131076 : 343] Z/6Z [-224, 12768] |
6.77022839261274 | [175 : -288 : -25312] [225 : 19775 : -32544] |
-56 |
-57 | [0, 3588, 0, -912, 0] 4942056726521856000/96848719 218196 |
0 | - Z/6Z [-228, 13224] |
- | - | -57 |
-58 | [0, 3709, 0, -928, 0] 2604981500967214869625/46289962052 229274 |
1 | [-696332552 : -146434734232 : 30400540561] Z/6Z [-232, 13688] |
15.0714682382382 | [111600756 : -198642325 : -14941163300] [27728847 : -3712351475 : 2085664428] |
-58 |
-59 | [0, 3832, 0, -944, 0] 12375543671015497936/199722375 60180 |
2 | [1444 : -104880 : 1], [3364 : -285360 : 1] Z/6Z [-236, 14160] |
3.41654720050278 4.14367873735443 |
[2 : -21 : -114] [7 : 38 : -399] [6 : -35 : -406] [15 : 174 : -1015] |
-59 |
-60 | [0, 3957, 0, -960, 0] 142256513008321688907/2088225200 126270 |
0 | - Z/6Z [-240, 14640] |
- | - | -60 |
-61 | [0, 4084, 0, -976, 0] 290150363043928735744/3879830885 264740 |
1 | [28938650 : 2190541865 : 17576] Z/6Z [-244, 15128] |
14.1664601050212 |
[19535516 : -196404156 : -1171388655] [66454388 : 396345565 : -3984738165] |
-61 |
-62 | [0, 4213, 0, -992, 0] 5594576132523476463625/68243827428 92442 |
0 | - Z/6Z [-248, 15624] |
- | - | -62 |
-63 | [0, 4344, 0, -1008, 0] 2251439055699625/25088 126 |
0 | - Z/6Z [-252, 16128] |
- | - | -63 |
-64 | [0, 4477, 0, -1024, 0] 8056051600393270819801/82115072000 9490 |
0 | - Z/6Z [-256, 16640] |
- | - | -64 |
-65 | [0, 4612, 0, -1040, 0] 601737260143037661184/5617851525 317460 |
0 | - Z/6Z [-260, 17160] |
- | - | -65 |
-66 | [0, 4749, 0, -1056, 0] 425042400204763799067/3639232300 66330 |
0 | - Z/6Z [-264, 17688] |
- | - | -66 |
-67 | [0, 4888, 0, -1072, 0] 53299431631709746000/419034683 86564 |
1 | [16428 : -1217744 : 27] Z/6Z [-268, 18224] |
4.89590624570529 | [15 : -348 : -740] [87 : 185 : -4292] |
-67 |
-68 | [0, 5029, 0, -1088, 0] 16182959645747940207625/116964972432 180642 |
0 | - Z/6Z [-272, 18768] |
- | - | -68 |
-69 | [0, 5172, 0, -1104, 0] 44322949895986741248/294851375 376740 |
1 | [3468 : -252280 : 27] Z/6Z [-276, 19320] |
3.84192826067535 | [3 : -68 : -156] [17 : 39 : -884] |
-69 |
-70 | [0, 5317, 0, -1120, 0] 22602412819916594979049/138547348100 392630 |
2 | [10 : 665 : 8], [648 : 50040 : 1] Z/6Z [-280, 19880] |
4.45890988114303 5.22888230822343 |
[9 : 275 : -900] [-11 : 36 : 1100] [28 : -667 : -1449] [116 : 252 : -6003] |
-70 |
-71 | [0, 5464, 0, -1136, 0] 6499095407581304809/36748890 2130 |
0 | - Z/6Z [-284, 20448] |
- | - | -71 |
-72 | [0, 5613, 0, -1152, 0] 42912679782639390625/224073792 1314 |
1 | [-84 : -8532 : 343] Z/6Z [-288, 21024] |
4.51978167651286 | [52 : 700 : -4277] [100 : -611 : -8225] |
-72 |
-73 | [0, 5764, 0, -1168, 0] 2292774030761728000000/11067123317 442964 |
0 |
- Z/6Z [-292, 21608] |
- | - | -73 |
-74 | [0, 5917, 0, -1184, 0] 42928092574972066968121/191745562500 92130 |
0 | - Z/6Z [-296, 22200] |
- | - | -74 |
-75 | [0, 6072, 0, -1200, 0] 7252939560652551792/30008125 23940 |
0 | - Z/6Z [-300, 22800] |
- | - | -75 |
-76 | [0, 6229, 0, -1216, 0] 58429541178747172156969/224139441680 248710 |
2 | [400 : -32560 : 1], [18845008 : -2293095056 : 2197] Z/6Z [-304, 23408] |
2.97261636156562 12.1982601758386 |
[1 : -36 : -45] [4 : 5 : -180] [-5053373 : 1136460 : -109032735] [1219556 : 26313420 : -117005021] |
-76 |
-77 | [0, 6388, 0, -1232, 0] 4248028911027748864000/15123201093 516516 |
0 | - Z/6Z [-308, 24024] |
- | - | -77 |
-78 | [0, 6549, 0, -1248, 0] 2922805445028622000875/9665536556 536094 |
1 | [-12990877176 : -1063298090184 : 1647212741] Z/6Z [-312, 24648] |
13.5095082695496 | [9762340 : -264208596 : -551048889] [35518805 : 17030020 : -961283217] |
-78 |
-79 | [0, 6712, 0, -1264, 0] 22328595172851474961/68651000 8690 |
0 | - Z/6Z [-316, 25280] |
- | - | -79 |
-80 | [0, 6877, 0, -1280, 0] 105803474625631920221209/302708793600 2670 |
0 | - Z/6Z [-320, 25920] |
- | - | -80 |
-81 | [0, 7044, 0, -1296, 0] 10475401104030908416/2791300 7380 |
0 | - Z/6Z [-324, 26568] |
- | - | -81 |
-82 | [0, 7213, 0, -1312, 0] 140862099071150894697625/349867316708 619346 |
0 | - Z/6Z [-328, 27224] |
- | - | -82 |
-83 | [0, 7384, 0, -1328, 0] 633293980892869714000/1467377667 160356 |
0 | - Z/6Z [-332, 27888] |
- | - | -83 |
-84 | [0, 7557, 0, -1344, 0] 6899613337905758373483/14925694000 332010 |
1 | [236992 : -21520640 : 343] Z/6Z [-336, 28560] |
5.93263379723439 | [77 : -2331 : -4554] [259 : 506 : -15318] |
84 |
-85 | [0, 7732, 0, -1360, 0] 13357340681706321608704/26998589525 687140 |
1 | [-60 : -5480 : 27] Z/6Z [-340, 29240] |
3.67657946188448 | [4 : -133 : -228] [7 : 12 : -399] |
-85 |
-86 | [0, 7909, 0, -1376, 0] 244802264768607231125929/462677377860 710790 |
1 | [96160712 : 42542055789 : 512] Z/6Z [-344, 29928] |
13.3309718051126 | [2214640 : 392828435 : -585149859] [1812880 : -2700432 : -478997253] |
-86 |
-87 | [0, 8088, 0, -1392, 0] 2531665445447410875/4477484 5742 |
0 | - Z/6Z [-348, 30624] |
- | - | -87 |
-88 | [0, 8269, 0, -1408, 0] 319741078614026236701625/529550153792 189926 |
1 | [-32 : -2912 : 1] Z/6Z [-352, 31328] |
2.8159809781317 | [1 : 9 : -90] [1 : -10 : -90] |
-88 |
-89 | [0, 8452, 0, -1424, 0] 22788453825192951463936/35368255125 37380 |
1 | [100 : -9240 : 1] Z/6Z [-356, 32040] |
2.1174709481781 | [1 : -20 : -76] [5 : 19 : -380] |
-89 |
-90 | [0, 8637, 0, -1440, 0] 569541582763202518561/828928100 90090 |
0 | - Z/6Z [-360, 32760] |
- | - | -90 |
-91 | [0, 8824, 0, -1456, 0] 1844276867950490221264/2518873175 41860 |
1 | [-7 : -665 : 1] Z/6Z [-364, 33488] |
2.83301362321699 | [3 : -102 : -187] [6 : 11 : -374] |
-91 |
-92 | [0, 9013, 0, -1472, 0] 536150914376867443851625/687615842448 432078 |
1 |
[66660617744 : -10537038516432 : 4173281] Z/6Z [-368, 34224] |
14.8106726248905 | [45830904 : 1457527240 : -5623815849] [46129720 : -177989847 : -5660482945] |
-92 |
-93 | [0, 9204, 0, -1488, 0] 1407487566456815616000/1696156351 891684 |
0 | - Z/6Z [-372, 34968] |
- | - | -93 |
-94 | [0, 9397, 0, -1504, 0] 688655360101930386055561/780303846500 919790 |
0 | - Z/6Z [-376, 35720] |
- | - | -94 |
-95 | [0, 9592, 0, -1520, 0] 190177723376764332769/202737600 7410 |
1 | [484 : -48576 : 1] Z/6Z [-380, 36480] |
2.85866273758706 | [1 : -45 : -55] [9 : 11 : -495] |
-95 |
-96 | [0, 9789, 0, -1536, 0] 32593339344811308900507/32710200320 61110 |
1 | [35208152148 : 3448011074516 : 4716275733] Z/6Z [-384, 37248] |
18.1250498623291 |
[3003037440 : -5581682432 : -627978474123] [4102539520 : 461564508405 : -857900229109] |
-96 |
-97 | [0, 9988, 0, -1552, 0] 62060036418810112000000/58668850373 143948 |
0 | - Z/6Z [-388, 38024] |
- | - | -97 |
-98 | [0, 10189, 0, -1568, 0] 1119047205729567539265625/997106420772 49434 |
0 | - Z/6Z [-392, 38808] |
- | - | -98 |
-99 | [0, 10392, 0, -1584, 0] 6749703004355978704/5671875 1980 |
0 | - Z/6Z [-396, 39600] |
- | - | -99 |
-100 | [0, 10597, 0, -1600, 0] 1416293608641272096072329/1123028090000 110090 |
1 | [10543167251152 : -1079242332937648 : 861985991413] Z/6Z [-400, 40400] |
18.0515117035529 |
[4450012553 : 219887106322 : -663397965750] [5089624981 : -15355365375 : -758749962750] |
-100 |
gp> read("./reciprocal3.gp") time = 150 ms. gp> e1=ec(-60) time = 131 ms. %5 = [0, 3957, 0, -960, 0, 15828, -1920, 0, -921600, 250571664, -3966412974912,230941001318400, 142256513008321688907/2088225200, [0.2425931636484586889415539985, 0.E-38, -3957.242593163648458688941554]~, 0.1982637253241678309476177803, 0.04993980928293878765131851380*I, 67.85936899391501106399315344, 1.247284603487413674594150081*I, 0.009901252630413902639922383745] gp> ar(e1,1.00001) time = 11,575 ms. %6 = -0.00003457918680392556967 gp> elllseries(e1,1.0000000001) time = 5,548 ms. %7 = 2.114813069393155319492478359 gp> e2=ec(-73) time = 79 ms. %8 = [0, 5764, 0, -1168, 0, 23056, -2336, 0, -1364224, 531635200, -12258027476992, 725294993702912, 2292774030761728000000/11067123317, [0.2026299342660894032421053265, 0.E-38, -5764.202629934266089403242105]~, 0.1716001749084163499881622984, 0.04137866133359280907867807562*I, 88.93624755196788393616521721, 3.137935112426557342923956153*I, 0.007100585522320650579108621231] gp> ar(e2,1.00001) time = 12,444 ms. %9 = -0.00004085370938634430243737582886 gp> elllseries(e2,1.0000000001) time = 6,119 ms. %10 = 1.372801398706468785437902081
Last Update: 2021.04.19 |
H.Nakao |