Rational points on curve: (x+y+z)(1/x+1/y+1/z)=n (n \in [-100..100])

(x+y+z)(1/x+1/y+1/z)=n (n \in [-100..100])の有理点


高さの大きい有理点(有理数解)を、赤色で強調した。

- En: Y2Z=X3+(n2-6n-3)X2Z+16nXZ2
y2=x3+(n2-6n-3)x2+16nx
 Cn: (x+y+z)(1/x+1/y+1/z)=n -
n [a1,a2,a3,a4,a6]
a1=a3=a6=0,
a2=n2-6n-3,
a4=16n
 		rank
of
En(Q)		 Tors(En(Q)) Tors(En(Q))
の生成元		 En(Q)/Tors(En(Q))
の生成元 [X:Y:Z]		 En(Q)/Tors(En(Q))
の生成元の高さ		 [x:y:z]
(最小解と次の解)		 n
2 [0, -11, 0, 32, 0] 0 Z/6Z [8, 8] - - - 2
3 [0, -12, 0, 48, 0] 0 Z/6Z [12, 24] - - - 3
4 [0, -11, 0, 64, 0] 0 Z/6Z [16, 48] - - - 4
5 [0, -8, 0, 80, 0] 0 Z/6Z [20, 80] - - - 5
6 [0, -3, 0, 96, 0] 0 Z/6Z [24, 120] - - - 6
7 [0, 4, 0, 112, 0] 0 Z/6Z [28, 168] - - - 7
8 [0, 13, 0, 128, 0] 0 Z/6Z [32, 224] - - - 8
10 [0, 37, 0, 160, 0] 0 Z/6Z×Z/2Z [-20, 60],
[0, 0]
 - - [1 : 1 : 2], [1 : 2 : 2]
 10
11 [0, 52, 0, 176, 0] 1 Z/6Z [44, 440] [-4 : -8 : 1] 0.673594857184961 [1 : 2 : 3], [2 : 3 : 6]
 11
12 [0, 69, 0, 192, 0] 0 Z/6Z [48, 528] - - - 12
13 [0, 88, 0, 208, 0] 0 Z/6Z [52, 624] - - - 13
14 [0, 109, 0, 224, 0] 1 Z/6Z [56, 728] [-4 : -28 : 1] 1.51870218672685 [2 : 3 : 10], [3 : 10 : 15]
 14
15 [0, 132, 0, 240, 0] 1 Z/6Z [60, 840] [-12 : -120 : 1] 1.03381685303953 [1 : 2 : 6], [1 : 3 : 6]
 15
16 [0, 157, 0, 256, 0] 0 Z/6Z [64, 960] - - - 16
17 [0, 184, 0, 272, 0] 0 Z/6Z [66, 1088] - - - 17
18 [0, 213, 0, 288, 0] 1 Z/6Z [72, 1224] [-36 : -468 : 1] 1.68788702948039 [2 : 3 : 15], [2 : 10 : 15]
 18
19 [0, 244, 0, 304, 0] 0 Z/6Z [76, 1368] - - - 19
20 [0, 277, 0, 320, 0] 0 Z/6Z [80, 1520] - - - 20
21 [0, 312, 0, 336, 0] 0 Z/6Z [84, 1680] - - - 21
22 [0, 349, 0, 352, 0] 0 Z/6Z [88, 1848] - - - 22
23 [0, 388, 0, 368, 0] 0 Z/6Z [92, 2024] - - - 23
24 [0, 429, 0, 384,0] 0 Z/6Z [96, 2208] - - - 24
25 [0, 472, 0, 400, 0] 0 Z/6Z [100, 2400] - - - 25
26 [0, 517, 0, 416, 0] 1 Z/6Z [104, 2600] [-1 : -10 : 1] 1.72166299585031 [1 : 6 : 14], [3 : 7 : 42]
 26
27 [0, 564, 0, 432, 0] 0 Z/6Z [108, 2808] - - - 27
28 [0, 613, 0, 448, 0] 0 Z/6Z [112, 3024] - - - 28
29 [0, 664, 0, 464, 0] 1 Z/6Z [116, 3248] [12 : 496 : 27] 3.06776193627718 [2 : -15 : 78], [-5 : 26 : -195]
 29
30 [0, 717, 0, 480, 0] 1 Z/6Z [120, 3480] [-216 : -4824 : 1] 3.39075763351921 [10 : 77 : 165], [14 : 30 : 231]
 30
31 [0, 772, 0, 496, 0] 1 Z/6Z [124, 3720] [484 : -17160 : 1] 2.75467200740796 [-1 : 12 : -44], [3 : -11 : 132]
 31
32 [0, 829, 0, 512, 0] 0 Z/6Z [128, 3968] - - - 32
33 [0, 888, 0, 528, 0] 0 Z/6Z [132, 4224] - - - 33
34 [0, 949, 0, 544, 0] 1 Z/6Z [136, 4488] [-24 : -440 : 27] 1.30258691809008 [1 : 6 : 21], [2 : 7 : 42]
 34
35 [0, 1012, 0, 560, 0] 1 Z/6Z [140, 4760] [-16 : -496 : 1] 2.86238816326288 [3 : 10 : 65], [6 : 39 : 130]
 35
36 [0, 1077, 0, 576, 0] 0 Z/6Z [144, 5040] - - - 36
37 [0, 1144, 0, 592, 0] 1 Z/6Z [148, 5328] [20 : 1392 : 125] 3.23846309225013 [2 : -35 : 110], [-7 : 22 : -385]
 37
38 [0, 1213, 0, 608, 0] 1 Z/6Z [152, 5624] [-676 : -15652 : 1] 5.25136679075421 [90 : 391 : 2210],
[207 : 1170 : 5083]
 38
39 [0, 1284, 0, 624, 0] 0 Z/6Z [156, 5928] - - - 39
40 [0, 1357, 0, 640, 0] 0 Z/6Z [160, 6240] - - - 40
41 [0, 1432, 0, 656, 0] 0 Z/6Z [164, 6560] - - - 41
42 [0, 1509, 0, 672, 0] 1 Z/6Z [168, 6888] [-8664 : -298376 : 27] 6.59921366460535 [561 : 6450 : 13889],
[1650 : 3553 : 40850]
 42
43 [0, 1588, 0, 688, 0] 1 Z/6Z [172, 7224] [6724 : -613032 : 1] 4.97538785845239 [-24 : 65 : -1640],
[39 : -984 : 2665]
 43
44 [0, 1669, 0, 704, 0] 1 Z/6Z [176, 7568] [14896 : 665392 : 6859] 7.52727627802677 [-777 : 40810 : -74518],
[1155 : -2109 : 110770]
 44
45 [0, 1752, 0, 720, 0] 0 Z/6Z [180, 7920] - - - 45
46 [0, 1837, 0, 736, 0] 0 Z/6Z [184, 8280] - - - 46
47 [0, 1924, 0, 752, 0] 0 Z/6Z [188, 8648] - - - 47
48 [0, 2013, 0, 768, 0] 1 Z/6Z [192, 9024] [9199680 : -423343936 : 1157625] 10.7297594067252 [70633 : -3329130 : 6685382],
[200445 : -402523 : 18972030]		 48
49 [0, 2104, 0, 784, 0] 0 Z/6Z [196, 9408] - - - 49
50 [0, 2197, 0, 800,0] 0 Z/6Z [200, 9800] - - - 50
51 [0, 2292, 0, 816, 0] 0 Z/6Z [204, 10200] - - - 51
52 [0, 2389, 0, 832, 0] 1 Z/6Z [208, 10608] [8 : 1887 : 512] 4.90720255014314 [5 : -464 : 720], [-29 : 45 : -4176]
 52
53 [0, 2488, 0, 848, 0] 1 Z/6Z [212, 11024] [3468 : -177616 : 27] 5.56336142430785 [15 : -1190 : 1974],
[85 : -141 : 11186]
 53
54 [0, 2589, 0, 864, 0] 1 Z/6Z [216, 11448] [-1764 : -50652 : 1] 4.51072900298448 [55 : 595 : 2002], [85 : 286 : 3094]
 54
55 [0, 2692, 0, 880, 0] 1 Z/6Z [220, 11880] [-80 : -4080 : 1] 2.22629739923563 [1 : 14 : 35], [2 : 5 : 70]
 55
56 [0, 2797, 0, 896, 0] 0 Z/6Z [224, 12320] - - - 56
57 [0, 3013, 0, 928, 0] 0 Z/6Z [228, 12768] - - - 57
58 [0, 3013, 0, 928, 0] 0 Z/6Z [232, 13224] - - - 58
59 [0, 3124, 0, 944, 0] 1 Z/6Z [236, 13688] [-36 : -1992 : 1] 3.09619783215107 [2 : 15 : 85], [6 : 34 : 255]
 59
60 [0, 3237, 0, 960, 0] 0 Z/6Z [240, 14160] - - - 60
61 [0, 3352, 0, 976, 0] 0 Z/6Z [244, 14640] - - - 61
62 [0, 3469, 0, 992, 0] 1 Z/6Z [248, 15128] [-637 : -4382 : 2197] 8.58198027418168 [5075 : 128050 : 160602],
[34475 : 43239 : 1090986]		 62
63 [0, 3588, 0, 1008, 0] 1 Z/6Z [252, 15624] [-3 : -171 : 1] 2.64589312579157 [5 : 119 : 170], [7 : 10 : 238] 63
64 [0, 3709, 0, 1024, 0] 1 Z/6Z [256, 16128] [16384 : -2322432 : 1] 3.32752420180303 [1 : -40 : 104], [-5 : 13 : -520]
 64
65 [0, 3832, 0, 1040, 0] 0 Z/6Z [260, 16640] - - - 65
66 [0, 3957, 0, 1056, 0] 0 Z/6Z [264, 17160] - - - 66
67 [0, 4084, 0, 1072, 0] 1 Z/6Z [268, 17688] [1764 : -134904 : 1] 4.26277557167492 [-5 : 68 : -420],
[17 : -105 : 1428]
 67
68 [0, 4213, 0, 1088, 0] 0 Z/6Z [272, 18224] - - - 68
69 [0, 4344, 0, 1104, 0] 1 Z/6Z [276, 18768] [96100 : -30456880 : 1] 9.16977000480395 [-4823 : 7458 : -930930],
[5989 : -747565 : 1155990]
 69
70 [0, 4477, 0, 1120, 0] 1 Z/6Z [280, 19320] [-1620 : -86580 : 1] 5.39008635301204 [70 : 551 : 3654], [95 : 630 : 4959] 70
71 [0, 4612, 0, 1136, 0] 1 Z/6Z [264, 19880] [3364 : -300440 : 1] 4.0600024608223 [-4 : 33 : -348], [11 : -116 : 957]
 71
72 [0, 4749, 0, 1152, 0] 0 Z/6Z [288, 20448] - - - 72
73 [0, 4888, 0, 1168, 0] 1 Z/6Z [292, 21024] [1444 : -114912 : 1] 2.97688197131706 [-1 : 20 : -95], [4 : -19 : 380]
 73
74 [0, 5029, 0, 1184, 0] 2 Z/6Z [296, 21608] [-1 : -62 : 1],
[1764 : 131068 : 729]
 3.82491508345059
7.10236671517554		 [5 : 22 : 270], [11 : 135 : 594],
[224 : -369 : 41760],
[-287 : 32480 : -53505]
 74
75 [0, 5172, 0, 1200, 0] 1 Z/6Z [300, 22200] [2429900 : 203275000 : 1331] 9.10891625435626 [10153 : -58656 : 957719],
[-13728 : 224147 : -1294944]
 75
76 [0, 5317, 0, 1216, 0] 1 Z/6Z [304, 22800] [704 : 61440 : 1331] 4.42130818678977 [14 : -99 : 1309],
[-18 : 238 : -1683]
 76
77 [0, 5464, 0, 1232, 0] 0 Z/6Z [308, 23408] - - - 77
78 [0, 5613, 0, 1248, 0] 0 Z/6Z [312, 24024] - - - 78
79 [0, 5764, 0, 1264, 0] 0 Z/6Z [316, 24648] - - - 79
80 [0, 5917, 0, 1280, 0] 0 Z/6Z [320, 25280] - - - 80
81 [0, 6072, 0, 1296, 0] 0 Z/6Z [324, 25920] - - - 81
82 [0, 6229, 0, 1312, 0] 1 Z/6Z [328, 26568] [-3872 : -187968 : 1] 5.60028104176275 [119 : 475 : 7106],
[175 : 2618 : 10450]
 82
83 [0, 6388, 0, 1328, 0] 0 Z/6Z [332, 27224] - - - 83
84 [0, 6549, 0, 1344, 0] 1 Z/6Z [336, 27888] [108 : 8820 : 1] 3.09288912090045 [2 : -78 : 247], [6 : -19 : 741]
 84
85 [0, 6712, 0, 1360, 0] 1 Z/6Z [340, 28560] [3 : 413 : 27] 4.57888865059071 [5 : -138 : 570], [-23 ; 95 : -2622]
 85
86 [0, 6877, 0, 1376, 0] 1 Z/6Z [344, 29240] [196 : 16492 : 1] 4.8221051731666 [3 : -336 : 592], [21 : -37 : 4144]
 86
87 [0, 7044, 0, 1392, 0] 0-2? Z/6Z [348, 29928] ??? ??? ??? 87
88 [0, 7213, 0, 1408, 0] 0 Z/6Z [352, 30624] - - - 88
89 [0, 7384, 0, 1424, 0] 0 Z/6Z [356, 31328] - - - 89
90 [0, 7557, 0, 1440, 0] 1 Z/6Z [360, 32040] [-67032 : -5766264 : 6859] 7.74994729753809 [1386 : 35226 : 81473],
[3762 : 8701 : 221141]
 90
91 [0, 7732, 0, 1456, 0] 0 Z/6Z [364, 32760] - - - 91
92 [0, 7909, 0, 1472, 0] 1 Z/6Z [368, 33488] [19600 : 1868048 : 15625] 6.71708479594771 [-203 : 8778 : -27550],
[231 : -725 : 31350]
 92
93 [0, 8088, 0, 1488, 0] 1 Z/6Z [372, 34224] [45783380 : 28012359824 : 125] 11.6324302693721 [-106865 : 144690 : -37646466],
[124285 : -32337349 : 43783194]

 93
94 [0, 8269, 0, 1504, 0] 1 Z/6Z [376, 34968] [1532644 : -275094123 : 64] 12.1699718804879 [-571064 : 1799160 : -79045681],
[1005720 : -44186077 : 139210005]
 94
95 [0, 8452, 0, 1520, 0] 1 Z/6Z [380, 35720] [-560 : -48560 : 343] 5.82566452834231 [102 : 2530 : 6545],
[138 : 357 : 8855]
 95
96 [0, 8637, 0, 1536, 0] 1 Z/6Z [384, 36480] [868332 : -175389076 : 27] 9.26295518168213 [-6816 : 18921 : -1024352],
[11289 : -611168 : 1696583]
 96
97 [0, 8824, 0, 1552, 0] 0 Z/6Z [388, 37248] - - - 97
98 [0, 9013, 0, 1568, 0] 1 Z/6Z [392, 38024] [-7396 : -297388 : 1] 7.55237294648677 [1221 : 2950 : 79550],
[1947 : 52503 : 126850]
 98
99 [0, 9204, 0, 1584, 0] 0 Z/6Z [396, 38808] - - - 99
100 [0, 9397, 0, 1600, 0] 1 Z/6Z [400, 39600] [4624 : -547536 : 1] 4.03959537308976 [-5 : 56 : -595], [8 : -85 : 952]
 100

mwrankによって、Mordell-Weil群のrankが決定できなかったのは、E87だけである。


- En: Y2Z=X3+(n2-6n-3)X2Z+16nXZ2
y2=x3+(n2-6n-3)x2+16nx
 Cn: (x+y+z)(1/x+1/y+1/z)=n -
n [a1,a2,a3,a4,a6]
a1=a3=a6=0,
a2=n2-6n-3,
a4=16n
 		rank
of
En(Q)		 Tors(En(Q)) Tors(En(Q))
の生成元		 En(Q)/Tors(En(Q))
の生成元 [X:Y:Z]		 En(Q)/Tors(En(Q))
の生成元の高さ		 [x:y:z]
(最小解と次の解)		 n
-1 [0, 4, 0, -16, 0] 0 Z/6Z [-4, 8] - - - -1
-2 [0, 13, 0, -32, 0] 0 Z/6Z [-8, 24] - - - -2
-3 [0, 24, 0, -48, 0] 0 Z/6Z [-12, 48] - - - -3
-4 [0, 37, 0, -64, 0] 1 Z/6Z [-16, 80] [64 : -640 : 1] 1.17046415359472 [1 : 2 : -6], [1 : -3 : -6] -4
-5 [0, 52, 0, -80, 0] 0 Z/6Z [-20, 120] - - - -5
-6 [0, 69, 0, -96, 0] 0 Z/6Z [-24, 168] - - - -6
-7 [0, 88, 0, -112, 0] 0 Z/6Z [-28, 224] - - - -7
-8 [0, 109, 0, -128, 0] 0 Z/6Z [-32, 288] - - - -8
-9 [0, 132, 0, -144, 0] 0 Z/6Z [-36, 360] - - - -9
-10 [0, 157, 0, -160, 0] 1 Z/6Z [-40, 440] [-8 : -104 : 1] 1.58004991707949 [1 : 3 : -12], [1 : -4 : -12]
 -10
-11 [0, 184, 0, -176, 0] 1 Z/6Z [-44, 528] [100 : -1680 : 1] 1.86793279055472 [1 : -6 : -10], [3 : 5 : -30]
 -11
-12 [0, 213, 0, -192, 0] 1 Z/6Z [-48, 624] [48 : -496 : 27] 2.9354641578894 [4 : 35 : -84], [-5 : 12 : 105]
 -12
-13 [0, 244, 0, -208, 0] 0 Z/6Z [-52, 728] - - - -13
-14 [0, 277, 0, -224, 0 0 Z/6Z [-56, 840] - - - -14
-15 [0, 312, 0, -240, 0] 0 Z/6Z [-60, 960] - - - -15
-16 [0, 349, 0, -256, 0] 0 Z/6Z [-64, 1088] - - - -16
-17 [0, 388, 0, -272, 0] 0 Z/6Z [-68, 1224] - - - -17
-18 [0, 429, 0, -288, 0] 1 Z/6Z [-72, 1368] [-6 : -171 : 8] 1.0648513558535 [1 : 4 : -20], [1 : -5 : -20]
 -18
-19 [0, 472, 0, -304, 0] 1 Z/6Z [-76, 1520] [484 : -14960 : 1] 2.96077212296937 [3 : -14 : -66], [7 : 33 : -154] -19
-20 [0, 517, 0, -320, 0] 1 Z/6Z [-80, 1680] [144 : -3696 : 1] 2.0018566081256 [1 : -10 : -15], [2 : 3 : -30]
 -20
-21 [0, 564, 0, -336, 0] 0 Z/6Z [-84, 1848] - - - -21
-22 [0, 613, 0, -352, 0] 1 Z/6Z [-88, 2024] [896 : 19648 : 343] 5.48934516696152 [60 : 2516 : -3885],
[-68 : 105 : 4403]
 -22
-23 [0, 664, 0, -368, 0] 0 Z/6Z [-92, 2208] - - - -23
-24 [0, 717, 0, -384, 0] 0 Z/6Z [-96, 2400] - - - -24
-25 [0, 772, 0, -400, 0] 1 Z/6Z [-100, 2600] [2500 : -143000 : 1] 2.71203065247207 [1 : 15 : -40], [3 : -8 : -120]
 -25
-26 [0, 829, 0, -416, 0] 0 Z/6Z [-104, 2808] - - - -26
-27 [0, 888, 0, -432, 0] 0 Z/6Z [-108, 3024] - - - -27
-28 [0, 949, 0, -448, 0] 2 Z/6Z [-112, 3248] [-16 : -496 : 1],
[12288 : -460288 : 27]
 2.13676074496033
4.25392692429032		 [1 : 5 : -30], [1 : -6 : -30],
[21 : -165 : -560], [33 : 112 : -880]
 -28
-29 [0, 1012, 0, -464, 0] 1 Z/6Z [-116, 3480] [242 : -7755 : 8] 4.13817729242365 [12 : -77 : -364],
[33 : 156 : -1001]
 -29
-30 [0, 1077, 0, -480, 0] 1 Z/6Z [-120, 3720] [-756 : -27180 : 343] 5.49123017151537 [156 : -2109 : -3367],
[228 : 364 : -4921]
 -30
-31 [0, 1144, 0, -496, 0] 1 Z/6Z [-124, 3968] [196 : -7168 : 1] 1.95523821419308 [1 : -21 : -15], [5 : 7 : -1057]
 -31
-32 [0, 1213, 0, -512, 0] 1 Z/6Z [-128, 4224] [5780 : -204204 : 125] 4.34954503066335 [15 : -136 : -440],
[51 : 165 : -1496]
 -32
-33 [0, 1284, 0, -528, 0] 0 Z/6Z [-132, 4488] - - - -33
-34 [0, 1357, 0, -544, 0] 0 Z/6Z [-136, 4760] - - - -34
-35 [0, 1432, 0, -560, 0] 0 Z/6Z [-140, 5040] - - - -35
-36 [0, 1509, 0, -576, 0] 1 Z/6Z [-144, 5328] [18928 : 746096 : 343] 6.82571727616647 [770 : -8151 : -24339],
[1430 : 4270 : -45201]
 -36
-37 [0, 1588, 0, -592, 0] 0 Z/6Z [-148, 5624] - - - -37
-38 [0, 1669, 0, -608, 0] 0 Z/6Z [-152, 5928] - - - -38
-39 [0, 1752, 0, -624, 0] 1 Z/6Z [-156, 6240] [64 : -2720 : 1] 2.68362485347205 [1 : -12 : -33], [4 : 11 : -132]
 -39
-40 [0, 1837, 0, -640, 0] 1 Z/6Z [-160, 6560] [32 : 1376 : 1] 2.34388546902757 [1 : 6 : -42], [1 : -7 : -42]
 -40
-41 [0, 1924, 0, -656, 0] 0 Z/6Z [-164, 6888] - - - -41
-42 [0, 2013, 0, -672, 0] 1 Z/6Z [-168, 7224] [-32856 : -927368 : 27] 7.66387594743436 [2460 : 11859 : -101639],
[-3540 : 30340: 146261]
 -42
-43 [0, 2104, 0, -688, 0] 1 Z/6Z [-172, 7568] [1887876 : 129335536 : 729] 8.45826732910586 [5166 : -23715 : -272510],
[11439 : 131446 : -603415]
 -43
-44 [0, 2197, 0, -704, 0] 1 Z/6Z [-176, 7920] [256 : -12672 : 1] 2.55634880034839 [1 : -21 : -28], [3 : 4 : -84]
 -44
-45 [0, 2292, 0, -720, 0] 0 Z/6Z [-180, 8280] - - - -45
-46 [0, 2389, 0, -736, 0] 0 Z/6Z [-184, 8648] - - - -46
-47 [0, 2488, 0, -752, 0] 0 Z/6Z [-188, 9024] - - - -47
-48 [0, 2589, 0, -768, 0] 1 Z/6Z [-192, 9408] [704 : -23744 : 1331] 3.83101824860721 [12 : 65 : -572], [-15 : 132 : 715] -48
-49 [0, 2692, 0, -784, 0] 0 Z/6Z [-196, 9800] - - - -49
-50 [0, 2797, 0, -800, 0] 1 Z/6Z [-200, 10200] [89100 : -4758300 : 1331] 5.21430844250903 [209, -2736, 63], [-1008, -3344, 77]
 -50
-51 [0, 2904, 0, -816, 0] 1 Z/6Z [-204, 10608] [7396 : -750608 : 1] 6.65642974042261 [266 : -825 : -19866],
[475 : 11438 : -35475]		 -51
-52 [0, 3013, 0, -832, 0] 2 Z/6Z [-208, 11024] [-1 : -62 : 1],
[67280 : -4008496 : 125]
 3.68629321398363
5.2029233790362		 [5 : 18 : -230], [9 : -115 : -414],
[45 : -770 : -1827], [110 : 261 : -4466]
 -52
-53 [0, 3124, 0, -848, 0] 0 Z/6Z [-212, 11448] - - - -53
-54 [0, 3237, 0, -864, 0] 1 Z/6Z [-216, 11880] [36 : -2052 : 1] 1.97239038546929 [1 : 7 : -56], [1 : -8 : -56]
 -54
-55 [0, 3352, 0, -880, 0] 0 Z/6Z [-220, 12320] - - - -55
-56 [0, 3469, 0, -896, 0] 1 Z/6Z [-224, 12768] [2268 : -131076 : 343] 6.77022839261274 [175 : -288 : -25312],
[225 : 19775 : -32544]		 -56
-57 [0, 3588, 0, -912, 0] 0 Z/6Z [-228, 13224] - - - -57
-58 [0, 3709, 0, -928, 0] 1 Z/6Z [-232, 13688] [-696332552 : -146434734232 : 30400540561] 15.0714682382382 [111600756 : -198642325 : -14941163300],
[27728847 : -3712351475 : 2085664428]
 -58
-59 [0, 3832, 0, -944, 0] 2 Z/6Z [-236, 14160] [1444 : -104880 : 1],
[3364 : -285360 : 1]
 3.41654720050278
4.14367873735443		 [2 : -21 : -114], [7 : 38 : -399],
[6 : -35 : -406], [15 : 174 : -1015]		 -59
-60 [0, 3957, 0, -960, 0] 0-2? Z/6Z [-240, 14640] ??? ??? ??? -60
-61 [0, 4084, 0, -976, 0] 1 Z/6Z [-244, 15128] [28938650 : 2190541865 : 17576] 14.1664601050212 [19535516 : -196404156 : -1171388655],
[66454388 : 396345565 : -3984738165]
 -61
-62 [0, 4213, 0, -992, 0] 0 Z/6Z [-248, 15624] - - - -62
-63 [0, 4344, 0, -1008, 0] 0 Z/6Z [-252, 16128] - - - -63
-64 [0, 4477, 0, -1024, 0] 0 Z/6Z [-256, 16640] - - - -64
-65 [0, 4612, 0, -1040, 0] 0 Z/6Z [-260, 17160] - - - -65
-66 [0, 4749, 0, -1056, 0] 0 Z/6Z [-264, 17688] - - - -66
-67 [0, 4888, 0, -1072, 0] 1 Z/6Z [-268, 18224] [16428 : -1217744 : 27] 4.89590624570529 [15 : -348 : -740],
[87 : 185 : -4292]
 -67
-68 [0, 5029, 0, -1088, 0] 0 Z/6Z [-272, 18768] - - - -68
-69 [0, 5172, 0, -1104, 0] 1 Z/6Z [-276, 19320] [3468 : -252280 : 27] 3.84192826067535 [3 : -68 : -156], [17 : 39 : -884]
 -69
-70 [0, 5317, 0, -1120, 0] 2 Z/6Z [-280, 19880] [10 : 665 : 8],
[648 : 50040 : 1]
 4.45890988114303
5.22888230822343
 [9 : 275 : -900], [-11 : 36 : 1100],
[28 : -667 : -1449], [116 : 252 : -6003]
 -70
-71 [0, 5464, 0, -1136, 0] 0 Z/6Z [-284, 20448] - - - -71
-72 [0, 5613, 0, -1152, 0] 1 Z/6Z [-288, 21024] [-84 : -8532 : 343] 4.51978167651286 [52 : 700 : -4277],
[100 : -611 : -8225]		 -72
-73 [0, 5764, 0, -1168, 0] 0-2? Z/6Z [-292, 21608] ??? ??? ??? -73
-74 [0, 5917, 0, -1184, 0] 0 Z/6Z [-296, 22200] - - - -74
-75 [0, 6072, 0, -1200, 0] 0 Z/6Z [-300, 22800] - - - -75
-76 [0, 6229, 0, -1216, 0] 2 Z/6Z [-304, 23408] [400 : -32560 : 1],
[18845008 : -2293095056 : 2197]
 2.97261636156562
12.1982601758386		 [1 : -36 : -45], [4 : 5 : -180],
[-5053373 : 1136460 : -109032735],
[1219556 : 26313420 : -117005021]
 		-76
-77 [0, 6388, 0, -1232, 0] 0 Z/6Z [-308, 24024] - - - -77
-78 [0, 6549, 0, -1248, 0] 1 Z/6Z [-312, 24648] [-12990877176 : -1063298090184 : 1647212741] 13.5095082695496 [9762340 : -264208596 : -551048889],
[35518805 : 17030020 : -961283217]		 -78
-79 [0, 6712, 0, -1264, 0] 0 Z/6Z [-316, 25280] - - - -79
-80 [0, 6877, 0, -1280, 0] 0 Z/6Z [-320, 25920] - - - -80
-81 [0, 7044, 0, -1296, 0] 0 Z/6Z [-324, 26568] - - - -81
-82 [0, 7213, 0, -1312, 0] 0 Z/6Z [-328, 27224] - - - -82
-83 [0, 7384, 0, -1328, 0] 0 Z/6Z [-332, 27888] - - - -83
-84 [0, 7557, 0, -1344, 0] 1 Z/6Z [-336, 28560] [236992 : -21520640 : 343] 5.93263379723439 [77 : -2331 : -4554],
[259 : 506 : -15318]
 84
-85 [0, 7732, 0, -1360, 0] 1 Z/6Z [-340, 29240] [-60 : -5480 : 27] 3.67657946188448 [4 : -133 : -228], [7 : 12 : -399]
 -85
-86 [0, 7909, 0, -1376, 0] 1 Z/6Z [-344, 29928] [96160712 : 42542055789 : 512] 13.3309718051126 [2214640 : 392828435 : -585149859]
[1812880 : -2700432 : -478997253]
 -86
-87 [0, 8088, 0, -1392, 0] 0 Z/6Z [-348, 30624] - - - -87
-88 [0, 8269, 0, -1408, 0] 1 Z/6Z [-352, 31328] [-32 : -2912 : 1] 2.8159809781317 [1 : 9 : -90], [1 : -10 : -90] -88
-89 [0, 8452, 0, -1424, 0] 1 Z/6Z [-356, 32040] [100 : -9240 : 1] 2.1174709481781 [1 : -20 : -76], [5 : 19 : -380] -89
-90 [0, 8637, 0, -1440, 0] 0 Z/6Z [-360, 32760] - - - -90
-91 [0, 8824, 0, -1456, 0] 1 Z/6Z [-364, 33488] [-7 : -665 : 1] 2.83301362321699 [3 : -102 : -187], [6 : 11 : -374] -91
-92 [0, 9013, 0, -1472, 0] 1 Z/6Z [-368, 34224] [66660617744 : -10537038516432 : 4173281] 14.8106726248905 [45830904 : 1457527240 : -5623815849],
[46129720 : -177989847 : -5660482945]
 -92
-93 [0, 9204, 0, -1488, 0] 0 Z/6Z [-372, 34968] - - - -93
-94 [0, 9397, 0, -1504, 0] 0 Z/6Z [-376, 35720] - - - -94
-95 [0, 9592, 0, -1520, 0] 1 Z/6Z [-380, 36480] [484 : -48576 : 1] 2.85866273758706 [1 : -45 : -55], [9 : 11 : -495]
 -95
-96 [0, 9789, 0, -1536, 0] 1 Z/6Z [-384, 37248] [35208152148 : 3448011074516 : 4716275733] 18.1250498623291 [3003037440 : -5581682432 : -627978474123],
[4102539520 : 461564508405 : -857900229109]
 -96
-97 [0, 9988, 0, -1552, 0] 0 Z/6Z [-388, 38024] - - - -97
-98 [0, 10189, 0, -1568, 0] 0 Z/6Z [-392, 38808] - - - -98
-99 [0, 10392, 0, -1584, 0] 0 Z/6Z [-396, 39600] - - - -99
-100 [0, 10597, 0, -1600, 0] 1 Z/6Z [-400, 40400] [10543167251152 : -1079242332937648 : 861985991413] 18.0515117035529 [4450012553 : 219887106322 : -663397965750],
[5089624981 : -15355365375 : -758749962750]
 -100

mwrankによって、Mordell-Weil群のrankが決定できなかったのは、E-60, E-73だけである。