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Rational Points on Ellptic Curves: y^2=x^3+n (n \in [1..1000])


[2021.08.08]y^2=x^3+n (n \in [1..1000])の有理点



■n=1,2,...,1000に対して、
     En: y2 = x3+n
の有理点を求める。

■n=1,2,...,1000に対して、楕円曲線Enのねじれ点群En(Q)torsを計算すると、
     En(Q)tors=Z/2Z if n=8,27,125,216,343,512,1000
                 =Z/3Z if n=4,9,16,25,36,49,81,100,121,144,169,196,225,256,289,324,361,400,441,484,529,625,676,784,841,900,961
                 =Z/6Z if n=1,64,729
                 ={O} if nが平方数,立方数でない
となる。

[pari/gpによる計算]
gp> ec(n)=ellinit([0,0,0,0,n]);
gp> tg(n1,n2)=local(e,g,o);for(i=n1,n2,e=ec(i);g=elltors(e);o=g[1];if(o>1,print("E_",i,"(Q)_{tors}=",g)));
gp> tg(1,1000)
E_1(Q)_{tors}=[6, [6], [[2, 3]]]
E_4(Q)_{tors}=[3, [3], [[0, 2]]]
E_8(Q)_{tors}=[2, [2], [[-2, 0]]]
E_9(Q)_{tors}=[3, [3], [[0, 3]]]
E_16(Q)_{tors}=[3, [3], [[0, 4]]]
E_25(Q)_{tors}=[3, [3], [[0, 5]]]
E_27(Q)_{tors}=[2, [2], [[-3, 0]]]
E_36(Q)_{tors}=[3, [3], [[0, 6]]]
E_49(Q)_{tors}=[3, [3], [[0, 7]]]
E_64(Q)_{tors}=[6, [6], [[8, 24]]]
E_81(Q)_{tors}=[3, [3], [[0, 9]]]
E_100(Q)_{tors}=[3, [3], [[0, 10]]]
E_121(Q)_{tors}=[3, [3], [[0, 11]]]
E_125(Q)_{tors}=[2, [2], [[-5, 0]]]
E_144(Q)_{tors}=[3, [3], [[0, 12]]]
E_169(Q)_{tors}=[3, [3], [[0, 13]]]
E_196(Q)_{tors}=[3, [3], [[0, 14]]]
E_216(Q)_{tors}=[2, [2], [[-6, 0]]]
E_225(Q)_{tors}=[3, [3], [[0, 15]]]
E_256(Q)_{tors}=[3, [3], [[0, 16]]]
E_289(Q)_{tors}=[3, [3], [[0, 17]]]
E_324(Q)_{tors}=[3, [3], [[0, 18]]]
E_343(Q)_{tors}=[2, [2], [[-7, 0]]]
E_361(Q)_{tors}=[3, [3], [[0, 19]]]
E_400(Q)_{tors}=[3, [3], [[0, 20]]]
E_441(Q)_{tors}=[3, [3], [[0, 21]]]
E_484(Q)_{tors}=[3, [3], [[0, 22]]]
E_512(Q)_{tors}=[2, [2], [[-8, 0]]]
E_529(Q)_{tors}=[3, [3], [[0, 23]]]
E_576(Q)_{tors}=[3, [3], [[0, 24]]]
E_625(Q)_{tors}=[3, [3], [[0, 25]]]
E_676(Q)_{tors}=[3, [3], [[0, 26]]]
E_729(Q)_{tors}=[6, [6], [[18, 81]]]
E_784(Q)_{tors}=[3, [3], [[0, 28]]]
E_841(Q)_{tors}=[3, [3], [[0, 29]]]
E_900(Q)_{tors}=[3, [3], [[0, 30]]]
E_961(Q)_{tors}=[3, [3], [[0, 31]]]
E_1000(Q)_{tors}=[2, [2], [[-10, 0]]]
time = 8 ms.

■nが6乗因子kをもつとき、つまり、n=k6・mのとき、
双有理変換φ: En → Em;      (x,y) → (X,Y)
       X=x/{k2}
       Y=y/{k3}
[
逆変換φ-1: Em → En;      (X,Y) → (x,y)
       x={k2}・X
       y={k3}・Y
]
により、
2つの楕円曲線EnとEmQ-isomorphicになる。

よって、nは6乗因子を含まないものだけを考えれば十分であるが、見易さのために、対象範囲内の全てのnについて、楕円曲線Enの有理点の一覧表をまとめる。

■楕円曲線EnのMordell-Weil群のrankと生成元を求めると以下のようになる。

- E~n: Y2Z=X3+nZ3
En: y2=x3+n
-
n [a1,a2,a3,a4,a6]
j(En)
Conductor of En
En(Q)tors
En(Q)torsの生成元
rank(En(Q)) En(Q)/En(Q)tors
の生成元 [X:Y:Z]
En(Q)/En(Q)tors
の生成元の高さ
En(Z)
Enの整点[x,y]
(where y >= 0)
n
1 [0, 0, 0, 0, 1]
0
36
Z/6Z
[2, 3]
0 - - [-1, 0], [0, 1], [2, 3] 1
2 [0, 0, 0, 0, 2]
0
1728
{O}
-
1 [-1 : 1 : 1] 0.75457690318122
[-1, 1] 2
3 [0, 0, 0, 0, 3]
0
3888
{O}
-
1 [1 : 2 : 1] 0.92191912512801
[1, 2] 3
4 [0, 0, 0, 0, 4]
0
108
Z/3Z
[0, 2]
0 - - [0, 2] 4
5 [0, 0, 0, 0, 5]
0
2700
{O}
-
1 [-1 : 2 : 1] 1.03610657165260
[-1, 2] 5
6 [0, 0, 0, 0, 6]
0
15552
{O}
-
0 - - - 6
7 [0, 0, 0, 0, 7]
0
21168
{O}
-
0 - - - 7
8 [0, 0, 0, 0, 8]
0
576
Z/2Z
[-2, 0]
1 [1 : 3 : 1] 0.65323467754297
[-2, 0], [1, 3], [2, 4], [46, 312] 8
9 [0, 0, 0, 0, 9]
0
972
Z/3Z
[0, 3]
1 [-2 : 1 : 1] 0.81469544056682
[-2, 1], [0, 3], [3, 6], [6, 15], [40, 253] 9
10 [0, 0, 0, 0, 10]
0
14400
{O}
-
1 [-1 : 3 : 1] 0.70567545070452
[-1, 3] 10
11 [0, 0, 0, 0, 11]
0
52272
{O}
-
1 [-14 : 19 : 8] 2.70829782852713
- 11
12 [0, 0, 0, 0, 12]
0
3888
{O}
-
1 [-2 : 2 : 1] 0.66847573678882
[-2, 2], [13, 47] 12
13 [0, 0, 0, 0, 13]
0
18252
{O}
-
0 - - - 13
14 [0, 0, 0, 0, 14]
0
84672
{O}
-
0 - - - 14
15 [0, 0, 0, 0, 15]
0
97200
{O}
-
2 [1 : 4 : 1],
[109 : 1138 : 1]
1.39508335033481,
4.70526471938057
[1, 4], [109, 1138] 15
16 [0, 0, 0, 0, 16]
0
27
Z/3Z
[0, 4]
0 - - [0, 4] 16
17 [0, 0, 0, 0, 17]
0
10404
{O}
-
2 [-2 : 3 : 1],
[-1 : 4 : 1]
0.45461686518421,
1.42510431540567
[-2, 3], [-1, 4], [2, 5], [4, 9], [8, 23], [43, 282], [52, 375], [5234, 378661] 17
18 [0, 0, 0, 0, 18]
0
15552
{O}
-
1 [7 : 19 : 1] 2.18134812974833
[7, 19] 18
19 [0, 0, 0, 0, 19]
0
51984
{O}
-
1 [5 : 12 : 1] 1.40089769000757
[5, 12] 19
20 [0, 0, 0, 0, 20]
0
2700
{O}
-
0 - - - 20
21 [0, 0, 0, 0, 21]
0
47628
{O}
-
0 - - - 21
22 [0, 0, 0, 0, 22]
0
209088
{O}
-
1 [3 : 7 : 1] 1.70861016239140
[3, 7] 22
23 [0, 0, 0, 0, 23]
0
228528
{O}
-
0 - - - 23
24 [0, 0, 0, 0, 24]
0
15552
{O}
-
2 [-2 : 4 : 1],
[1 : 5 : 1]
0.87788972028431,
1.54316286139803
[-2, 4], [1, 5], [10, 32], [8158, 736844] 24
25 [0, 0, 0, 0, 25]
0
2700
Z/3Z
[0, 5]
0 - - [0, 5] 25
26 [0, 0, 0, 0, 26]
0
97344
{O}
-
1 [-1 : 5 : 1] 1.56257198911012
[-1, 5] 26
27 [0, 0, 0, 0, 27]
0
144
Z/2Z
[-3, 0]
0 - - [-3, 0] 27
28 [0, 0, 0, 0, 28]
0
7056
{O}
-
1 [2 : -6 : 1] 0.41462415202842
[-3, 1], [2, 6] 28
29 [0, 0, 0, 0, 29]
0
90828
{O}
-
0 - - - 29
30 [0, 0, 0, 0, 30]
0
388800
{O}
-
1 [19 : 83 : 1] 3.04553796383648
[19, 83] 30
31 [0, 0, 0, 0, 31]
0
415152
{O}
-
1 [-3 : 2 : 1] 1.68822040963350
[-3, 2] 31
32 [0, 0, 0, 0, 32]
0
1728
{O}
-
0 - - - 32
33 [0, 0, 0, 0, 33]
0
117612
{O}
-
1 [-2 : 5 : 1] 1.20722936899394
[-2, 5] 33
34 [0, 0, 0, 0, 34]
0
499392
{O}
-
0 - - - 34
35 [0, 0, 0, 0, 35]
0
176400
{O}
-
1 [1 : 6 : 1] 1.11453406621689
[1, 6] 35
36 [0, 0, 0, 0, 36]
0
972
Z/3Z
[0, 6]
1 [-3 : 3 : 1] 0.99978932437879
[-3, 3], [0, 6], [4, 10], [12, 42] 36
37 [0, 0, 0, 0, 37]
0
49284
{O}
-
2 [-1 : 6 : 1],
[3 : 8 : 1]
1.12809941041083,
1.82280884124843
[-1, 6], [3, 8], [243, 3788] 37
38 [0, 0, 0, 0, 38]
0
623808
{O}
-
1 [11 : 37 : 1] 2.58887689334079
[11, 37] 38
39 [0, 0, 0, 0, 39]
0
657072
{O}
-
1 [434 : 3197 : 8] 5.41837396831351
- 39
40 [0, 0, 0, 0, 40]
0
43200
{O}
-
1 [6 : 16 : 1] 1.46375595197452
[6, 16] 40
41 [0, 0, 0, 0, 41]
0
181548
{O}
-
1 [2 : 7 : 1] 1.30252922330015
[2, 7] 41
42 [0, 0, 0, 0, 42]
0
762048
{O}
-
0 - - - 42
43 [0, 0, 0, 0, 43]
0
798768
{O}
-
2 [-3 : 4 : 1],
[18 : 59 : 8]
1.78491369500229,
3.18209445444608
[-3, 4] 43
44 [0, 0, 0, 0, 44]
0
17424
{O}
-
1 [-2 : 6 : 1] 0.51663312954498
[-2, 6], [5, 13] 44
45 [0, 0, 0, 0, 45]
0
24300
{O}
-
0 - - - 45
46 [0, 0, 0, 0, 46]
0
304704
{O}
-
1 [-14 : 51 : 8] 2.60297991556194
- 46
47 [0, 0, 0, 0, 47]
0
954288
{O}
-
1 [34 : 89 : 8] 3.38676609979565
- 47
48 [0, 0, 0, 0, 48]
0
3888
{O}
-
1 [1 : 7 : 1] 1.76580333576025
[1, 7] 48
49 [0, 0, 0, 0, 49]
0
5292
Z/3Z
[0, 7]
0 - - [0, 7] 49
50 [0, 0, 0, 0, 50]
0
43200
{O}
-
1 [-1 : 7 : 1] 1.77581329740530
[-1, 7] 50
51 [0, 0, 0, 0, 51]
0
1123632
{O}
-
0 - - - 51
52 [0, 0, 0, 0, 52]
0
18252
{O}
-
1 [-3 : 5 : 1] 1.84182497128130
[-3, 5] 52
53 [0, 0, 0, 0, 53]
0
101124
{O}
-
0 - - - 53
54 [0, 0, 0, 0, 54]
0
1728
{O}
-
1 [3 : 9 : 1] 0.44985894522670
[3, 9] 54
55 [0, 0, 0, 0, 55]
0
435600
{O}
-
1 [9 : 28 : 1] 2.46418376249214
[9, 28] 55
56 [0, 0, 0, 0, 56]
0
84672
{O}
-
1 [2 : 8 : 1] 1.16270026077160
[2, 8] 56
57 [0, 0, 0, 0, 57]
0
350892
{O}
-
2 [-2 : 7 : 1],
[4 : 11 : 1]
1.37845694997101,
1.55269184783923
[-2, 7], [4, 11], [7, 20] 57
58 [0, 0, 0, 0, 58]
0
1453248
{O}
-
1 [1446 : 4087 : 216] 5.85547955296906
- 58
59 [0, 0, 0, 0, 59]
0
1503792
{O}
-
0 - - - 59
60 [0, 0, 0, 0, 60]
0
97200
{O}
-
0 - - - 60
61 [0, 0, 0, 0, 61]
0
401868
{O}
-
1 [-30 : 23 : 8] 3.29984249460462
- 61
62 [0, 0, 0, 0, 62]
0
553536
{O}
-
1 [2 : 63 : 8] 2.67543221280610
- 62
63 [0, 0, 0, 0, 63]
0
190512
{O}
-
2 [-3 : 6 : 1],
[1 : 8 : 1]
1.16740456186110,
1.85413136382490
[-3, 6], [1, 8] 63
64 [0, 0, 0, 0, 64]
0
36
Z/6Z
[8, 24]
0 - - [-4, 0], [0, 8], [8, 24] 64
65 [0, 0, 0, 0, 65]
0
456300
{O}
-
2 [-4 : 1 : 1],
[-1 : 8 : 1]
1.47800929931260,
1.86181926031200
[-4, 1], [-1, 8], [14, 53], [584, 14113] 65
66 [0, 0, 0, 0, 66]
0
1881792
{O}
-
1 [2 : 65 : 8] 3.24555216402642
- 66
67 [0, 0, 0, 0, 67]
0
1939248
{O}
-
1 [294 : 1801 : 216] 5.46738523212918
- 67
68 [0, 0, 0, 0, 68]
0
31212
{O}
-
1 [-4 : 2 : 1] 1.02905048526023
[-4, 2], [152, 1874] 68
69 [0, 0, 0, 0, 69]
0
514188
{O}
-
1 [-15 : 224 : 27] 4.07330690108667
- 69
70 [0, 0, 0, 0, 70]
0
2116800
{O}
-
0 - - - 70
71 [0, 0, 0, 0, 71]
0
725904
{O}
-
1 [5 : 14 : 1] 2.14892978254364
[5, 14] 71
72 [0, 0, 0, 0, 72]
0
15552
{O}
-
1 [-2 : 8 : 1] 1.22144449790466
[-2, 8] 72
73 [0, 0, 0, 0, 73]
0
191844
{O}
-
2 [2 : 9 : 1],
[-4 : 3 : 1]
0.92394317160039,
0.96257318809237
[-4, 3], [2, 9], [3, 10], [6, 17], [72, 611], [356, 6717] 73
74 [0, 0, 0, 0, 74]
0
2365632
{O}
-
1 [21 : 233 : 27] 4.09980835941542
- 74
75 [0, 0, 0, 0, 75]
0
97200
{O}
-
0 - - - 75
76 [0, 0, 0, 0, 76]
0
155952
{O}
-
1 [-3 : 7 : 1] 1.95698235940070
[-3, 7] 76
77 [0, 0, 0, 0, 77]
0
640332
{O}
-
1 [-305 : 988 : 125] 5.16524084733908
- 77
78 [0, 0, 0, 0, 78]
0
2628288
{O}
-
0 - - - 78
79 [0, 0, 0, 0, 79]
0
2696112
{O}
-
2 [-30 : 41 : 8],
[45 : 302 : 1]
3.37614418975677,
3.86541416561391
[45, 302] 79
80 [0, 0, 0, 0, 80]
0
225
{O}
-
1 [4 : 12 : 1] 0.15364003501329
[-4, 4], [1, 9], [4, 12], [44, 292] 80
81 [0, 0, 0, 0, 81]
0
972
Z/3Z
[0, 9]
0 - - [0, 9] 81
82 [0, 0, 0, 0, 82]
0
968256
{O}
-
1 [-1 : 9 : 1] 1.38884891673285
[-1, 9] 82
83 [0, 0, 0, 0, 83]
0
2976048
{O}
-
1 [15846 : 135737 : 216] 7.91550908242369
- 83
84 [0, 0, 0, 0, 84]
0
47628
{O}
-
0 - - - 84
85 [0, 0, 0, 0, 85]
0
780300
{O}
-
0 - - - 85
86 [0, 0, 0, 0, 86]
0
3195072
{O}
-
0 - - - 86
87 [0, 0, 0, 0, 87]
0
3269808
{O}
-
0 - - - 87
88 [0, 0, 0, 0, 88]
0
209088
{O}
-
0 - - - 88
89 [0, 0, 0, 0, 89]
0
285156
{O}
-
2 [-2 : 9 : 1],
[-4 : 5 : 1]
0.97074612242338,
1.57021917516742
[-4, 5], [-2, 9], [10, 33], [55, 408] 89
90 [0, 0, 0, 0, 90]
0
388800
{O}
-
0 - - - 90
91 [0, 0, 0, 0, 91]
0
1192464
{O}
-
1 [-3 : 8 : 1] 2.01229376795131
[-3, 8] 91
92 [0, 0, 0, 0, 92]
0
228528
{O}
-
1 [2 : 10 : 1] 1.31279848407488
[2, 10] 92
93 [0, 0, 0, 0, 93]
0
934092
{O}
-
0 - - - 93
94 [0, 0, 0, 0, 94]
0
3817152
{O}
-
1 [3 : 11 : 1] 2.06108190143438
[3, 11] 94
95 [0, 0, 0, 0, 95]
0
3898800
{O}
-
0 - - - 95
96 [0, 0, 0, 0, 96]
0
15552
{O}
-
0 - - - 96
97 [0, 0, 0, 0, 97]
0
1016172
{O}
-
1 [18 : 77 : 1] 2.5870983635244
[18, 77] 97
98 [0, 0, 0, 0, 98]
0
28224
{O}
-
1 [7 : 21 : 1] 0.52431451180667
[7, 21] 98
99 [0, 0, 0, 0, 99]
0
470448
{O}
-
1 [1 : 10 : 1] 2.00181012583338
[1, 10] 99
100 [0, 0, 0, 0, 100]
0
900
Z/3Z
[0, 10]
1 [-4 : 6 : 1] 0.59341039923109
[-4, 6], [0, 10], [5, 15], [20, 90], [24, 118], [2660, 137190] 100
101 [0, 0, 0, 0, 101]
0
1101708
{O}
-
2 [-1 : 10 : 1],
[95 : 926 : 1]
2.00675079147932,
4.58404107330309
[-1, 10], [95, 926] 101
102 [0, 0, 0, 0, 102]
0
4494528
{O}
-
1 [17549 : 124675 : 12167] 8.29171665229353
- 102
103 [0, 0, 0, 0, 103]
0
4583088
{O}
-
1 [39 : 278 : 27] 4.22114032161449
- 103
104 [0, 0, 0, 0, 104]
0
292032
{O}
-
0 - - - 104
105 [0, 0, 0, 0, 105]
0
1190700
{O}
-
1 [4 : 13 : 1] 1.69125237968437
[4, 13] 105
106 [0, 0, 0, 0, 106]
0
4853952
{O}
-
2 [15 : 59 : 1],
[-51 : 269 : 27]
2.90537583953471,
4.23327675453680
[15, 59] 106
107 [0, 0, 0, 0, 107]
0
1648656
{O}
-
1 [13 : 48 : 1] 2.24509336669847
[13, 48] 107
108 [0, 0, 0, 0, 108]
0
432
{O}
-
1 [6 : 18 : 1] 0.15010689521329
[-3, 9], [-2, 10], [6, 18], [366, 7002] 108
109 [0, 0, 0, 0, 109]
0
427716
{O}
-
0 - - - 109
110 [0, 0, 0, 0, 110]
0
5227200
{O}
-
1 [649 : 13967 : 1331] 6.82639516275280
- 110
111 [0, 0, 0, 0, 111]
0
5322672
{O}
-
1 [-2365 : 13664 : 1331] 6.84473391694478
- 111
112 [0, 0, 0, 0, 112]
0
21168
{O}
-
1 [9 : 29 : 1] 2.53926798575395
[9, 29] 112
113 [0, 0, 0, 0, 113]
0
1379052
{O}
-
3 [2 : 11 : 1],
[-4 : 7 : 1],
[8 : 25 : 1]
1.60741760647698,
1.64131903293877,
2.00561989408730
[-4, 7], [2, 11], [8, 25], [11, 38], [26, 133], [422, 8669] 113
114 [0, 0, 0, 0, 114]
0
5614272
{O}
-
0 - - - 114
115 [0, 0, 0, 0, 115]
0
5713200
{O}
-
0 - - - 115
116 [0, 0, 0, 0, 116]
0
30276
{O}
-
0 - - - 116
117 [0, 0, 0, 0, 117]
0
164268
{O}
-
1 [3 : 12 : 1] 1.38976930536409
[3, 12] 117
118 [0, 0, 0, 0, 118]
0
2005056
{O}
-
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1 [2800697003811477 : 20221430657567426 : 54048739866129] 25.1409910689030
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1 [31127604486 : 173532262897 : 5213714904] 17.7182909350128
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1 [52878 : 246625 : 2744] 8.55832460712204
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1 [6768555137 : 60471500159 : 1939096223] 17.0206786205616
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1 [-551460 : -6538627 : 216000] 10.9393032687183
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1 [9905 : 88254 : 125] 7.11817629455457
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1 [38672 : 172937 : 4096] 8.46805112614903
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1 [-6 : 27 : 1] 0.67621077516461
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1 [2428951 : -11608532 : 300763] 11.2918965627959
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1 [-5 : 29 : 1] 2.78253961221747
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1 [69 : 574 : 1] 4.32260340886497
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1 [-2 : 31 : 1] 2.29769190799500
[-2, 31] 969
970 [0, 0, 0, 0, 970]
0
406468800
{O}
-
0 - - - 970
971 [0, 0, 0, 0, 971]
0
135769104
{O}
-
1 [265 : 4314 : 1] 5.05341221290617
[265, 4314] 971
972 [0, 0, 0, 0, 972]
0
3888
{O}
-
0 - - - 972
973 [0, 0, 0, 0, 973]
0
34082244
{O}
-
2 [11 : 48 : 1],
[18 : 251 : 8]
2.43662102967300,
4.15072954557906
[11, 48] 973
974 [0, 0, 0, 0, 974]
0
409828032
{O}
-
1 [16915 : 156509 : 4913] 8.44412821491303
- 974
975 [0, 0, 0, 0, 975]
0
16426800
{O}
-
0 - - - 975
976 [0, 0, 0, 0, 976]
0
100467
{O}
-
1 [12 : 52 : 1] 1.63706284310101
[12, 52] 976
977 [0, 0, 0, 0, 977]
0
103089132
{O}
-
1 [14 : 61 : 1] 2.63587968444244
[14, 61] 977
978 [0, 0, 0, 0, 978]
0
413201088
{O}
-
0 - - - 978
979 [0, 0, 0, 0, 979]
0
414046512
{O}
-
2 [75 : 854 : 27],
[8100 : 91147 : 64]
4.96861156793386,
7.66163762236480
- 979
980 [0, 0, 0, 0, 980]
0
44100
{O}
-
0 - - - 980
981 [0, 0, 0, 0, 981]
0
11548332
{O}
-
1 [15 : 66 : 1] 2.40318647295838
[15, 66] 981
982 [0, 0, 0, 0, 982]
0
138862656
{O}
-
1 [82 : 363 : 8] 3.79773998915812
- 982
983 [0, 0, 0, 0, 983]
0
417436848
{O}
-
1 [-165013 : 849466 : 29791] 9.66154590297228
- 983
984 [0, 0, 0, 0, 984]
0
26142912
{O}
-
1 [98 : 425 : 8] 4.42041658264977
- 984
985 [0, 0, 0, 0, 985]
0
104784300
{O}
-
2 [-9 : 16 : 1],
[1011 : 32146 : 1]
2.83473453761714,
6.92474805808403
[-9, 16], [1011, 32146] 985
986 [0, 0, 0, 0, 986]
0
419988672
{O}
-
0 - - - 986
987 [0, 0, 0, 0, 987]
0
420841008
{O}
-
0 - - - 987
988 [0, 0, 0, 0, 988]
0
26355888
{O}
-
2 [-3 : 31 : 1],
[42 : 274 : 1]
2.77240595648653,
3.19186092426420
[-3, 31], [42, 274] 988
989 [0, 0, 0, 0, 989]
0
35212356
{O}
-
0 - - - 989
990 [0, 0, 0, 0, 990]
0
47044800
{O}
-
1 [33032 : -265807 : 512] 8.42124146917990
- 990
991 [0, 0, 0, 0, 991]
0
141419664
{O}
-
1 [-20473565 : 261389892 : 8365427] 12.8468360725565
- 991
992 [0, 0, 0, 0, 992]
0
1660608
{O}
-
1 [291 : 1279 : 27] 5.17898969351681
- 992
993 [0, 0, 0, 0, 993]
0
106493292
{O}
-
0 - - - 993
994 [0, 0, 0, 0, 994]
0
426831552
{O}
-
1 [25163580 : 136100483 : 3796416] 12.9480954264623
- 994
995 [0, 0, 0, 0, 995]
0
427690800
{O}
-
0 - - - 995
996 [0, 0, 0, 0, 996]
0
6696108
{O}
-
1 [-8 : 22 : 1] 1.90147192285593
[-8, 22] 996
997 [0, 0, 0, 0, 997]
0
107352972
{O}
-
3 [3 : 32 : 1],
[-78 : 67 : 8],
[114 : 499 : 8]
2.77969044355994,
4.23410888836960,
4.49674910361627
[3, 32] 997
998 [0, 0, 0, 0, 998]
0
143424576
{O}
-
1 [490 : 31593 : 1000] 6.82025939194805
- 998
999 [0, 0, 0, 0, 999]
0
197136
{O}
-
0 - - - 999
1000 [0, 0, 0, 0, 1000]
0
14400
Z/2Z
[-10, 0]
1 [-6 : 28 : 1] 2.11069738234011
[-10, 0], [-6, 28], [65, 525] 1000


[参考文献]


Last Update: 2021.09.26
H.Nakao

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