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Rational Points on Elliptic Curves:y^2=x^4+n, y^2=x^3-4nx(n \in [1..100])

[2001.06.24]y^2=x^4+n,y^2=x^3-4nx (n \in [1..100])の有理点

■楕円曲線y2=x4+n (n \in [1..100])の有理点を求める。
100以下の自然数nに対して、楕円曲線
     En: y2=x4+n ----- (1)
の有理点を求める。
特に、nが4乗数d4の場合、(1)の整点は、(0,±d2)のみである。
双有理変換\phi:(x,y)--->(X,Y)
     X=1/{2(y-x2)} --------- (2)
     Y=x/{y-x2} --------- (3)
および、双有理変換\phi-1:(X,Y)--->(x,y)
     x=Y/{2X} --------- (4)
     y={2X+Y2}/{4X2} --------- (5)
を考える。楕円曲線Enを双有理変換\phiで写すと、
     Y2=4nX3-X --------- (6)
となる。さらに、(6)を双有理変換(4nX,(4n)2Y)-->(x,y)で写すと、楕円曲線
    C-4n:   y2=x3-4nx --------- (7)
に有理同型となる。

楕円曲線C-4nの判別式Δは、
     Δ=-16*4*(-4n)3=212n3≠ 0
なので、C-4nは非特異である。
また、その有理点のなす群C-4n(Q)は、nが平方数d2なら位数2のねじれ点(0,0),(±2d,0)を持ち、nが非平方数なら位数2のねじれ点(0,0)を持つ。

CremonaのmwrankでC-4nの有理点群のrankと生成元を求めると、以下のようになる。また、Noam Elkies/Colin Stahlke/Michael Stollのratpoints-1.4でEnの高さ1000以下の有理点を求めると、以下のようになる。
rankが決定できなかったのは、C-4*62,C-4*82の２個である。
[2004.01.12追記]
C-4*62,C-4*82のL-関数のs=1における値を計算すると、どちらも0でないことが分かるので、
     rank(C-4*62) = 0,
     rank(C-4*82) = 0
である。よって、
     rank(E62) = 0,
     rank(E82) = 0
より、E62, E82の有理点はねじれ点に限る。
つまり、n=62, 82のとき、(1)は有理数解(x,y)を持たないことが分かる。


n rank(C-4n) C-4n(Q)/C-4n(Q)torsの生成元
[X:Y:Z] 		C-4n(Q)/C-4n(Q)torsの生成元の高さ Enの有理点(x,y)
(x,y>=0,無限遠点を除く)
1 0 - - -
2 0 - - -
3 0 [-2 : -4 : 1] 0.250591196023589 (1,2),
(1/2,7/4),
(11/3,122/9),
(47/28,2593/784),
...
4 0 - - -
5 1 [-4 : -4 : 1] 1.12202430642277 (1/2,9/4),
(79/36,6881/1296),
...
6 0 - - -
7 0 - - -
8 1 [-4 : -8 : 1] 0.608709031976981 (1,3),
(7/6,113/36),
(239/13,57123/169),
...
9 1 [-3 : -9 : 1] 0.888625874839619 (0,3),
(2,5),
(3/2,15/4),
(60/7,3603/49),
(7/20,1201/400),
...
10 0 - - -
11 0 - - -
12 0 - - -
13 1 [-4 : -12 : 1] 1.49620470504506 (3/2,17/4),
(127/204,150913/41616),
...
14 2 [8 : 8 : 1]
[9 : -15 : 1]		 1.36853485730167
2.49647635928722		 (1/2,15/4),
(11/4,135/16),
(5/6,137/36),
(181/16,32775/256),
(223/60,51521/3600),
(155/78,33097/6084),
...
15 1 [-6 : -12 : 1] 0.567382170117069 (1,4),
(7/4,79/16),
(191/33,36724/1089),
...
16 0 - - -
17 0 - - -
18 1 [9 : -9 : 1]
1.42948269736774 (1/2,17/4),
(287/68,84673/4624),
...
19 1 [-2 : -12 : 1] 1.04937960084785 (3,10),
(31/30,4039/900),
...
20 1 [-4 : -16 : 1] 0.63552871444455 (2,6),
(1/6,161/36),
(358/143,157446/20449),
...
21 1 [-3 : -15 : 1] 1.80046628587121 (5/2,31/4),
(289/620,1763521/384400),
...
22 0 - - -
23 0 - - -
24 0 - - -
25 0 - - -
26 0 - - -
27 0 - - -
28 1 [16 : -48 : 1] 1.63428203191366 (3/2,23/4),
(367/276,424993/76176),
...
29 1 [-4 : -20 : 1] 1.83414963768783 (5/2,33/4),
(161/660,2345921/435600),
...
30 0 - - -
31 1 [18 : 60 : 1] 1.91182249502089 (5/3,56/9),
(943/840,4027999/705600),
...
32 0 - - -
33 2 [-2 : -16 : 1]
[12 : 12 : 1]		 1.47660732522772
1.57737105330718		 (2,7),
(4,17),
(1/2,23/4),
(31/6,983/36),
(32/15,1649/225),
(17/28,4513/784),
(58/63,23047/3969),
(527/92,281953/8464),
(223/136,117313/18496),
(664/817,3859697/667489),
...
34 2 [-8 : -24 : 1]
[98 : 105 : 8]		 1.67223799803014
4.22523597238742		 (3/2,25/4),
(37/6,1385/36),
(15/28,4577/784),
(495/184,314657/33856),
(463/300,566881/90000),
...
35 1 [-10 : -20 : 1] 0.756308496004588 (1,6),
(17/6,359/36),
(971/253,1016046/64009),
...
36 0 - - -
37 1 [-180 : -1812 : 125] 5.07622795241565 (151/30,23449/900),
...
38 0 - - -
39 2 [-3 : -21 : 1]
[16 : -40 : 1]		 2.07685863329456
2.36375717513366		 (1/2,25/4),
(7/2,55/4),
(5/4,103/16),
(353/42,125095/1764),
(115/48,19543/2304),
(623/100,393121/10000),
...
40 1 [-4 : -24 : 1] 1.28152870217857 (3,11),
(41/66,27601/4356),
...
41 0 - - -
42 0 - - -
43 0 - - -
44 0 - - -
45 0 - - -
46 2 [72 : 600 : 1]
[25 : -105 : 1]		 3.31766254492884
3.42860629739401		 (25/6,671/36),
(45/8,2071/64),
(21/10,809/100),
...
47 1 [50 : 340 : 1] 2.81339161393482 (17/5,336/25),
...
48 1 [-8 : -32 : 1] 0.250591196023589 (1,7),
(2,8),
(22/3,488/9),
(47/14,2593/196),
(26/475,1563176/225625),
...
49 1 [18 : -48 : 1] 2.13412084904475 (0,7),
(4/3,65/9),
(21/4,455/16),
...
50 0 - - -
51 1 [-2 : -20 : 1] 1.50699989031181 (5,26),
(287/130,146119/16900),
...
52 0 - - -
53 1 [-4 : -28 : 1] 2.09971127652012 (7/2,57/4),
...
54 0 - - -
55 2 [20 : 60 : 1]
[16 : -24 : 1]		 1.7705031138084
2.40591531831169		 (3/2,31/4),
(3/4,119/16),
(19/6,449/36),
(181/18,32849/324),
(799/372,1208641/138384),
...
56 1 [32 : 160 : 1] 1.91733469610728 (5/2,39/4),
(271/780,4553441/608400),
...
57 0 - - -
58 0 - - -
59 0 - - -
60 1 [16 : -16 : 1] 1.72419250264738 (1/2,31/4),
(959/124,927361/15376),
...
61 1 [-1620 : -3924 : 125] 5.5730412073318 (109/90,64369/8100),
...
62 0 - - -
63 2 [18 : 36 : 1]
[-3 : -27 : 1]		 0.89250642214409
1.74705455122838		 (1,8),
(3,12),
(3/2,33/4),
(9/2,87/4),
(1/4,127/16),
(19/3,368/9),
(81/5,6564/25),
(53/5,2816/25),
(81/5,6564/25),
(103/44,18673/1936),
(447/70,203559/4900),
(57/77,47172/5929),
(333/143,196572/20449),
(617/348,1033873/121104),
...
64 0 - - -
65 2 [-16 : -8 : 1]
[90 : 840 : 1]		 2.43166732920862
2.73235845463796		 (2,9),
(14/3,209/9),
(1/4,129/16),
(74/21,6529/441),
(49/36,10721/1296),
(199/40,41649/1600),
...
66 2 [-8 : -40 : 1]
[25 : -95 : 1]		 1.94114550189285
3.46324967869655		 (5/2,41/4),
(19/10,889/100),
(25/24,4721/576),
(695/44,483281/1936),
(431/820,5465761/672400),
...
67 1 [-150 : -980 : 27] 3.91852062035625 (49/15,3026/225),
...
68 2 [-4 : -32 : 1]
[36 : -192 : 1]		 1.1721830987007
2.03181433819186		 (4,18),
(1/2,33/4),
(8/3,98/9),
(103/10,10641/100),
(32/21,3778/441),
(47/36,10913/1296),
(596/247,615858/61009),
...
69 1 [48 : 312 : 1] 2.77003877954295 (13/4,215/16),
...
70 0 - - -
71 0 - - -
72 0 - - -
73 2 [-16 : -24 : 1]
[18 : 24 : 1]		 2.4742985316412
2.18333078885445		 (6,37),
(2/3,77/9),
(3/4,137/16),
(171/80,62009/6400),
(422/189,353357/35721),
...
74 0 - - -
75 0 - - -
76 0 - - -
77 1 [-700 : -2905 : 64] 4.90060304991852 (83/40,15639/1600),
...
78 0 - - -
79 1 [18 : 12 : 1] 2.0205565288006 (1/3,80/9),
...
80 1 [-16 : -32 : 1] 1.12202430642277 (1,9),
(79/18,6881/324),
...
81 0 - - -
82 0 - - -
83 1 [-18 : -12 : 1] 2.0328344240424 (1/3,82/9),
...
84 1 [-12 : -48 : 1] 0.872242314444939 (2,10),
(17/10,961/100),
...
85 1 [24 : 72 : 1] 1.32698264398174 (9/2,89/4),
...
86 0 - - -
87 0 - - -
88 1 [-8 : -48 : 1] 1.36905981185627 (3,13),
(7/78,57073/6084),
...
89 2 [-16 : -40 : 1]
[98 : 952 : 1]		 2.54835316936709
3.62762503229393		 (2/3,85/9),
(5/4,153/16),
(34/7,1245/49),
(374/23,139965/529),
...
90 1 [24 : 72 : 1] 1.32698264398174 (3/2,39/4),
(151/52,34321/2704),
...
91 0 - - -
92 1 [6480 : 43344 : 125] 5.93075954207064 (301/90,119351/8100),
...
93 1 [-3 : -33 : 1] 2.47749579697918 (11/2,127/4),
...
94 2 [32 : 144 : 1]
[49 : -315 : 1]		 2.6564126487925
4.05106968901552		 (9/4,175/16),
(45/14,2777/196),
(229/18,52535/324),
(571/918,8177015/842724),
...
95 1 [20 : 20 : 1] 1.83753066055259 (1/2,39/4),
...
96 0 - - -
97 0 - - -
98 1 [-7 : -49 : 1] 1.18373727529848 (7/2,63/4),
(17/36,12833/1296),
...
99 2 [-2 : -28 : 1]
[36 : -180 : 1]		 1.82202800791029
2.00485405799203		 (1,10),
(7,50),
(5/12,1433/144),
(49/10,2599/100),
(53/21,5210/441),
(829/171,746290/29241),
(959/940,8839681/883600),
...
100 1 [-16 : -48 : 1] 1.8994821725318 (0,10),
(3/2,41/4),
(20/3,410/9),
...

[参考文献]

Last Update: 2005.08.21
H.Nakao

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