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


[2002.07.13](x+y+z)(1/x+1/y+1/z)=n (n \in [-100..-1])の有理点


■n = -1,-2,...,-100について、Diophantus方程式
     Cn: (x+y+z)(1/x+1/y+1/z) = n
で表される曲線の有理点(Diophantus方程式の有理数解)を求める。

Cnと双有理変換可能な楕円曲線
     En: y2=x3+(n2-6n-3)x2+16nx
のねじれ点群は、Z/6Zである。
En(Q)/En(Q)torsの生成元の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
(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

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。

[2002.08.03追記]
■pari/gpで楕円曲線のねじれ点群を正確に求めるには、elltors()関数に引数flag=1を与えて、Nagell-Lutzの定理を使う必要がある。
実際に計算すると、結果は、このようになる。

elltors()関数のflag引数を省略する(flag=0)と、Doudのアルゴリズム--あるgood reductionに対する#E(Fp)を計算する--が使われる。しかし、n=-21,-39,-45,-56,-66,-75,-84の場合には、En(Q)tors=Z/2Z, 生成元が(0,0)となり、計算結果は正しくない。

また、このNagell-Lutzの定理に基づいた計算では、stackも多く必要とするので、~/.gprc内でparisize=20Mと指定する。

[2008.06.15追記]
L-関数を使うと、楕円曲線E-60,E-73のrankを決定できる。
L(E-60,1)!=0より、rank(E-60) = 0である。
同様に、L(E-73,1)!=0より、rank(E-73) = 0である。
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

[2021.04.19追記]
参考文献[5]Bremner他を追加した。

[参考文献]



Last Update: 2021.04.19
H.Nakao

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