Rational Points on Elliptic Curves: x^4-1785y^2=1, v^2=u^3+4*1785^2u
[2003.08.16]x^4-1785y^2=1, v^2=u^3+4*1785^2uの有理点
■Diophantus方程式
C: x4-1785y2 = 1 ----- (1)
で表される楕円曲線の有理点(x,y)を求める。
■曲線Cは、自明な整点(±1,0)を持つ。
また、参考文献[4]のCohnの定理
---------------------------------------------------------------
Theorem 1.1(Cohn)
dを平方数でない正整数, X,Yを正整数とする。
Pell方程式
v2-du2 = 1
の基本解(fundamental solution)をa+b*sqrt(d)とする。
このとき、方程式
X4-dY2 = 1
の可能な解は、
X2 = a
または、
X2 = 2a2-1
で与えられるものに限る。
さらに、両方が解になるのは、d=1785の場合に限る。
----------------------------------------------------------------
をd=1785について適用すると、曲線Cの自明でない整点は、(±13,±4),(±239,±1352)に限ることが分かる。
[証明]
Pell方程式
v2-1785u2 = 1
の基本解を連分数を使って求めると、169+4sqrt(1785)である。
Theorem 1.1より、X4-1785Y2=1の正整数解X,Yは、
X2 = 169
または、
X2 = 2*1692-1
を満たすので、
X = 13, Y = 4;
または、
X = 239, Y = 1352
に限る。
よって、曲線Cの整点は、(±1,0),(±13,±4),(±239,±1352)に限ることが分かる。
gp> read("pell.gp")
time = 42 ms.
gp> pell(1785,100)
realprecision = 105 significant digits (100 digits displayed)
time = 44 ms.
%1 = [169, 4, 1]
gp> factor(169)
time = 0 ms.
%2 =
[13 2]
gp> factor(2*169^2-1)
time = 2 ms.
%3 =
[239 2]
■双有理変換(x,y)→(x,1785y)[逆変換は、(u,v)→(u,v/1785)]によって、曲線Cは、曲線
C~: y2 = 1785(x4-1) ----- (2)
に写される。曲線C~は、自明な有理点(±1,0)を持つ。
参考文献[3](Prop. 2.1, 2.2)を参考にして、双有理変換φ: C~→E
φ(x,y) = (-2*sqrt(1785)y+2*1785x2, 4*1785xy-4*1785sqrt(1785)x3)
と、逆有理変換φ-1: E→C~
φ-1(u,v) = (-v/{2sqrt(1785)u}, {v2-2u3}/{4sqrt(1785)u2}
を定義すると、φによって、曲線C~は、楕円曲線
E: v2 = u3+4*17852u ------ (3)
に写される。
ただし、双有理変換φは、Q(sqrt(1785))-isomorphicであるが、Q-isomorphicではない。
曲線C~の有理点の１つ(1,0)を使って、Q-isomorphoicな双有理変換φ^:C~→Eを
φ^(x,y) = φ(x,y)-φ(1,0)
で定義する(右辺の+,-は楕円曲線E上の加法,減法)。また、
φ^-1(u,v) = φ-1((u,v)+φ(1,0))
である。
asirを使って、φ(1, 0)と (u,v) = φ^(x,y)と(x, y) = φ^-1(u,v)をそれぞれ計算すると、以下のようになる。
φ(1, 0) = (3570, -7140sqrt(1785))
u = 3570(x+1)/(x-1)
v = -7140y/(x-1)2
x = (u+3570)/(u-3570)
y = -7140v/(u-3570)2
よって、双有理変換ψは、Q-isomorphicである。
[asirでの計算結果]
bash-2.05a$ asir
This is Risa/Asir, Version 20011226 (Kobe Distribution).
Copyright (C) 1994-2000, all rights reserved, FUJITSU LABORATORIES LIMITED.
Copyright 2000,2001, Risa/Asir committers, http://www.openxm.org/.
GC 5.3, copyright 1999, H-J. Boehm, A. J. Demers, Xerox, SGI, HP.
PARI 2.2.1(alpha), copyright (C) 2000,
C. Batut, K. Belabas, D. Bernardi, H. Cohen and M. Olivier.
[0] load("sp")$
0
[101] load("./de1785.asir")$
...省略...
[180] [X0,-Y0];
[3570,(-7140*#0)]
[181] XXXnf;
[[3570,1],[x+1,1]]
[182] XXXdf;
[[1,1],[x-1,1]]
[183] YYYnf;
[[-7140,1],[y,1]]
[184] YYYdf;
[[1,1],[x-1,2]]
[185] XX3nf;
[[-1,1],[u+3570,1]]
[186] XX3df;
[[-1,1],[u-3570,1]]
[187] YY3nf;
[[7140,1],[v,1]]
[188] YY3df;
[[-1,1],[u-3570,2]]
[189] fctr(nm(XXX^3+4*1785^2*XXX-YYY^2));
[[50979600,1],[x-1,4],[1785*x^4-y^2-1785,1]]
[190] fctr(nm(1785*(XX3^4-1)-YY3^2));
[[50979600,1],[u-3570,4],[u^3+12744900*u-v^2,1]]
[191] quit;
bash-2.05a$
■楕円曲線Cと楕円曲線Eの間の双有理変換を求める。
曲線Cと曲線C~の間の有理変換と上記の有理変換を合成すると、以下のようになる。
ψ:C→E; (x,y)→(u,v)
u = 3570(x+1)/(x-1)
v = -12744900y/(x-1)2
ψ-1:E→C; (u,v)→(x,y)
x = (u+3570)/(u-3570)
y = -4v/(u-3570)2
■楕円曲線Eのねじれ点群Etors(Q)は、 Z/2Zである。
pari/gpで計算すると、以下のようになる。
Etors(Q) = Z/2Z = { (0,0), O}
[pari/gpでの計算結果]
gp> e=ellinit([0,0,0,4*1785^2,0])
time = 130 ms.
%1 = [0, 0, 0, 12744900, 0, 0, 25489800, 0, -162432476010000, -611755200, 0, -132491882463990336000000, 1728, [0.E-28, 0.E-28 - 3570.000000000000000000000000*I, 0.E-28 + 3570.000000000000000000000000*I]~, 0.06206162031382841605542621374, -0.03103081015691420802771310687 + 0.03103081015691420802771310687*I, -50.62053871142309457575145006 + 5.75398543 E-28*I, 25.31026935571154728787572502 - 75.93080806713464186362717509*I, 0.001925822357988899951677897977]
gp> elltors(e,0)
time = 64 ms.
%2 = [2, [2], [[0, 0]]]
■楕円曲線EのMordell-Weil群E(Q)をCremonaのmwrank3で計算すると、rankは2であり、その生成元は
P1(85/9, 296225/27),
P2(3600, 304200)
である。
E(Q) = Z/2Z×Z×Z
[mwrank3での計算結果]
bash-2.05a$ mwrank3
Program mwrank: uses 2-descent (via 2-isogeny if possible) to
determine the rank of an elliptic curve E over Q, and list a
set of points which generate E(Q) modulo 2E(Q).
and finally search for further points on the curve.
For more details see the file mwrank.doc.
For details of algorithms see the author's book.
Please acknowledge use of this program in published work,
and send problems to John.Cremona@nottingham.ac.uk.
Version compiled on Feb 11 2003 at 17:40:15 by GCC 3.2.1
using base arithmetic option LiDIA_ALL (LiDIA bigints and multiprecision floating point)
Using LiDIA multiprecision floating point with 15 decimal places.
Enter curve: [0, 0, 0, 12744900, 0]
Curve [0,0,0,12744900,0] :
1 points of order 2:
[0 : 0 : 1]
Using 2-isogenous curve [0,0,0,-50979600,0]
-------------------------------------------------------
First step, determining 1st descent Selmer groups
-------------------------------------------------------
After first local descent, rank bound = 2
rk(S^{phi}(E'))= 1
rk(S^{phi'}(E))= 3
-------------------------------------------------------
Second step, determining 2nd descent Selmer groups
-------------------------------------------------------
After second local descent, rank bound = 2
rk(phi'(S^{2}(E)))= 1
rk(phi(S^{2}(E')))= 3
rk(S^{2}(E))= 3
rk(S^{2}(E'))= 4
Third step, determining E(Q)/phi(E'(Q)) and E'(Q)/phi'(E(Q))
-------------------------------------------------------
1. E(Q)/phi(E'(Q))
-------------------------------------------------------
(c,d) =(0,12744900)
(c',d')=(0,-50979600)
First stage (no second descent yet)...
(85,0,0,0,149940): (x:y:z) = (1:3485:3)
Curve E Point [255 : 296225 : 27], height = 5.93610421376098
After first global descent, this component of the rank = 1
-------------------------------------------------------
2. E'(Q)/phi'(E(Q))
-------------------------------------------------------
First stage (no second descent yet)...
(15,0,0,0,-3398640): (x:y:z) = (34:4080:1)
Curve E' Point [17340 : 2080800 : 1], height = 2.65159037828925
Curve E Point [3600 : 304200 : 1], height = 5.3031807565785
(119,0,0,0,-428400): (x:y:z) = (12:1428:1)
Curve E' Point [17136 : 2039184 : 1], height = 2.65159037828925
Curve E Point [28322 : 2393209 : 8], height = 5.3031807565785
After first global descent, this component of the rank = 3
-------------------------------------------------------
Summary of results:
-------------------------------------------------------
rank(E) = 2
#E(Q)/2E(Q) = 8
Information on III(E/Q):
#III(E/Q)[phi'] = 1
#III(E/Q)[2] = 1
Information on III(E'/Q):
#phi'(III(E/Q)[2]) = 1
#III(E'/Q)[phi] = 1
#III(E'/Q)[2] = 1
-------------------------------------------------------
List of points on E = [0,0,0,12744900,0]:
I. Points on E mod phi(E')
Point [255 : 296225 : 27], height = 5.93610421376098
II. Points on phi(E') mod 2E
Point [3600 : 304200 : 1], height = 5.3031807565785
Point [28322 : 2393209 : 8], height = 5.3031807565785
-------------------------------------------------------
Computing full set of 4 coset representatives for
2E(Q) in E(Q) (modulo torsion), and sorting into height order....done.
Rank = 2
After descent, rank of points found is 2
Generator 1 is [255 : 296225 : 27]; height 5.93610421376098
Generator 2 is [3600 : 304200 : 1]; height 5.3031807565785
The rank has been determined unconditionally.
The basis given is for a subgroup of full rank of the Mordell-Weil group
(modulo torsion), possibly of index greater than 1.
Regulator (of this subgroup) = 11.9160953537247
(20 seconds)
Enter curve: [0,0,0,0,0]
bash-2.05a$
■pari/gpで、楕円曲線E: v2 = u3+12744900uの有理点をいくつか計算すると、以下のようになる。
gp> read("de1785.gp")
time = 23 ms.
gp> v=ratpointE(10,50);
[4165, 354025]
[4165, -354025]
[3060, 260100]
[3060, -260100]
[3600, 304200]
[3600, -304200]
[14161/4, -2393209/8]
[14161/4, 2393209/8]
[85/9, -296225/27]
[1349460, 1567622700]
[1349460, -1567622700]
[85/9, 296225/27]
[1085365, 1130748625]
[1085365, -1130748625]
[149940/12769, -17651686500/1442897]
[149940/12769, 17651686500/1442897]
[169/4, -185653/8]
[169/4, 185653/8]
[50979600/169, 364019833800/2197]
[50979600/169, -364019833800/2197]
[1742400/361, 2860743600/6859]
[1742400/361, -2860743600/6859]
[5112121/1936, 19437091219/85184]
[5112121/1936, -19437091219/85184]
[7944335161/1617984, -875468715193709/2058075648]
[7944335161/1617984, 875468715193709/2058075648]
[1456185600/561001, -94491884728800/420189749]
[1456185600/561001, 94491884728800/420189749]
[173973586885/55905529, 110433518866998775/418005640333]
[173973586885/55905529, -110433518866998775/418005640333]
[171070918740/41770369, -93864323892258900/269961894847]
[171070918740/41770369, 93864323892258900/269961894847]
[5824113422400/57121, 14055449413161920400/13651919]
[57121/456976, -389905632719/308915776]
[5824113422400/57121, -14055449413161920400/13651919]
[57121/456976, 389905632719/308915776]
[832176427860/145709041, -895250942481134700/1758853833911]
[606878155765/271953081, -891937912611928175/4484778258771]
[606878155765/271953081, 891937912611928175/4484778258771]
[832176427860/145709041, 895250942481134700/1758853833911]
[193943334565/1394761, 85438845975508775/1647212741]
[193943334565/1394761, -85438845975508775/1647212741]
[209130464340/2281686289, -3726283434154161300/108989308966663]
[209130464340/2281686289, 3726283434154161300/108989308966663]
[20415551689/60516, 2917198597535963/14886936]
[20415551689/60516, -2917198597535963/14886936]
[771270368400/20415551689, -64011348332629349400/2917035271979387]
[771270368400/20415551689, 64011348332629349400/2917035271979387]
[198270176139540/46808620609, -3651142486314646730700/10127185494618977]
[198270176139540/46808620609, 3651142486314646730700/10127185494618977]
[194957904836485/64794175209, 4223935711363935566825/16493162916925323]
[194957904836485/64794175209, -4223935711363935566825/16493162916925323]
[206379180294085/2094089727409, 107438977429638195682025/3030350962264381673]
[206379180294085/2094089727409, -107438977429638195682025/3030350962264381673]
[313987813727705460/2427990356401, 176008892992316756385293700/3783297001334394601]
[313987813727705460/2427990356401, -176008892992316756385293700/3783297001334394601]
[40808859980181685/7272390447169, 9770808412048896431923375/19611724397327187553]
[22253514768337140/9798045613489, -6184516944416545545106500/30669675812567838487]
[40808859980181685/7272390447169, -9770808412048896431923375/19611724397327187553]
[22253514768337140/9798045613489, 6184516944416545545106500/30669675812567838487]
[675885123840384241/101216678422500, 630085760458043304571286039/1018305575771324625000]
[91095010580250000/47728629605281, -58315608234090571624065000/329737580933377869071]
[91095010580250000/47728629605281, 58315608234090571624065000/329737580933377869071]
[675885123840384241/101216678422500, -630085760458043304571286039/1018305575771324625000]
[26583866701086409/97975260100, -4334757572432726491415077/30667236163901000]
[26583866701086409/97975260100, 4334757572432726491415077/30667236163901000]
[1248684892448490000/26583866701086409, 106059697332341196940602309000/4334382416847054103991173]
[1248684892448490000/26583866701086409, -106059697332341196940602309000/4334382416847054103991173]
[41571066447445921/551472582544, 8485417865412137793845519/409530157468164928]
[41571066447445921/551472582544, -8485417865412137793845519/409530157468164928]
[7028462917265025600/41571066447445921, -393890989478699850136607134800/8475918045028050793896719]
[7028462917265025600/41571066447445921, 393890989478699850136607134800/8475918045028050793896719]
[2531446619999945635765/1469102352134641, 127366171052741423022173871985775/56309034540765229520311]
[2531446619999945635765/1469102352134641, -127366171052741423022173871985775/56309034540765229520311]
[220277206679068071540/29781724941175831009, -1577988292466446400159204204144700/162526720577863684471111364527]
[220277206679068071540/29781724941175831009, 1577988292466446400159204204144700/162526720577863684471111364527]
[2334205593091296961/14589487773450304, 79655085080735189684576801759/1762219759396328743633408]
[2334205593091296961/14589487773450304, -79655085080735189684576801759/1762219759396328743633408]
[185941562723846779449600/2334205593091296961, -80260231410183805779022722610912800/3566224325474635842627924959]
[185941562723846779449600/2334205593091296961, 80260231410183805779022722610912800/3566224325474635842627924959]
[16002590840631918201721/3372668780858598400, -2533360616925110892298743593687731/6193847936330282725019648000]
[16002590840631918201721/3372668780858598400, 2533360616925110892298743593687731/6193847936330282725019648000]
[3035401902772738560000/1130046666240513961, -278153326772723968932334305168000/1201280887948430772758509291]
[3035401902772738560000/1130046666240513961, 278153326772723968932334305168000/1201280887948430772758509291]
[38322651409273364442001/10468488243564714276, -10478884825836027334016163993062951/33870838998733774700951957976]
[9421639419208242848400/2706210819099877441, 1309859835613452829623013609008600/4451869641500150332422990239]
[38322651409273364442001/10468488243564714276, 10478884825836027334016163993062951/33870838998733774700951957976]
[9421639419208242848400/2706210819099877441, -1309859835613452829623013609008600/4451869641500150332422990239]
[223311555032516148848400/32832134831718099361, 119180723881392332653202088670859400/188125940416934130352753765391]
[464935861351960005051121/248123950036129054276, -682138639055534036012051336264061719/3908436222007145380853820782024]
[464935861351960005051121/248123950036129054276, 682138639055534036012051336264061719/3908436222007145380853820782024]
[223311555032516148848400/32832134831718099361, -119180723881392332653202088670859400/188125940416934130352753765391]
[61471546868230777555370485/10574600108165615374609, 565590494498306113179705764042675075825/1087416527365147366932783243638023]
[61471546868230777555370485/10574600108165615374609, -565590494498306113179705764042675075825/1087416527365147366932783243638023]
[32358276330986783046303540/14759074878326717300209, 351731230291325715817612019594839724700/1793034740191584537479075955887177]
[32358276330986783046303540/14759074878326717300209, -351731230291325715817612019594839724700/1793034740191584537479075955887177]
[318865836925215175100485/3750942730356576559809, 7563725119514654623245224590366295575/229726264140796110480298338694623]
[318865836925215175100485/3750942730356576559809, -7563725119514654623245224590366295575/229726264140796110480298338694623]
[562416352989665089377761460/3751362787355472648241, 13341664084957776681617449731783151835100/229764854769980953244394115898711]
[562416352989665089377761460/3751362787355472648241, -13341664084957776681617449731783151835100/229764854769980953244394115898711]
[390611577872271689404885/27333534858951466375129, -60987390467592615162127793444952999775/4519014132136866701786192592937117]
[4098390216751182868286842260/4595430327909078698881, 262375596365956136855145776701576415229300/311522399339929523305349305505471]
[4098390216751182868286842260/4595430327909078698881, -262375596365956136855145776701576415229300/311522399339929523305349305505471]
[390611577872271689404885/27333534858951466375129, 60987390467592615162127793444952999775/4519014132136866701786192592937117]
[17282712356708605275915685/66828733966471123841089, -994429326949495595285501887590854693575/17276065843987294298009831470665313]
[10020300370932680308732884660/203326027725983591481361, 1005674951653027929148277230307320248180900/91683136930564058035853815364340359]
[10020300370932680308732884660/203326027725983591481361, -1005674951653027929148277230307320248180900/91683136930564058035853815364340359]
[17282712356708605275915685/66828733966471123841089, 994429326949495595285501887590854693575/17276065843987294298009831470665313]
[45469098769120488414776858965/5671792559704824697721641, 10613511499789235918891253369706160359601375/13507672047083872665222092672032560811]
[45469098769120488414776858965/5671792559704824697721641, -10613511499789235918891253369706160359601375/13507672047083872665222092672032560811]
[17355685232696763575028221460/10916950484782830351687121, 5620500807826046913340141908738131596255500/36070487403007093869365609603119961831]
[17355685232696763575028221460/10916950484782830351687121, -5620500807826046913340141908738131596255500/36070487403007093869365609603119961831]
[1436584049243509110119968426740/182700163579434834052789969, 1891013205845434127372711071135306872352343100/2469496439258588256228211922355167558297]
[760946181308346083829870220885/469471911517486637294107329, -1605657093667030825948969327912236739221428225/10172188989227657375172480136996289941983]
[760946181308346083829870220885/469471911517486637294107329, 1605657093667030825948969327912236739221428225/10172188989227657375172480136996289941983]
[1436584049243509110119968426740/182700163579434834052789969, -1891013205845434127372711071135306872352343100/2469496439258588256228211922355167558297]
[747332514811847085663204778801/384456859254447876457774756, -1351001275138093095605309248995406377046691401/7538265347936707002509448841956241282904]
[346011173329003088811997280400/52773992995681596332406241, 231749163327411171008976927019012445944877400/383380423362687726295893286705878174289]
[747332514811847085663204778801/384456859254447876457774756, 1351001275138093095605309248995406377046691401/7538265347936707002509448841956241282904]
[346011173329003088811997280400/52773992995681596332406241, -231749163327411171008976927019012445944877400/383380423362687726295893286705878174289]
[6126563568332226628146202689600/1226510349809250819430394281, -18639227327415486937410994171786184452119241200/42954317800928616941679436874949798348821]
[17368613063648800853953813413241/6807292853702474031273558544, -124488063088927266638434564486806485393628931661/561644683088936043700488133289837579867072]
[6126563568332226628146202689600/1226510349809250819430394281, 18639227327415486937410994171786184452119241200/42954317800928616941679436874949798348821]
[17368613063648800853953813413241/6807292853702474031273558544, 124488063088927266638434564486806485393628931661/561644683088936043700488133289837579867072]
[4412919347848475042971655229121/24735895118875769635673508096, -185737858878063283839655072333403431891731893281/3890374825171411893996937289377777839353856]
[4412919347848475042971655229121/24735895118875769635673508096, 185737858878063283839655072333403431891731893281/3890374825171411893996937289377777839353856]
[315256509700559796429695293332710400/4412919347848475042971655229121, 177230183598136724277929907541251914746843507454174400/9270197467446230494994369463433033527661186081]
[315256509700559796429695293332710400/4412919347848475042971655229121, -177230183598136724277929907541251914746843507454174400/9270197467446230494994369463433033527661186081]
[10493582592347119877601719069106565/222421233834775208489608919961, 1078016698216136085169965764359083390546903557573975/104897314137934247227513006141699234100955459]
[10493582592347119877601719069106565/222421233834775208489608919961, -1078016698216136085169965764359083390546903557573975/104897314137934247227513006141699234100955459]
[33349839801186194760931961458952340/123453912851142586795314341989489, -80716024208299419446228022008814344930032050898068300/1371694158223690242426249126006527964170372918263]
[33349839801186194760931961458952340/123453912851142586795314341989489, 80716024208299419446228022008814344930032050898068300/1371694158223690242426249126006527964170372918263]
[19339264612859848512460628397794283060/6111005139416497322503830638873281, -128209027768916681809555307462927916841561420925187553500/477715104864334938501947728217645391610514883951071]
[25452336405669711348228454610907215365/6320021115313675984464257646338001, 171596824209633530107189146865420894823658386871070088375/502432580748271501902094219379186934872250166475751]
[25452336405669711348228454610907215365/6320021115313675984464257646338001, -171596824209633530107189146865420894823658386871070088375/502432580748271501902094219379186934872250166475751]
[19339264612859848512460628397794283060/6111005139416497322503830638873281, 128209027768916681809555307462927916841561420925187553500/477715104864334938501947728217645391610514883951071]
[730188031451956321839713961897769/21759489958643598229502287572196, -2099198309109767924626918935504807474426832377217547/101501641774443620180201397511353996819660014744]
[730188031451956321839713961897769/21759489958643598229502287572196, 2099198309109767924626918935504807474426832377217547/101501641774443620180201397511353996819660014744]
[277322523573916795075183704878880800400/730188031451956321839713961897769, 4618456959522276861756947563985244270260534434166625576600/19731134871520987718337174308928942571878774674603]
[277322523573916795075183704878880800400/730188031451956321839713961897769, -4618456959522276861756947563985244270260534434166625576600/19731134871520987718337174308928942571878774674603]
[3354978880224413562307958994005222400/351688637465588088520326708437161, 6561392932059907002020351090614949995060130125127966400/6595344809413843376132278799243398034530143221909]
[4980262795150192921536346518178636921/3727754311360459513675509993339136, -31710492545806138692506717741792059002648706930127448269/227599299859951198775518570493252058753795196399616]
[4980262795150192921536346518178636921/3727754311360459513675509993339136, 31710492545806138692506717741792059002648706930127448269/227599299859951198775518570493252058753795196399616]
[3354978880224413562307958994005222400/351688637465588088520326708437161, -6561392932059907002020351090614949995060130125127966400/6595344809413843376132278799243398034530143221909]
[14436440575386855835655220492111071365/280165345883467093796842844792521, 54983200185569308775077863613884168634410382239864814825/4689446909955510472517502060193729472015353864069]
[14436440575386855835655220492111071365/280165345883467093796842844792521, -54983200185569308775077863613884168634410382239864814825/4689446909955510472517502060193729472015353864069]
[42007991961767056043898616148190598740/169840477357492421595943770495424369, -3939265034453485385602190614856123601173157164663765006100/69994159478106569839837936975079223855053175714902153]
[42007991961767056043898616148190598740/169840477357492421595943770495424369, 3939265034453485385602190614856123601173157164663765006100/69994159478106569839837936975079223855053175714902153]
[7083476067469938194402119748322430081/277889669100101909895212743744, 18852530437654167071691135054867451500014323461668009279/146490210290853948565899036485809276753833472]
[7083476067469938194402119748322430081/277889669100101909895212743744, -18852530437654167071691135054867451500014323461668009279/146490210290853948565899036485809276753833472]
[3541676043713888831423496897742905600/7083476067469938194402119748322430081, -47590368564791353114491439994153896088355712562085754687200/18852530252757951474958993691552830791264896155714524479]
[3541676043713888831423496897742905600/7083476067469938194402119748322430081, 47590368564791353114491439994153896088355712562085754687200/18852530252757951474958993691552830791264896155714524479]
[766963194037247286789848347993523142565/6360868236154762932982391377708281, 21249663667887475229364167342289987872163472339291679769625/507311373615164192980405782573655639823994306803571]
[766963194037247286789848347993523142565/6360868236154762932982391377708281, -21249663667887475229364167342289987872163472339291679769625/507311373615164192980405782573655639823994306803571]
[953748583329045154171379763173579653140/9023096400438203373998215858747331089, -995247378731530732585674579618226684319366133005125599775500/27104000453943500495791662933806935181514294158289160313]
[953748583329045154171379763173579653140/9023096400438203373998215858747331089, 995247378731530732585674579618226684319366133005125599775500/27104000453943500495791662933806935181514294158289160313]
[2955729685362154870267070961745553626165/1278740373648337132530284352844784041, -295668991652034721397483559317851361557224036771307798603825/1446017557060261161437117084429321936255107494362352661]
[3912945543363911625542670119705039165460/709658988082150028875647289734826801, 291593968000561997592919942304123441516035490639479111293300/597825673029205393072727672381069366946830257899580201]
[2955729685362154870267070961745553626165/1278740373648337132530284352844784041, 295668991652034721397483559317851361557224036771307798603825/1446017557060261161437117084429321936255107494362352661]
[3912945543363911625542670119705039165460/709658988082150028875647289734826801, -291593968000561997592919942304123441516035490639479111293300/597825673029205393072727672381069366946830257899580201]
[2631468311635951434716535947648381929/6904867681278999140083986548931876, -40214553043863327746701453943481976165014367203791189867/573763663458101233051524381899586984413335231875624]
[2631468311635951434716535947648381929/6904867681278999140083986548931876, 40214553043863327746701453943481976165014367203791189867/573763663458101233051524381899586984413335231875624]
[88001848111132716140456400167481866432400/2631468311635951434716535947648381929, 26254169599004766127069941362375607727150426712396113294194600/4268715557142325380200562084973011693292161261279478283]
[88001848111132716140456400167481866432400/2631468311635951434716535947648381929, -26254169599004766127069941362375607727150426712396113294194600/4268715557142325380200562084973011693292161261279478283]
[27067678457786613542889872786151185929/780180032628289370368774517974564, 141567523489700045738766907957506879468065086250985802117/21791756959971756554127309384853675206209219319512]
[27067678457786613542889872786151185929/780180032628289370368774517974564, -141567523489700045738766907957506879468065086250985802117/21791756959971756554127309384853675206209219319512]
[9943316497844285196412994354134020723600/27067678457786613542889872786151185929, -9686613648080943389304833763509485302193085026708425838618200/140823947208695534262035801195975234611050737388411746533]
[9943316497844285196412994354134020723600/27067678457786613542889872786151185929, 9686613648080943389304833763509485302193085026708425838618200/140823947208695534262035801195975234611050737388411746533]
[34189357685275127387339411573297456943052165/7936561705106641211378794104979450757041, 259636640454610366118251727866250873692173950614898990038854013175/707047507663043605284540620484084444932105959755572689445911]
[24285878817626322106819109961237119316545460/8208729336200510777272367724681262171201, -187564495074003258706391782213681054025067427469922277130741564700/743727609354252916562470107540211938564376624879642975515551]
[34189357685275127387339411573297456943052165/7936561705106641211378794104979450757041, -259636640454610366118251727866250873692173950614898990038854013175/707047507663043605284540620484084444932105959755572689445911]
[24285878817626322106819109961237119316545460/8208729336200510777272367724681262171201, 187564495074003258706391782213681054025067427469922277130741564700/743727609354252916562470107540211938564376624879642975515551]
[173678959323641572996977184912140858539041/2059976176156798853206853762656889616, 72445230110138900558420383683898039463391054772016966624253361/2956604929124717863120782412436331936939980094380022336]
[173678959323641572996977184912140858539041/2059976176156798853206853762656889616, -72445230110138900558420383683898039463391054772016966624253361/2956604929124717863120782412436331936939980094380022336]
[26254190367500785704236030519685792466958400/173678959323641572996977184912140858539041, -3179828688392976400866066610468983617553771788256733033943709536400/72380372238566636331381753198324087756152742599633471536010161]
[26254190367500785704236030519685792466958400/173678959323641572996977184912140858539041, 3179828688392976400866066610468983617553771788256733033943709536400/72380372238566636331381753198324087756152742599633471536010161]
[1433765373726904359169531204865867825130628885/919594403221089565035563287008788901583729, -135647213124439975684355752088644625932509403884918038856011945199775/881849512465935594896591245298710491517720786374742951511287433]
[2813958873856534069008823658246894038846210740/344241386248956628852228380520016284545169, 162886058744478438062633308762798751679301886470011008905718791284900/201973607670432990847733888149673071258225487769768244125973447]
[1433765373726904359169531204865867825130628885/919594403221089565035563287008788901583729, 135647213124439975684355752088644625932509403884918038856011945199775/881849512465935594896591245298710491517720786374742951511287433]
[2813958873856534069008823658246894038846210740/344241386248956628852228380520016284545169, -162886058744478438062633308762798751679301886470011008905718791284900/201973607670432990847733888149673071258225487769768244125973447]
time = 10,045 ms.
■pari/gpで、楕円曲線C: x4-1785y2 = 1の有理点をいくつか計算すると、以下のようになる。
gp> ratpointC(v)
[-13, -4]
[13, -4]
[-13, 4]
[13, 4]
[-239, -1352]
[-239, 1352]
[239, -1352]
[239, 1352]
[379/377, 492/142129]
[379/377, -492/142129]
[-379/377, -492/142129]
[-379/377, 492/142129]
[12811/12727, -626020/161976529]
[-12811/12727, 626020/161976529]
[12811/12727, 626020/161976529]
[-12811/12727, -626020/161976529]
[14449/14111, -1485224/199120321]
[-14449/14111, -1485224/199120321]
[14449/14111, 1485224/199120321]
[-14449/14111, 1485224/199120321]
[-101039/15121, 241573904/228644641]
[-101039/15121, -241573904/228644641]
[101039/15121, -241573904/228644641]
[101039/15121, 241573904/228644641]
[-115298639/18219599, -314552985163872/331953787720801]
[-115298639/18219599, 314552985163872/331953787720801]
[115298639/18219599, -314552985163872/331953787720801]
[115298639/18219599, 314552985163872/331953787720801]
[627825757/43040591, 9329413691898028/1852492473629281]
[-627825757/43040591, 9329413691898028/1852492473629281]
[-627825757/43040591, -9329413691898028/1852492473629281]
[627825757/43040591, -9329413691898028/1852492473629281]
[-1631461441/1631347199, -1054304830872176/2661293683685145601]
[-1631461441/1631347199, 1054304830872176/2661293683685145601]
[1631461441/1631347199, 1054304830872176/2661293683685145601]
[1631461441/1631347199, -1054304830872176/2661293683685145601]
[-2340266251/2223106327, -55863406005301516/4942201741147430929]
[2340266251/2223106327, 55863406005301516/4942201741147430929]
[2340266251/2223106327, -55863406005301516/4942201741147430929]
[-2340266251/2223106327, 55863406005301516/4942201741147430929]
[2651681773/611755199, -166191066923333748/374244423503529601]
[-2651681773/611755199, -166191066923333748/374244423503529601]
[-2651681773/611755199, 166191066923333748/374244423503529601]
[2651681773/611755199, 166191066923333748/374244423503529601]
[-20631593809/20199509569, -2870523419975387592/408020186828122565761]
[-20631593809/20199509569, 2870523419975387592/408020186828122565761]
[20631593809/20199509569, 2870523419975387592/408020186828122565761]
[20631593809/20199509569, -2870523419975387592/408020186828122565761]
[-716425395517/61104706991, 12148183473150830667084/3733785216455964274081]
[-716425395517/61104706991, -12148183473150830667084/3733785216455964274081]
[716425395517/61104706991, -12148183473150830667084/3733785216455964274081]
[716425395517/61104706991, 12148183473150830667084/3733785216455964274081]
[90379758907579/85523778194777, 86075915250655858301382052/7314316636709413852950079729]
[90379758907579/85523778194777, -86075915250655858301382052/7314316636709413852950079729]
[-90379758907579/85523778194777, 86075915250655858301382052/7314316636709413852950079729]
[-90379758907579/85523778194777, -86075915250655858301382052/7314316636709413852950079729]
[-112220661977437/24951976611409, -297711184969236690257871580/622601136816301762188965281]
[112220661977437/24951976611409, 297711184969236690257871580/622601136816301762188965281]
[-112220661977437/24951976611409, 297711184969236690257871580/622601136816301762188965281]
[112220661977437/24951976611409, -297711184969236690257871580/622601136816301762188965281]
[-8716207275703439/2643206570353439, -1790571938691395628030537077400/6986540973559589473913379126721]
[-8716207275703439/2643206570353439, 1790571938691395628030537077400/6986540973559589473913379126721]
[8716207275703439/2643206570353439, -1790571938691395628030537077400/6986540973559589473913379126721]
[8716207275703439/2643206570353439, 1790571938691395628030537077400/6986540973559589473913379126721]
[-26933638379643409/26234095022529409, 5427289870988670876311333007080/688227741651102312507284269889281]
[-26933638379643409/26234095022529409, -5427289870988670876311333007080/688227741651102312507284269889281]
[26933638379643409/26234095022529409, -5427289870988670876311333007080/688227741651102312507284269889281]
[26933638379643409/26234095022529409, 5427289870988670876311333007080/688227741651102312507284269889281]
[-43539823567128001/39602309327763841, -25205492527477753885452834222512/1568342904091890928046873467073281]
[-43539823567128001/39602309327763841, 25205492527477753885452834222512/1568342904091890928046873467073281]
[43539823567128001/39602309327763841, 25205492527477753885452834222512/1568342904091890928046873467073281]
[43539823567128001/39602309327763841, -25205492527477753885452834222512/1568342904091890928046873467073281]
[-29843427239965485931/29720022642386176087, -2702727496218386269549249815315312836/883279745863946984263027055678170631569]
[-29843427239965485931/29720022642386176087, 2702727496218386269549249815315312836/883279745863946984263027055678170631569]
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[-181033074272521344902925652844825954468161/166324844374761801091028716979455762609921, 415911459393319683889744332025599857706007227527808763257356768397472227167393424/27663953856288732347695169859248034686158756860618766713139425892653920001607626241]
[181033074272521344902925652844825954468161/166324844374761801091028716979455762609921, 415911459393319683889744332025599857706007227527808763257356768397472227167393424/27663953856288732347695169859248034686158756860618766713139425892653920001607626241]
[181033074272521344902925652844825954468161/166324844374761801091028716979455762609921, -415911459393319683889744332025599857706007227527808763257356768397472227167393424/27663953856288732347695169859248034686158756860618766713139425892653920001607626241]
[7927256123069233792178978385692847401318557/3107876715583840988247781058412619417686191, 1469721661867191832004589175999800941601629835698106948341237696194303254635699708348/9658897679268202850616605103275415120731558689772988770694927419791847350212152088481]
[-7927256123069233792178978385692847401318557/3107876715583840988247781058412619417686191, -1469721661867191832004589175999800941601629835698106948341237696194303254635699708348/9658897679268202850616605103275415120731558689772988770694927419791847350212152088481]
[7927256123069233792178978385692847401318557/3107876715583840988247781058412619417686191, -1469721661867191832004589175999800941601629835698106948341237696194303254635699708348/9658897679268202850616605103275415120731558689772988770694927419791847350212152088481]
[-7927256123069233792178978385692847401318557/3107876715583840988247781058412619417686191, 1469721661867191832004589175999800941601629835698106948341237696194303254635699708348/9658897679268202850616605103275415120731558689772988770694927419791847350212152088481]
time = 52 ms.
[参考文献]
- [1]Joseph H.Silverman, John Tate(著), 足立 恒雄, 木田 雅成, 小松 啓一, 田谷 久雄(訳), "楕円曲線論入門", シュプリンガー・フェアラーク東京, 1995, ISBN4-431-70683-6, {3900円}.
- [2]Joseph H. Silverman, "The Arithmetic of Elliptic Curves", GTM 106, Springer-Verlag New York Inc., 1986, ISBN0-387-96203-4.
- [3]長尾 孝一, "rankの高い楕円曲線の構成法について", p1-2.
- [4]Michael A. Bennett, Gary Walsh, "The Diophantine Equation b^2X^4-dY^2=1", 1991, p1-10.
- [5]Michael A. Bennett, Gary Walsh, "Simultaneous quadratic quations with few or no solutions", 1999, p1-10.
- [6]J.H.E.Cohn, "The Diophantine Equation x^4+1=Dy^2",Math. of Comp, Vol.66(1997), No.219, p1347-1351.
- [7]Gary Walsh, "A note on a theorem of Ljunggren and the Diophantine equations x^2-kxy^2+y^4=1,4", Arch. Math. 73(1999), p119-125.
| Last Update: 2005.06.12 |
| H.Nakao |