Rational Points on Elliptic Curves: 37y^2-4x^4=1, y^2=148x^4+37

[2002.10.13]37y^2-4x^4=1, y^2=148x^4+37の有理点

y^2=148x^4+37とy^2=x^3-21904xの間の双有理変換を計算するプログラム(asir)
37y^2-4x^4=1の有理点を計算するプログラム(pari/gp)

■Diophantus方程式
     C: 37y²-4x⁴ = 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、整点(±21,±145)を持つ。
     37(y²-145²) = 4(x⁴-21⁴) ----- (2)

■双有理変換(x,y)→(x,37y)[逆変換は、(u,v)→(u,v/37)]によって、曲線Cは、曲線
     C^~: y² = 148x⁴+37 ----- (3)
に写される。曲線C^~は、有理点(±21,±5365)を持つ。

sqrt(148)=2sqrt(37)より、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) = (-4sqrt(37)y+296x², 592xy-1184sqrt(37)x³)
と、逆有理変換φ^-1: E→C^~
     φ^-1(u,v) = (-v/{4sqrt(37)u}, {v²-2u³}/{8sqrt(37)u²}
を定義すると、φによって、曲線C^~は、楕円曲線
     E: y² = x³-21904x ------ (4)
つまり、
     E: y² = x(x+148)(x-148)
に写される。
ただし、双有理変換φは、Q(sqrt(37))-isomorphicであるが、Q-isomorphicではない。
曲線C^~の有理点の１つ(21,5365)を使って、Q-isomorphoicな双有理変換ψ:C^~→Eを
     ψ(x,y) = φ(x,y)-φ(21,5365)
で定義する(右辺の+,-は楕円曲線E上の加法,減法)。また、
     ψ^-1(u,v) = φ^-1((u,v)+φ(21,5365))
である。
asirを使って、φ(21, 5365)と (u,v) = ψ(x,y)と(x, y) = ψ^-1(u,v)をそれぞれ計算すると、以下のようになる。
     φ(21, 5365) = (-21460sqrt(37)+130536, -10965024sqrt(37)+66697680)
     u = -296(x+21)²/(1764x²-145y+1)
     v = -592(x+21)(450660x³-37044xy+y-5365)/(1764x²-145y+1)²

     x = (21u²-10730v+459984)/(u²-261072u-21904)
     y = 37(145u⁴+37855440u³-148176u²v+(-49728v+829185557760)u+3245647104v-69568856320)/(u²-261072u-21904)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("./de37.asir")$
[180] [X0,-Y0];
[(-21460*#0+130536),(-10965024*#0+66697680)]
[181] XXXnf;
[[-296,1],[x+21,2]]
[182] XXXdf;
[[1,1],[1764*x^2-145*y+1,1]]
[183] YYYnf;
[[-592,1],[x+21,1],[450660*x^3-37044*y*x+y-5365,1]]
[184] YYYdf;
[[1,1],[1764*x^2-145*y+1,2]]
[185] XX3nf;
[[1,1],[21*u^2-10730*v+459984,1]]
[186] XX3df;
[[1,1],[u^2-261072*u-21904,1]]
[187] YY3nf;
[[-37,1],[145*u^4+37855440*u^3-148176*v*u^2+(-49728*v+829185557760)*u+3245647104*v-69568856320,1]]
[188] YY3df;
[[-1,1],[u^2-261072*u-21904,2]]
[189] fctr(nm(XXX^3-21904*XXX-YYY^2));
[[-175232,1],[x+21,2],[1764*x^2-145*y+1,4],[1372263228*x^2+148176*x-112799125*y+2333773,1],[148*x^4-y^2+37,1]]
[190] fctr(nm(148*XX3^4+37-YY3^2));
[[-43808,1],[u^2-261072*u-21904,4],[343065807*u^4+44781920085242*u^3+(-350579680500*v+75144825256032)*u^2-980903177547140768*u+44782380616250*v^2-7679097321672000*v+164597902313709312,1],[u^3-21904*u-v^2,1]]

■楕円曲線Eのねじれ点群E_tors(Q)は、 Z/2Z×Z/2Zである。

pari/gpで計算すると、以下のようになる。
E_tors(Q) = Z/2Z×Z/2Z = {(±148,0), (0,0), O}
[pari/gpでの計算結果]

gp> e=ellinit([0,0,0,-21904,0])
time = 109 ms.
%1 = [0, 0, 0, -21904, 0, 0, -43808, 0, -479785216, 1051392, 0, 672589783760896, 1728, [148.0000000000000000000000000, 0.E-28, -148.0000000000000000000000000]~, 0.2155318032909931925100877945, 0.2155318032909931925100877945*I, -7.288002525892373744956404749, -21.86400757767712123486921424*I, 0.04645395822986738396952767419]
gp> elltors(e,1)
time = 105 ms.
%2 = [4, [2, 2], [[0, 0], [148, 0]]]
gp> elladd(e,[148, 0], [0, 0])
time = 0 ms.
%3 = [-148, 0]
gp> sqrt(21904)
time = 1 ms.
%4 = 148.0000000000000000000000000

■楕円曲線EのMordell-Weil群E(Q)をCremonaのmwrankで計算すると、rankは1であり、その生成元は(-1023120/3048625, -261382128/3048625) = (-7056/21025, -261382128/3048625) = (-{2⁴・3²・7²}/{145²},-{2⁴・3・7・881・883}/{145³})である。

E(Q) = Z/2Z×Z/2Z×Z

21904=148²=2²・37なので、このことは、148が合同数(よって37も合同数)であることを意味している。

[mwrankでの計算結果]

bash-2.05a$ mwrank
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 Apr 17 2002 at 20:36:44 by GCC egcs-2.91.66 19990314 (egcs-1.1.2 release)
using base arithmetic option LiDIA (LiDIA bigints and multiprecision floating point)
Using LiDIA multiprecision floating point with 15 decimal places.
Enter curve: [0,0,0,-21904,0]

Curve [0,0,0,-21904,0] :    Working with minimal curve [0,0,0,-1369,0]
      [u,r,s,t] = [2,0,0,0]
3 points of order 2:
[37 : 0 : 1], [-37 : 0 : 1], [0 : 0 : 1]
Using 2-isogenous curve [0,0,0,-15059,-709142]
-------------------------------------------------------
First step, determining Selmer group
-------------------------------------------------------
-------------------------------------------------------
Rank <= 1
-------------------------------------------------------
Second step, determining E(Q)/phi(E'(Q)) and E'(Q)/phi'(E(Q))
-------------------------------------------------------
1. E(Q)/phi(E'(Q))
-------------------------------------------------------
First stage (no second descent yet)...
(-1,0,111,0,-2738):  no rational point found (hlim=6)
(-2,0,111,0,-1369):  no rational point found (hlim=6)
(37,0,111,0,74):  no rational point found (hlim=6)
(74,0,111,0,37):  no rational point found (hlim=6)
After first descent, this component of the rank
        has lower bound 0
       and upper bound 1
       (difference =   1)
Second descent will attempt to reduce this
Second stage (using second descent)...
d1=-1:
 (x:y:z) = (883:37002:145)
     Curve E         Point [-113054905 : -32672766 : 3048625], height = 13.5654207002856
Second descent successfully found rational point for d1=-1

-------------------------------------------------------
2. E'(Q)/phi'(E(Q))
-------------------------------------------------------
This component of the rank is 0

-------------------------------------------------------
Summary of results:
-------------------------------------------------------
  rank(E) = 1
     #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,-1369,0]:

I.  Points on E mod phi(E')
Point [-255780 : -32672766 : 3048625], height = 13.5654207002856

II.  Points on phi(E') mod 2E
--none (modulo 2-torsion).

-------------------------------------------------------
Computing full set of 2 coset representatives for
2E(Q) in E(Q) (modulo torsion), and sorting into height order....done.

Rank = 1
After descent, rank of points found is 1
Transferring points back to original curve [0,0,0,-21904,0]

Generator 1 is [-1023120 : -261382128 : 3048625]; height 13.5654207002856

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) = 13.5654207002856

 (1.17 seconds)

[2002.10.20 追記]
楕円曲線y² = x³-21904xは、双有理変換(x,y)→(4x,8y)[逆変換(u,v)→(u/4,v/8)]により、y² = x³-1369xにQ-isomorphicである。

■pari/gpで、楕円曲線C^~: y² = 148x⁴+37の有理点をいくつか計算すると、以下のようになる。

gp> read("./de37.gp")
time = 103 ms.
gp> rp1(3)
[0]

[-21, -5365]

[-1/42, 5365/882]

[1/42, -5365/882]

[21, -5365]

[-21, 5365]

[-1/42, -5365/882]

[1/42, 5365/882]

[-12708382726809/1815501916871, -1964870981459593403751737605/3296047210162275394430641]

[12708382726809/1815501916871, 1964870981459593403751737605/3296047210162275394430641]

[1815501916871/25416765453618, -1964870981459593403751737605/323005983062114708654644962]

[-1815501916871/25416765453618, 1964870981459593403751737605/323005983062114708654644962]

[-12708382726809/1815501916871, 1964870981459593403751737605/3296047210162275394430641]

[12708382726809/1815501916871, -1964870981459593403751737605/3296047210162275394430641]

[1815501916871/25416765453618, 1964870981459593403751737605/323005983062114708654644962]

[-1815501916871/25416765453618, -1964870981459593403751737605/323005983062114708654644962]

[4653871650202792108551303355338854769/1108153566259737423427108398342515689, 263593142325668140730615935006970654631797608022994274057065951883841868325/1228004326414174260056267754512167498989300524730567952554629441211144721]

[-4653871650202792108551303355338854769/1108153566259737423427108398342515689, -263593142325668140730615935006970654631797608022994274057065951883841868325/1228004326414174260056267754512167498989300524730567952554629441211144721]

[-1108153566259737423427108398342515689/9307743300405584217102606710677709538, 263593142325668140730615935006970654631797608022994274057065951883841868325/43317042673122518779404782302426856618580023830861211342259445088948086722]

[1108153566259737423427108398342515689/9307743300405584217102606710677709538, -263593142325668140730615935006970654631797608022994274057065951883841868325/43317042673122518779404782302426856618580023830861211342259445088948086722]

[4653871650202792108551303355338854769/1108153566259737423427108398342515689, -263593142325668140730615935006970654631797608022994274057065951883841868325/1228004326414174260056267754512167498989300524730567952554629441211144721]

[-4653871650202792108551303355338854769/1108153566259737423427108398342515689, 263593142325668140730615935006970654631797608022994274057065951883841868325/1228004326414174260056267754512167498989300524730567952554629441211144721]

[-1108153566259737423427108398342515689/9307743300405584217102606710677709538, -263593142325668140730615935006970654631797608022994274057065951883841868325/43317042673122518779404782302426856618580023830861211342259445088948086722]

[1108153566259737423427108398342515689/9307743300405584217102606710677709538, 263593142325668140730615935006970654631797608022994274057065951883841868325/43317042673122518779404782302426856618580023830861211342259445088948086722]

[-1031230861330553908881077192069704645439808380625433030904777384054012541/343849697192637207586050019103540266072953638589444128389642150848935279, -12957244720921714931866396406266692470080572272779755875718344151091278859596068748576684866174679852195823622052445263997483960173435839756444245/118232614259468299955887894737743991261228883275976947056490633428113283831680565031732910137833704547062601889401240235367859361510807930807841]

[1031230861330553908881077192069704645439808380625433030904777384054012541/343849697192637207586050019103540266072953638589444128389642150848935279, 12957244720921714931866396406266692470080572272779755875718344151091278859596068748576684866174679852195823622052445263997483960173435839756444245/118232614259468299955887894737743991261228883275976947056490633428113283831680565031732910137833704547062601889401240235367859361510807930807841]

[343849697192637207586050019103540266072953638589444128389642150848935279/2062461722661107817762154384139409290879616761250866061809554768108025082, -12957244720921714931866396406266692470080572272779755875718344151091278859596068748576684866174679852195823622052445263997483960173435839756444245/2126874178721112210467926890895134540036197232703926668882211620586697606476661598642598394531527162443016577402317538093761134202525685170553362]

[-343849697192637207586050019103540266072953638589444128389642150848935279/2062461722661107817762154384139409290879616761250866061809554768108025082, 12957244720921714931866396406266692470080572272779755875718344151091278859596068748576684866174679852195823622052445263997483960173435839756444245/2126874178721112210467926890895134540036197232703926668882211620586697606476661598642598394531527162443016577402317538093761134202525685170553362]

time = 189 ms.

■pari/gpで、楕円曲線C: 37y²-4x⁴ = 1の有理点をいくつか計算すると、以下のようになる。

gp> rp2(3)
[0]

[-21, -145]

[-1/42, 145/882]

[1/42, -145/882]

[21, -145]

[-21, 145]

[-1/42, -145/882]

[1/42, 145/882]

[-12708382726809/1815501916871, -53104621120529551452749665/3296047210162275394430641]

[12708382726809/1815501916871, 53104621120529551452749665/3296047210162275394430641]

[1815501916871/25416765453618, -53104621120529551452749665/323005983062114708654644962]

[-1815501916871/25416765453618, 53104621120529551452749665/323005983062114708654644962]

[-12708382726809/1815501916871, 53104621120529551452749665/3296047210162275394430641]

[12708382726809/1815501916871, -53104621120529551452749665/3296047210162275394430641]

[1815501916871/25416765453618, 53104621120529551452749665/323005983062114708654644962]

[-1815501916871/25416765453618, -53104621120529551452749665/323005983062114708654644962]

[4653871650202792108551303355338854769/1108153566259737423427108398342515689, 7124138981774814614340971216404612287345881297918764163704485186049780225/1228004326414174260056267754512167498989300524730567952554629441211144721]

[-4653871650202792108551303355338854769/1108153566259737423427108398342515689, -7124138981774814614340971216404612287345881297918764163704485186049780225/1228004326414174260056267754512167498989300524730567952554629441211144721]

[-1108153566259737423427108398342515689/9307743300405584217102606710677709538, 7124138981774814614340971216404612287345881297918764163704485186049780225/43317042673122518779404782302426856618580023830861211342259445088948086722]

[1108153566259737423427108398342515689/9307743300405584217102606710677709538, -7124138981774814614340971216404612287345881297918764163704485186049780225/43317042673122518779404782302426856618580023830861211342259445088948086722]

[4653871650202792108551303355338854769/1108153566259737423427108398342515689, -7124138981774814614340971216404612287345881297918764163704485186049780225/1228004326414174260056267754512167498989300524730567952554629441211144721]

[-4653871650202792108551303355338854769/1108153566259737423427108398342515689, 7124138981774814614340971216404612287345881297918764163704485186049780225/1228004326414174260056267754512167498989300524730567952554629441211144721]

[-1108153566259737423427108398342515689/9307743300405584217102606710677709538, -7124138981774814614340971216404612287345881297918764163704485186049780225/43317042673122518779404782302426856618580023830861211342259445088948086722]

[1108153566259737423427108398342515689/9307743300405584217102606710677709538, 7124138981774814614340971216404612287345881297918764163704485186049780225/43317042673122518779404782302426856618580023830861211342259445088948086722]

[-1031230861330553908881077192069704645439808380625433030904777384054012541/343849697192637207586050019103540266072953638589444128389642150848935279, -350195803268154457618010713682883580272447899264317726370766058137602131340434290502072563950667023032319557352768790918850917842525292966390385/118232614259468299955887894737743991261228883275976947056490633428113283831680565031732910137833704547062601889401240235367859361510807930807841]

[1031230861330553908881077192069704645439808380625433030904777384054012541/343849697192637207586050019103540266072953638589444128389642150848935279, 350195803268154457618010713682883580272447899264317726370766058137602131340434290502072563950667023032319557352768790918850917842525292966390385/118232614259468299955887894737743991261228883275976947056490633428113283831680565031732910137833704547062601889401240235367859361510807930807841]

[343849697192637207586050019103540266072953638589444128389642150848935279/2062461722661107817762154384139409290879616761250866061809554768108025082, -350195803268154457618010713682883580272447899264317726370766058137602131340434290502072563950667023032319557352768790918850917842525292966390385/2126874178721112210467926890895134540036197232703926668882211620586697606476661598642598394531527162443016577402317538093761134202525685170553362]

[-343849697192637207586050019103540266072953638589444128389642150848935279/2062461722661107817762154384139409290879616761250866061809554768108025082, 350195803268154457618010713682883580272447899264317726370766058137602131340434290502072563950667023032319557352768790918850917842525292966390385/2126874178721112210467926890895134540036197232703926668882211620586697606476661598642598394531527162443016577402317538093761134202525685170553362]

time = 38 ms.
gp> f(x,y)=37*y^2-4*x^4-1
time = 0 ms.
gp> f(21, -145)
time = 0 ms.
%1 = 0
gp> f(1/42, 145/882)
time = 0 ms.
%2 = 0
gp> f(12708382726809/1815501916871, 53104621120529551452749665/3296047210162275394430641)
time = 0 ms.
%3 = 0
gp> f(1815501916871/25416765453618, -53104621120529551452749665/323005983062114708654644962)
time = 0 ms.
%4 = 0
gp> f(343849697192637207586050019103540266072953638589444128389642150848935279/2062461722661107817762154384139409290879616761250866061809554768108025082, -350195803268154457618010713682883580272447899264317726370766058137602131340434290502072563950667023032319557352768790918850917842525292966390385/2126874178721112210467926890895134540036197232703926668882211620586697606476661598642598394531527162443016577402317538093761134202525685170553362)
time = 1 ms.
%5 = 0

[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.

Last Update: 2005.06.12

H.Nakao