Rational Points on Elliptic Curves: v^2=u^4-8u^3-26u^2-24u+9, y^2=x^3-x^2-4x+32

Rational Points on Elliptic Curves: v^2=u^4-8u^3-26u^2-24u+9, y^2=x^3+x^2-4x+32

[2004.04.04]v^2=u^4-8u^3-26u^2-24u+9, y^2=x^3+x^2-4x+32の有理点

v^2=u^4-8u^3-26u^2-24u+9, y^2=x^3-x^2-4x+32の有理点を計算するプログラム(pari/gp)
v^2=u^4-8u^3-26u^2-24u+9のグラフ(jpeg)
y^2=x^3+x^2-4x+32のグラフ(jpeg)
y^2-8xy-48y=x^3-42x^2-36x+1512のグラフ(jpeg)

■参考文献[1]に記述されている楕円曲線
     C: v²=u⁴-8u³-26u²-24u+9
で表される曲線の有理点(u,v)を求める。

■楕円曲線CをWeierstrass長形式に変換すると、以下のようになる。
     E₁: y²-8xy-48y = x³-42x²-36x+1512
ここで、双有理変換φ:C→E₁は、
     x = {6(-4u + v + 3)}/u²,
     y = {36(-7u² - 4u + v + 3)}/u³
であり、逆変換φ^-1:E₁→Cは、
     u = {6(x-42)}/y,
     v = {3(2x³ - 168x² + 8xy + 3528x-y² - 336y)}/y²
である。

[pari/gpによる計算]

gp>  read("./de13.gp")
time = 78 ms.
gp>  f1([u,v])
time = 32 ms.
%1 = [(-24*u + (6*v + 18))/u^2, (-252*u^2 - 144*u + (36*v + 108))/u^3]
gp>  %1[1]*u^2/6
time = 9 ms.
%2 = -4*u + (v + 3)
gp>  %1[2]*u^3/36
time = 11 ms.
%3 = -7*u^2 - 4*u + (v + 3)
gp>  g1([x,y])
time = 19 ms.
%4 = [6/y*x - 252/y, 6/y^2*x^3 - 504/y^2*x^2 + ((24*y + 10584)/y^2)*x + ((-3*y - 1008)/y)]
gp>  %4[1]*y/6
time = 3 ms.
%5 = x - 42
gp>  %4[2]*y^2/3
time = 20 ms.
%6 = 2*x^3 - 168*x^2 + (8*y + 3528)*x + (-y^2 - 336*y)

■楕円曲線E₁をWeierstrass標準形に変換すると、以下のようになる。
     E: Y² = X³+X²-4X+32
ここで、双有理変換ψ=[1/2,-5/2,-2,-3]:E₁→Eは、
     X = {x-10}/4,
     Y = {-4x+y-24}/8
であり、逆変換ψ^-1=[2,10,4,64]:E→E₁は、
     x = 2(2X+5),
     y = 8(2X+Y+8)
である。

[pari/gpによる計算]

gp>  f2([x,y])
time = 1 ms.
%7 = [1/4*x - 5/2, -1/2*x + (1/8*y - 3)]
gp>  %7[1]*4
time = 0 ms.
%8 = x - 10
gp>  %7[2]*8
time = 0 ms.
%9 = -4*x + (y - 24)
gp>  g2([X,Y])
time = 1 ms.
%10 = [4*X + 10, 16*X + (8*Y + 64)]
gp>  %10[1]/2
time = 0 ms.
%11 = 2*X + 5
gp>  %10[2]/8
time = 0 ms.
%12 = 2*X + (Y + 8)

■双有理変換ξ=ψ o φ:C→Eを求めると、以下のようになる。
     X = {-5u² - 12u + 3v + 9}/{2u²},
     Y = {3(-2u³ - 13u² -2uv - 18u + 3v + 9)}/{2u³}

逆変換ξ^-1:E→Cを求めると、
     u = {3(X - 8)}/{2X + Y + 8},
     v = {3(2X³ - 23X² - 16X -Y² - 48Y)}/{(2X+Y+8)²}
となる。
[pari/gpによる計算]

gp>  f([u,v])
time = 24 ms.
%13 = [(-5*u^2 - 12*u + (3*v + 9))/(2*u^2), (-6*u^3 - 39*u^2 + (-6*v - 54)*u + (9*v + 27))/(2*u^3)]
gp>  %13[1]*2*u^2
time = 1 ms.
%14 = -5*u^2 - 12*u + (3*v + 9)
gp>  %13[2]*2*u^3/3
time = 7 ms.
%15 = -2*u^3 - 13*u^2 + (-2*v - 18)*u + (3*v + 9)
gp>  g([X,Y])
time = 32 ms.
%16 = [(3*X - 24)/(2*X + (Y + 8)), (6*X^3 - 69*X^2 - 48*X + (-3*Y^2 - 144*Y))/(4*X^2 + (4*Y + 32)*X + (Y^2 + 16*Y + 64))]
gp>  %16[1]*(2*X+Y+8)/3
time = 4 ms.
%17 = X - 8
gp>  %16[2]*(2*X+Y+8)^2/3
time = 4 ms.
%18 = 2*X^3 - 23*X^2 - 16*X + (-Y^2 - 48*Y)

■楕円曲線E:Y²=X³+X²-4X+32のj-不変量j(E), 判別式Δ(E), conductor N(E)をpari/gpで計算すると、
     j(E) = -35152/1863,
     Δ(E) = -486928 = -2⁸・3⁴・23,
     N(E) = 552
となる。

[pari/gpでの計算結果]

gp>  e=ellinit([0,1,0,-4,32]) 
time = 214 ms.
%1 = [0, 1, 0, -4, 32, 4, -8, 128, 112, 208, -28864, -476928, -35152/1863, [-4.000000000000000000000000000, 1.500000000000000000000000000 - 2.397915761656359770798719032*I, 1.500000000000000000000000000 + 2.397915761656359770798719032*I]~, 2.446511007542945518026373136, -1.223255503771472759013186568 + 0.6481161030709770672174658388*I, -2.300446229736951659977971306 + 2.18809717 E-29*I, 1.150223114868475829988985653 - 1.893532824928403369155354473*I, 1.585623180328983630515214926]
gp>  e.j
time = 0 ms.
%2 = -35152/1863
gp>  ellglobalred(e)
time = 5 ms.
%3 = [552, [1, 0, 0, 0], 16]
gp>  e.disc
time = 0 ms.
%4 = -476928
gp>  factor(e.disc)
time = 0 ms.
%5 = 
[-1 1]

[2 8]

[3 4]

[23 1]

■楕円曲線Eのねじれ点群E_tors(Q)は、位数4の巡回群に同型である。

pari/gpで計算すると、以下のようになる。
E_tors(Q) = Z/4Z = { (2,6),(-4,0),(2,-6),O }

[pari/gpでの計算結果]

gp>  e=ellinit([0,1,0,-4,32])
time = 206 ms.
%19 = [0, 1, 0, -4, 32, 4, -8, 128, 112, 208, -28864, -476928, -35152/1863, [-4.000000000000000000000000000, 1.500000000000000000000000000 - 2.397915761656359770798719032*I, 1.500000000000000000000000000 + 2.397915761656359770798719032*I]~, 2.446511007542945518026373136, -1.223255503771472759013186568 + 0.6481161030709770672174658388*I, -2.300446229736951659977971306 + 2.18809717 E-29*I, 1.150223114868475829988985653 - 1.893532824928403369155354473*I, 1.585623180328983630515214926]
gp>  elltors(e,1)
time = 80 ms.
%20 = [4, [4], [[2, 6]]]
gp>  for(i=1,4,print(ellpow(e,[2,6],i)))
[2, 6]
[-4, 0]
[2, -6]
[0]
time = 4 ms.

■楕円曲線EのMordell-Weil群E(Q)をCremonaのmwrank3で計算すると、rankは1であり、その生成元は(-2,6)である。

E(Q) = Z×Z/4Z = {m(2,6)+n(-2,6): n \in Z, m=0,1,2,3 }

[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: [0,1,0,-4,32]

Curve [0,1,0,-4,32] :
1 points of order 2:
[-4 : 0 : 1]

Using 2-isogenous curve [0,22,0,-23,0]
-------------------------------------------------------
First step, determining 1st descent Selmer groups
-------------------------------------------------------
After first local descent, rank bound = 1
rk(S^{phi}(E'))=        2
rk(S^{phi'}(E))=        1

-------------------------------------------------------
Second step, determining 2nd descent Selmer groups
-------------------------------------------------------
After second local descent, rank bound = 1
rk(phi'(S^{2}(E)))=     2
rk(phi(S^{2}(E')))=     1
rk(S^{2}(E))=   2
rk(S^{2}(E'))=  3

Third step, determining E(Q)/phi(E'(Q)) and E'(Q)/phi'(E(Q))
-------------------------------------------------------
1. E(Q)/phi(E'(Q))
-------------------------------------------------------
(c,d)  =(-11,36)
(c',d')=(22,-23)
First stage (no second descent yet)...
(2,0,-11,0,18):  (x:y:z) = (1:3:1)
        Curve E         Point [2 : 6 : 1], height = 0.827078390038745
After first global descent, this component of the rank = 2
-------------------------------------------------------
2. E'(Q)/phi'(E(Q))
-------------------------------------------------------
This component of the rank is 0

-------------------------------------------------------
Summary of results:
-------------------------------------------------------
        rank(E) = 1
        #E(Q)/2E(Q) = 4

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 = [0,1,0,-4,32]:

I.  Points on E mod phi(E')
Point [-2 : 6 : 1], height = 0.827078390038745

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

Generator 1 is [-2 : 6 : 1]; height 0.827078390038745

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) = 0.827078390038745

 (9 seconds)
Enter curve: [0,0,0,0,0]

bash-2.05a$

■pari/gpで、楕円曲線Eの有理点をいくつか計算すると、以下のようになる。

gp>  rpE(5)
[0]
[2, 6]
[-4, 0]
[2, -6]
[-2, 6]
[-1, -6]
[14, -54]
[8, 24]
[-2, -6]
[8, -24]
[14, 54]
[-1, 6]
[28/9, -208/27]
[146, 1770]
[17/16, 351/64]
[-94/25, -354/125]
[28/9, 208/27]
[-94/25, 354/125]
[17/16, -351/64]
[146, -1770]
[2686/529, 157878/12167]
[-4024/1369, 263208/50653]
[-82/2401, -666954/117649]
[3623/121, -221334/1331]
[2686/529, -157878/12167]
[3623/121, 221334/1331]
[-82/2401, 666954/117649]
[-4024/1369, -263208/50653]
[-302351/97344, -149119255/30371328]
[1931378/413449, -3112510782/265847707]
[3156284/87025, 5677652448/25672375]
[90578/597529, 2589061362/461889917]
[-302351/97344, 149119255/30371328]
[90578/597529, -2589061362/461889917]
[3156284/87025, -5677652448/25672375]
[1931378/413449, 3112510782/265847707]
[2222041534/21911761, -105240287465034/102568953241]
[1171898807/347337769, 53496171771402/6473334000853]
[-4224965458/1154844289, 130467235694742/39245073473087]
[753053384/853749961, -136487648900184/24945720110459]
[2222041534/21911761, 105240287465034/102568953241]
[753053384/853749961, 136487648900184/24945720110459]
[-4224965458/1154844289, -130467235694742/39245073473087]
[1171898807/347337769, -53496171771402/6473334000853]
time = 54 ms.

■pari/gpで、楕円曲線E₁の有理点をいくつか計算すると、以下のようになる。

gp>  rpE1(5)
[0]
[18, 144]
[-6, 0]
[18, 48]
[2, 80]
[6, 0]
[66, -144]
[42, 384]
[2, -16]
[42, 0]
[66, 720]
[6, 96]
[202/9, 1408/27]
[594, 16560]
[57/4, 999/8]
[-126/25, -2352/125]
[202/9, 4736/27]
[-126/25, 3312/125]
[57/4, 297/8]
[594, -11760]
[16034/529, 3030160/12167]
[-2406/1369, 2965248/50653]
[23682/2401, 2129616/117649]
[15702/121, -1047840/1331]
[16034/529, 504112/12167]
[15702/121, 2493504/1331]
[23682/2401, 12800880/117649]
[-2406/1369, -1246080/50653]
[-58991/24336, -94815655/3796416]
[11860002/413449, 11984183856/265847707]
[13495386/87025, 61961912064/25672375]
[6337602/597529, 51393714288/461889917]
[-58991/24336, 203422855/3796416]
[6337602/597529, 9968732496/461889917]
[13495386/87025, -28880527104/25672375]
[11860002/413449, 61784356368/265847707]
[9107283746/21911761, -668935863982384/102568953241]
[8160972918/347337769, 1191713599282752/6473334000853]
[-5351418942/1154844289, 1258190569288080/39245073473087]
[11549713146/853749961, 856680365101440/24945720110459]
[9107283746/21911761, 1014908735458160/102568953241]
[11549713146/853749961, 3040482747504384/24945720110459]
[-5351418942/1154844289, -829285201827792/39245073473087]
[8160972918/347337769, 335774850940320/6473334000853]
time = 41 ms.

■pari/gpで、楕円曲線Cの有理点をいくつか計算すると、以下のようになる。

gp>  rpC(5)
[0]
[-1, -4]
[0]
[-3, 12]
[-3, -12]
[0]
[-1, 4]
[0, -3]
[15, 132]
[0]
[1/5, -44/25]
[-9/4, -111/16]
[-9/4, 111/16]
[1/5, 44/25]
[-4/3, -37/9]
[15, -132]
[-99/148, -87657/21904]
[-245/23, -74548/529]
[-148/33, 29219/1089]
[-69/245, 223644/60025]
[-69/245, -223644/60025]
[-148/33, -29219/1089]
[-245/23, 74548/529]
[-99/148, 87657/21904]
[-53337/31507, 4671141396/992691049]
[3245/11544, -22206263/133263936]
[-31507/17779, -1557047132/316092841]
[34632/3245, 66618789/10530025]
[34632/3245, -66618789/10530025]
[-31507/17779, 1557047132/316092841]
[3245/11544, 22206263/133263936]
[-53337/31507, -4671141396/992691049]
[-27348984/5497915, -994312149211221/30227069347225]
[-26268859/3009883, 876380957054108/9059395673689]
[-5497915/9116328, 331437383070407/83107436203584]
[-9029649/26268859, -2629142871162324/690052953161881]
[-9029649/26268859, 2629142871162324/690052953161881]
[-5497915/9116328, -331437383070407/83107436203584]
[-26268859/3009883, -876380957054108/9059395673689]
[-27348984/5497915, 994312149211221/30227069347225]
[22350291123/98649760445, 14228109659884619200164/9731775235855886598025]
[-47425826404/33837281289, -4782631699047467865491/1144961605030909501521]
[98649760445/7450097041, -4742703219961539733388/55503945920316955681]
[-101511843867/47425826404, 14347895097142403596473/2249209010102343571216]
time = 55 ms.

■A.Bremner[1]は、以下のDiophantus方程式系S
     x₁²+x₂²+x₃² = y₁²+y₂²+y₃²,
     x₁³+x₂³+x₃³ = y₁³+y₂³+y₃³,
     x₁⁴+x₂⁴+x₃⁴ = y₁⁴+y₂⁴+y₃⁴
の自明でない整数解 (x₁,x₂,x₃; y₁,y₂,y₃) の一部[x₁+x₂=±(y₁+y₂)を満たすもの]を求めている。

ここで、整数解(x₁,x₂,x₃; y₁,y₂,y₃)が自明であるとは、 x₁,x₂,x₃がy₁,y₂,y₃を置換したものであるということとする。

Diophantus方程式系Sは同次形なので、Sの整数解(x₁,x₂,x₃; y₁,y₂,y₃)!=(0,0,0;0,0,0)を射影空間P⁵(Q)の有理点(x₁:x₂:x₃:y₁:y₂:y₃)と見なして良い。

Bremnerは、Diophantus方程式系Sについて、[1]で以下のことを証明している。

x₁+x₂=y₁+y₂を満たす整数解は、自明なものに限る。
x₁+x₂=-(y₁+y₂)を満たす整数解は、楕円曲線Cの有理点P(u,v)に対応する有理点Q(x₁:x₂:x₃:y₁:y₂:y₃)に限る[本質的に一致する]。
     x₁=(3+u²)v-27+18u²+8u³+u⁴,
     x₂=-(3+u²)v+9+24u+18u²-3u⁴,
     x₃=2(3-u²)v+18+12u+12u²+4u³+2u⁴,
     y₁=(3+u²)v+27-18u²-8u³-u⁴,
     y₂=-(3+u²)v-9-24u-18u²+3u⁴,
     y₃=2(3-u²)v-18-12u-12u²-4u³-2u⁴

この有理点Q(x₁:x₂:x₃:y₁:y₂:y₃)が、実際に、Diophantus方程式系Sの解であり、x₁+x₂=-(y₁+y₂)を満たすことは、以下のように簡単に確認できる。
[asirによる計算]

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] X1=(3+u^2)*v-27+18*u^2+8*u^3+u^4;
u^4+8*u^3+(v+18)*u^2+3*v-27
[1] X2=-(3+u^2)*v+9+24*u+18*u^2-3*u^4;
-3*u^4+(-v+18)*u^2+24*u-3*v+9
[2] X3=2*(3-u^2)*v+18+12*u+12*u^2+4*u^3+2*u^4;
2*u^4+4*u^3+(-2*v+12)*u^2+12*u+6*v+18
[3] Y1=(3+u^2)*v+27-18*u^2-8*u^3-u^4;
-u^4-8*u^3+(v-18)*u^2+3*v+27
[4] Y2=-(3+u^2)*v-9-24*u-18*u^2+3*u^4;
3*u^4+(-v-18)*u^2-24*u-3*v-9
[5] Y3=2*(3-u^2)*v-18-12*u-12*u^2-4*u^3-2*u^4;
-2*u^4-4*u^3+(-2*v-12)*u^2-12*u+6*v-18
[6] X1+X2+Y1+Y2;
0
[7] F2=X1^2+X2^2+X3^2-(Y1^2+Y2^2+Y3^2);
0
[8] F3=X1^3+X2^3+X3^3-(Y1^3+Y2^3+Y3^3);
-36*u^12+144*u^11+1944*u^10+5616*u^9+(36*v^2+7812)*u^8+(144*v^2+7200)*u^7+(144*v^2+9936)*u^6+(144*v^2+21600)*u^5+(216*v^2+70308)*u^4+(432*v^2+151632)*u^3+(1296*v^2+157464)*u^2+(3888*v^2+34992)*u+2916*v^2-26244
[9] fctr(F3);
[[-36,1],[u^2+3,1],[u^4-2*u^2+9,1],[u+1,1],[u+3,1],[u^4-8*u^3-26*u^2-24*u-v^2+9,1]]
[10] F4=X1^4+X2^4+X3^4-(Y1^4+Y2^4+Y3^4);
96*v*u^14-576*v*u^13-4704*v*u^12-3840*v*u^11+(-96*v^3+26208*v)*u^10+(-192*v^3+90816*v)*u^9+(672*v^3+123552*v)*u^8+1920*v^3*u^7+(2880*v^3-370656*v)*u^6-817344*v*u^5+(-8640*v^3-707616*v)*u^4+(-17280*v^3+311040*v)*u^3+(-18144*v^3+1143072*v)*u^2+(15552*v^3+419904*v)*u+23328*v^3-209952*v
[11] fctr(F4);
[[96,1],[v,1],[u-1,1],[u+1,1],[u^2+2*u+3,1],[u^2-3,1],[u^2+3,1],[u-3,1],[u+3,1],[u^4-8*u^3-26*u^2-24*u-v^2+9,1]]
[12] quit;
bash-2.05a$

よって、楕円曲線Cの有理点(u,v)から、Sの整数解(x₁:x₂:x₃:y₁:y₂:y₃)をいくつか計算すると、以下のようになる。
[pari/gpによる計算]

gp>  rpS(10)
[-1, 0, 2, -1, 2, 0]
[776, -1233, 410, 815, -358, -1224]
[-931219912, 378382959, -156845590, 357088490, 195748463, -932263416]
[147729270939734015, -220931652694759344, 207749597674213298, -156840985477974094, 230043367232999423, -190461498615919440]
[-43992346521996843142080660865, -29910196385462462465784243024, 122741732517246503591277139538, 122563310654814476373862529663, -48660767747355170765997625774, 22490972972129645890059151440]
[-14153014093421377052189631057444575761819432, 7163833468298268650660674774340400158811759, 19736348798529055286396719073791446186775370, -13031913215316551317153327809205551969223057, 20021093840439659718682284092309727572230730, -8395047419047777542267556863556589393918616]
[2169563594807642920865024879670798387367521482213257599724136, -4556067701983900569194627711004319355341304579542997151226833, -210459342831468725785492243422367783084868082948503491570630, 2168617382531060875735837854151383678082798245948937638602042, 217886724645196772593764977182137289890984851380801912900655, -4556169032791484907564968484169682434039225041750082485261864]
[-558008934247048080809648502838496642508316518575593201918221426957488782222304641, 293193749230864193656855256848957756778007055996729869794562086643005859666402720, 29412090918398092017275871399888222871219678253177441097079791505875058320862562, 293317526893397899186631378645201874368921689957655156252342466807300133187038562, -28502341877214012033838132655662988638612227378791824128683126492817210631136641, -557991094287735594639028388330963163775232824379585805722951814029962377899567520]
[10919054776498828657107554553522006679785665907192223279293339122777839408328960148139343506913617153919, -30022391586983904349423688319589337480859208126755786471766519132663294257830394125090381745156390652320, 48203382740859947007964100405993683870107847102176868807323668875690012571697735488025757975571430648482, -29258882166154502706453886647672179596292405822883314142609178280655655348131081458189874722692160254081, 48362218976639578398770020413739510397365948042446877335082358290541110197632515435140912960934933752482, -12212675854163454650224818736663818898196059475570523205715265098020706532616838486380415702360147280480]
[-781669846687056834258938818334703500422583723203238495576713260686376383508497585129247231016035174739870335128013710631669989464, -1829146191835628395157681557525846526292301854056096671343028677126070428730944813809302911335545050389798574512133080417707606993, 4497930421089589015274212956296739190650805117671106860328342156463727370299141926191300553683971459995139432105636374306539149050, 4494583316984713624527088476189880196718658526812408045906114397285877174779961998465201623504446359226162744251558438482880406575, -1883767278462028395110468100329330170003772949553072878986372459473430362540519599526651481152866134096493834611411647433502810118, 662044757224289197914492053692339408645129409894758452031955625680307102240057678914369046441300842288719990052047602820959760536]
time = 84 ms.
gp>  check(776, -1233, 410, 815, -358, -1224)
(776)^2+(-1233)^2+(410)^2=2290565
(815)^2+(-358)^2+(-1224)^2=2290565
(776)^3+(-1233)^3+(410)^3=-1338306761
(815)^3+(-358)^3+(-1224)^3=-1338306761
(776)^4+(-1233)^4+(410)^4=2702152188497
(815)^4+(-358)^4+(-1224)^4=2702152188497
time = 33 ms.
gp>  check(-931219912, 378382959, -156845590, 357088490, 195748463, -93263416)
(-931219912)^2+(378382959)^2+(-156845590)^2=1034944727269331525
(357088490)^2+(195748463)^2+(-932263416)^2=1034944727269331525
(-931219912)^3+(378382959)^3+(-156845590)^3=-757210471081504090752837449
(357088490)^3+(195748463)^3+(-932263416)^3=-757210471081504090752837449
(-931219912)^4+(378382959)^4+(-156845590)^4=773088603061190653280586680548073297
(357088490)^4+(195748463)^4+(-932263416)^4=773088603061190653280586680548073297
time = 15 ms.
gp>  check(147729270939734015, -220931652694759344, 207749597674213298,-156840985477974094, 230043367232999423, -190461498615919440)
(147729270939734015)^2+(-220931652694759344)^2+(207749597674213298)^2=113794627988620596008386624223367365
(-156840985477974094)^2+(230043367232999423)^2+(-190461498615919440)^2=113794627988620596008386624223367365
(147729270939734015)^3+(-220931652694759344)^3+(207749597674213298)^3=1406635621459258552349138244035639562245368050893383
(-156840985477974094)^3+(230043367232999423)^3+(-190461498615919440)^3=1406635621459258552349138244035639562245368050893383
(147729270939734015)^4+(-220931652694759344)^4+(207749597674213298)^4=4721554537285354826871731566299595910206345120544004071191664672017937
(-156840985477974094)^4+(230043367232999423)^4+(-190461498615919440)^4=4721554537285354826871731566299595910206345120544004071191664672017937
time = 24 ms.
gp>  check(-43992346521996843142080660865, -2991019638546246246578424304, 122741732517246503591277139538, 122563310654814476373862529663, -48660767747355170765997625774, 22490972972129645890059151440)
(-43992346521996843142080660865)^2+(-29910196385462462465784243024)^2+(122741732517246503591277139538)^2=17895479301663667206897891286167186665939630775515358066245
(122563310654814476373862529663)^2+(-48660767747355170765997625774)^2+(22490972972129645890059151440)^2=17895479301663667206897891286167186665939630775515358066245
(-43992346521996843142080660865)^3+(-29910196385462462465784243024)^3+(122741732517246503591277139538)^3=1737271797935630445059134566660998783740857975432365145566097204377867243658298079236423
(122563310654814476373862529663)^3+(-48660767747355170765997625774)^3+(22490972972129645890059151440)^3=1737271797935630445059134566660998783740857975432365145566097204377867243658298079236423
(-43992346521996843142080660865)^4+(-29910196385462462465784243024)^4+(122741732517246503591277139538)^4=231516115138179887508356548349405433657995018567555627057863973509842586364768738328995843390407868782022797198571537
(122563310654814476373862529663)^4+(-48660767747355170765997625774)^4+(22490972972129645890059151440)^4=231516115138179887508356548349405433657995018567555627057863973509842586364768738328995843390407868782022797198571537
time = 6 ms.
gp>  check(-14153014093421377052189631057444575761819432, 716383346829868650660674774340400158811759, 19736348798529055286396719073791446186775370, -13031913215316551317153327809205551969223057, 20021093840439659718682284092309727572230730, -8395047419047777542267556863556589393918616)
(-14153014093421377052189631057444575761819432)^2+(7163833468298268650660674774340400158811759)^2+(19736348798529055286396719073791446186775370)^2=641151781787293808409378861039211600858074189717594021922550838472275872935100608113605
(-13031913215316551317153327809205551969223057)^2+(20021093840439659718682284092309727572230730)^2+(-8395047419047777542267556863556589393918616)^2=641151781787293808409378861039211600858074189717594021922550838472275872935100608113605
(-14153014093421377052189631057444575761819432)^3+(7163833468298268650660674774340400158811759)^3+(19736348798529055286396719073791446186775370)^3=5220463306923174486721886383473996673860688361561566626127983331427050239554484391921590447183658311942280405492148436640982681911
(-13031913215316551317153327809205551969223057)^3+(20021093840439659718682284092309727572230730)^3+(-8395047419047777542267556863556589393918616)^3=5220463306923174486721886383473996673860688361561566626127983331427050239554484391921590447183658311942280405492148436640982681911
(-14153014093421377052189631057444575761819432)^4+(7163833468298268650660674774340400158811759)^4+(19736348798529055286396719073791446186775370)^4=194485541586336751006603521900613877579526087261529327110413593874230262942382546310149079873734361805772048868964219461351980561978610755444219486640680849318169582640089937
(-13031913215316551317153327809205551969223057)^4+(20021093840439659718682284092309727572230730)^4+(-8395047419047777542267556863556589393918616)^4=194485541586336751006603521900613877579526087261529327110413593874230262942382546310149079873734361805772048868964219461351980561978610755444219486640680849318169582640089937
time = 5 ms.

よって、Diophantus方程式系Sの自明でない整数解(x₁,x₂,x₃; y₁,y₂,y₃)で、x₁+x₂=-(y₁+y₂)を満たすものは、以下のように無数に存在することが分かる。

(776, -1233, 410; 815, -358, -1224),
(-931219912, 378382959, -156845590; 357088490, 195748463, -932263416),
(147729270939734015, -220931652694759344, 207749597674213298;
-156840985477974094, 230043367232999423, -190461498615919440),
(-43992346521996843142080660865, -29910196385462462465784243024, 122741732517246503591277139538;
122563310654814476373862529663, -48660767747355170765997625774, 22490972972129645890059151440),
(-14153014093421377052189631057444575761819432, 7163833468298268650660674774340400158811759, 19736348798529055286396719073791446186775370;
-13031913215316551317153327809205551969223057, 20021093840439659718682284092309727572230730, -8395047419047777542267556863556589393918616),
.....

[参考文献]

[1]Andrew Bremner, "A Diophantine system" :
Miles Ried, Alexei Skorobogatov(編), "Number Theory and Algebraic Geometry", Cambridge University Press, LMSLNS 303, 2003, ISBN0-521-54518-8, p63-67.
[2]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}.
[3]Joseph H. Silverman, "The Arithmetic of Elliptic Curves", GTM 106, Springer-Verlag New York Inc., 1986, ISBN0-387-96203-4.

Last Update: 2005.06.12

H.Nakao