Rational Points on Elliptic Curve: x^4+z^4=97y^2

[2005.08.20]x^4+z^4=97y^2の有理点

x^4+z^4=97y^2, v^2=u^3-194^2*uの有理点を求めるプログラム(pari/gp)

Diophantus方程式
     C: x⁴+z⁴=97y²
で定義される楕円曲線Cの有理点(x:y:z)を求める。
ただし、(x,y,z)は(0,0,0)以外の3組の有理数とし、0でない任意の有理数dに対して、
     (x, y, z)～(dx, d²y, dz)
とする。～はQ³-{(0,0,0)}上の同値関係である。
Q³-{(0,0,0)}/～の元(x:y:z)を、
     (x:y:z) = {(dx,d²y,dz) | d ∈ Q-{0}}
とする。

■一般に、正有理数a,bに対して、正有理数cを
     c = a⁴+b⁴ ------- (*)
とする。
このとき、楕円曲線
     C_c: x⁴+z⁴=cy²
は、楕円曲線
     E_c: y² = x³-4c²x
にQ-isomorphicである。

楕円曲線C_cは、双有理変換φ_a,b
     u = (a²z²+b²x²+cy)/(ax-bz)², ------ (1)
     v = (az³+bx³+(a³z+b³x)y)/(ax-bz)³ ----- (2)
[逆変換φ_a,b^-1は、
     x = {a³u+bcv+ab²}w ----- (3)
     y = {cu³+3a²b²u²+2ab(a⁴-b⁴)v-a²b²}w² ----- (4)
     z = -{b³u-acv+a²b}w ----- (5)
ただし、wは0でない任意の有理数とする。
]
によって、楕円曲線
     E'_c: 2cv² = u³-u
に写される。また、楕円曲線E'_cは、双有理変換φ_a,b^-1によって、楕円曲線C_cに写される。

[証明]
(x:y:z)を楕円曲線C_cの有理点とすると、
     x⁴+z⁴-cy² = 0
である。(u,v)=φ_a,b(x:y:z)とすると、(*),(1),(2)より、
     u³-u-2cv² = -(3a⁴b²+b⁶x²-4a³b³xz+(a⁸+2a⁴b⁴+b⁸)y+(a⁶+3a²b⁴)z²)(x⁴-(a⁴+b⁴)y²+z⁴)/(ax-bz)⁶ = -(b²(3a⁴+b⁴)x²-4a³b³xz+c²y+a²(a⁴+3b⁴)z²)(x⁴-cy²+z⁴)/(ax-bz)⁶ = 0
となる。
よって、(u,v)は楕円曲線E_cの有理点である。

[asirによる計算]

[0] U=((a*z)^2+(b*x)^2+(a^4+b^4)*y)/(a*x-b*z)^2;
(b^2*x^2+(a^4+b^4)*y+a^2*z^2)/(a^2*x^2-2*b*a*z*x+b^2*z^2)
[1] V=(a*z^3+b*x^3+(a^3*z+b^3*x)*y)/(a*x-b*z)^3;
(b*x^3+b^3*y*x+a^3*z*y+a*z^3)/(a^3*x^3-3*b*a^2*z*x^2+3*b^2*a*z^2*x-b^3*z^3)
[2] F=U^3-U-2*(a^4+b^4)*V^2;
((-3*b^2*a^12-b^6*a^8)*x^14+(28*b^3*a^11+8*b^7*a^7)*z*x^13+((-a^16-2*b^4*a^12-b^8*a^8)*y+(-a^14-119*b^4*a^10-28*b^8*a^6)*z^2)*x^12+((8*b*a^15+16*b^5*a^11+8*b^9*a^7)*z*y+(8*b*a^13+304*b^5*a^9+56*b^9*a^5)*z^3)*x^11+((3*b^2*a^16+4*b^6*a^12+b^10*a^8)*y^2+(-28*b^2*a^14-56*b^6*a^10-28*b^10*a^6)*z^2*y+(-31*b^2*a^12-519*b^6*a^8-70*b^10*a^4)*z^4)*x^10+((-28*b^3*a^15-36*b^7*a^11-8*b^11*a^7)*z*y^2+(56*b^3*a^13+112*b^7*a^9+56*b^11*a^5)*z^3*y+(84*b^3*a^11+624*b^7*a^7+56*b^11*a^3)*z^5)*x^9+((a^20+3*b^4*a^16+3*b^8*a^12+b^12*a^8)*y^3+(a^18+120*b^4*a^14+147*b^8*a^10+28*b^12*a^6)*z^2*y^2+(-a^16-72*b^4*a^12-141*b^8*a^8-70*b^12*a^4)*z^4*y+(-a^14-189*b^4*a^10-546*b^8*a^6-28*b^12*a^2)*z^6)*x^8+((-8*b*a^19-24*b^5*a^15-24*b^9*a^11-8*b^13*a^7)*z*y^3+(-8*b*a^17-312*b^5*a^13-360*b^9*a^9-56*b^13*a^5)*z^3*y^2+(8*b*a^15+72*b^5*a^11+120*b^9*a^7+56*b^13*a^3)*z^5*y+(8*b*a^13+360*b^5*a^9+360*b^9*a^5+8*b^13*a)*z^7)*x^7+((28*b^2*a^18+84*b^6*a^14+84*b^10*a^10+28*b^14*a^6)*z^2*y^3+(28*b^2*a^16+546*b^6*a^12+588*b^10*a^8+70*b^14*a^4)*z^4*y^2+(-28*b^2*a^14-84*b^6*a^10-84*b^10*a^6-28*b^14*a^2)*z^6*y+(-28*b^2*a^12-546*b^6*a^8-189*b^10*a^4-b^14)*z^8)*x^6+((-56*b^3*a^17-168*b^7*a^13-168*b^11*a^9-56*b^15*a^5)*z^3*y^3+(-56*b^3*a^15-672*b^7*a^11-672*b^11*a^7-56*b^15*a^3)*z^5*y^2+(56*b^3*a^13+120*b^7*a^9+72*b^11*a^5+8*b^15*a)*z^7*y+(56*b^3*a^11+624*b^7*a^7+84*b^11*a^3)*z^9)*x^5+((70*b^4*a^16+210*b^8*a^12+210*b^12*a^8+70*b^16*a^4)*z^4*y^3+(70*b^4*a^14+588*b^8*a^10+546*b^12*a^6+28*b^16*a^2)*z^6*y^2+(-70*b^4*a^12-141*b^8*a^8-72*b^12*a^4-b^16)*z^8*y+(-70*b^4*a^10-519*b^8*a^6-31*b^12*a^2)*z^10)*x^4+((-56*b^5*a^15-168*b^9*a^11-168*b^13*a^7-56*b^17*a^3)*z^5*y^3+(-56*b^5*a^13-360*b^9*a^9-312*b^13*a^5-8*b^17*a)*z^7*y^2+(56*b^5*a^11+112*b^9*a^7+56*b^13*a^3)*z^9*y+(56*b^5*a^9+304*b^9*a^5+8*b^13*a)*z^11)*x^3+((28*b^6*a^14+84*b^10*a^10+84*b^14*a^6+28*b^18*a^2)*z^6*y^3+(28*b^6*a^12+147*b^10*a^8+120*b^14*a^4+b^18)*z^8*y^2+(-28*b^6*a^10-56*b^10*a^6-28*b^14*a^2)*z^10*y+(-28*b^6*a^8-119*b^10*a^4-b^14)*z^12)*x^2+((-8*b^7*a^13-24*b^11*a^9-24*b^15*a^5-8*b^19*a)*z^7*y^3+(-8*b^7*a^11-36*b^11*a^7-28*b^15*a^3)*z^9*y^2+(8*b^7*a^9+16*b^11*a^5+8*b^15*a)*z^11*y+(8*b^7*a^7+28*b^11*a^3)*z^13)*x+(b^8*a^12+3*b^12*a^8+3*b^16*a^4+b^20)*z^8*y^3+(b^8*a^10+4*b^12*a^6+3*b^16*a^2)*z^10*y^2+(-b^8*a^8-2*b^12*a^4-b^16)*z^12*y+(-b^8*a^6-3*b^12*a^2)*z^14)/(a^14*x^14-14*b*a^13*z*x^13+91*b^2*a^12*z^2*x^12-364*b^3*a^11*z^3*x^11+1001*b^4*a^10*z^4*x^10-2002*b^5*a^9*z^5*x^9+3003*b^6*a^8*z^6*x^8-3432*b^7*a^7*z^7*x^7+3003*b^8*a^6*z^8*x^6-2002*b^9*a^5*z^9*x^5+1001*b^10*a^4*z^10*x^4-364*b^11*a^3*z^11*x^3+91*b^12*a^2*z^12*x^2-14*b^13*a*z^13*x+b^14*z^14)
[3] F1=nm(F);
(-3*b^2*a^12-b^6*a^8)*x^14+(28*b^3*a^11+8*b^7*a^7)*z*x^13+((-a^16-2*b^4*a^12-b^8*a^8)*y+(-a^14-119*b^4*a^10-28*b^8*a^6)*z^2)*x^12+((8*b*a^15+16*b^5*a^11+8*b^9*a^7)*z*y+(8*b*a^13+304*b^5*a^9+56*b^9*a^5)*z^3)*x^11+((3*b^2*a^16+4*b^6*a^12+b^10*a^8)*y^2+(-28*b^2*a^14-56*b^6*a^10-28*b^10*a^6)*z^2*y+(-31*b^2*a^12-519*b^6*a^8-70*b^10*a^4)*z^4)*x^10+((-28*b^3*a^15-36*b^7*a^11-8*b^11*a^7)*z*y^2+(56*b^3*a^13+112*b^7*a^9+56*b^11*a^5)*z^3*y+(84*b^3*a^11+624*b^7*a^7+56*b^11*a^3)*z^5)*x^9+((a^20+3*b^4*a^16+3*b^8*a^12+b^12*a^8)*y^3+(a^18+120*b^4*a^14+147*b^8*a^10+28*b^12*a^6)*z^2*y^2+(-a^16-72*b^4*a^12-141*b^8*a^8-70*b^12*a^4)*z^4*y+(-a^14-189*b^4*a^10-546*b^8*a^6-28*b^12*a^2)*z^6)*x^8+((-8*b*a^19-24*b^5*a^15-24*b^9*a^11-8*b^13*a^7)*z*y^3+(-8*b*a^17-312*b^5*a^13-360*b^9*a^9-56*b^13*a^5)*z^3*y^2+(8*b*a^15+72*b^5*a^11+120*b^9*a^7+56*b^13*a^3)*z^5*y+(8*b*a^13+360*b^5*a^9+360*b^9*a^5+8*b^13*a)*z^7)*x^7+((28*b^2*a^18+84*b^6*a^14+84*b^10*a^10+28*b^14*a^6)*z^2*y^3+(28*b^2*a^16+546*b^6*a^12+588*b^10*a^8+70*b^14*a^4)*z^4*y^2+(-28*b^2*a^14-84*b^6*a^10-84*b^10*a^6-28*b^14*a^2)*z^6*y+(-28*b^2*a^12-546*b^6*a^8-189*b^10*a^4-b^14)*z^8)*x^6+((-56*b^3*a^17-168*b^7*a^13-168*b^11*a^9-56*b^15*a^5)*z^3*y^3+(-56*b^3*a^15-672*b^7*a^11-672*b^11*a^7-56*b^15*a^3)*z^5*y^2+(56*b^3*a^13+120*b^7*a^9+72*b^11*a^5+8*b^15*a)*z^7*y+(56*b^3*a^11+624*b^7*a^7+84*b^11*a^3)*z^9)*x^5+((70*b^4*a^16+210*b^8*a^12+210*b^12*a^8+70*b^16*a^4)*z^4*y^3+(70*b^4*a^14+588*b^8*a^10+546*b^12*a^6+28*b^16*a^2)*z^6*y^2+(-70*b^4*a^12-141*b^8*a^8-72*b^12*a^4-b^16)*z^8*y+(-70*b^4*a^10-519*b^8*a^6-31*b^12*a^2)*z^10)*x^4+((-56*b^5*a^15-168*b^9*a^11-168*b^13*a^7-56*b^17*a^3)*z^5*y^3+(-56*b^5*a^13-360*b^9*a^9-312*b^13*a^5-8*b^17*a)*z^7*y^2+(56*b^5*a^11+112*b^9*a^7+56*b^13*a^3)*z^9*y+(56*b^5*a^9+304*b^9*a^5+8*b^13*a)*z^11)*x^3+((28*b^6*a^14+84*b^10*a^10+84*b^14*a^6+28*b^18*a^2)*z^6*y^3+(28*b^6*a^12+147*b^10*a^8+120*b^14*a^4+b^18)*z^8*y^2+(-28*b^6*a^10-56*b^10*a^6-28*b^14*a^2)*z^10*y+(-28*b^6*a^8-119*b^10*a^4-b^14)*z^12)*x^2+((-8*b^7*a^13-24*b^11*a^9-24*b^15*a^5-8*b^19*a)*z^7*y^3+(-8*b^7*a^11-36*b^11*a^7-28*b^15*a^3)*z^9*y^2+(8*b^7*a^9+16*b^11*a^5+8*b^15*a)*z^11*y+(8*b^7*a^7+28*b^11*a^3)*z^13)*x+(b^8*a^12+3*b^12*a^8+3*b^16*a^4+b^20)*z^8*y^3+(b^8*a^10+4*b^12*a^6+3*b^16*a^2)*z^10*y^2+(-b^8*a^8-2*b^12*a^4-b^16)*z^12*y+(-b^8*a^6-3*b^12*a^2)*z^14
[4] F2=dn(F);
a^14*x^14-14*b*a^13*z*x^13+91*b^2*a^12*z^2*x^12-364*b^3*a^11*z^3*x^11+1001*b^4*a^10*z^4*x^10-2002*b^5*a^9*z^5*x^9+3003*b^6*a^8*z^6*x^8-3432*b^7*a^7*z^7*x^7+3003*b^8*a^6*z^8*x^6-2002*b^9*a^5*z^9*x^5+1001*b^10*a^4*z^10*x^4-364*b^11*a^3*z^11*x^3+91*b^12*a^2*z^12*x^2-14*b^13*a*z^13*x+b^14*z^14
[5] fctr(F1);
[[-1,1],[a*x-b*z,8],[(3*b^2*a^4+b^6)*x^2-4*b^3*a^3*z*x+(a^8+2*b^4*a^4+b^8)*y+(a^6+3*b^4*a^2)*z^2,1],[x^4+(-a^4-b^4)*y^2+z^4,1]]
[6] fctr(F2);
[[1,1],[a*x-b*z,14]]

同様に、(u,v)を楕円曲線E'_cの有理点とすると、
     u³-u-2cv² = 0
である。
(x:y:z)=φ_a,b^-1(u,v)とすると、(*),(3),(4),(5)より、
     x⁴+z⁴-cy² = x⁴+z⁴-(a⁴+b⁴)y² = -w⁴((a⁴+b⁴)u+2a²b²)³(u³-u-2(a⁴+b⁴)v²)
         = -w⁴(cu+2a²b²)³(u³-u-2cv²) = 0
となる。
よって、(x:y:z)は楕円曲線C_cの有理点である。

[asirによる計算]

[7] X=(a^3*u+b*(a^4+b^4)*v+a*b^2)*w;
a^3*w*u+(b*a^4+b^5)*w*v+b^2*a*w
[8] Y=((a^4+b^4)*u^3+3*a^2*b^2*u^2-(a^4+b^4)^2*v^2+2*a*b*(a^4-b^4)*v-a^2*b^2)*w^2;
(a^4+b^4)*w^2*u^3+3*b^2*a^2*w^2*u^2+(-a^8-2*b^4*a^4-b^8)*w^2*v^2+(2*b*a^5-2*b^5*a)*w^2*v-b^2*a^2*w^2
[9] Z=-(b^3*u-a*(a^4+b^4)*v+a^2*b)*w;
-b^3*w*u+(a^5+b^4*a)*w*v-b*a^2*w
[10] G=X^4+Z^4-(a^4+b^4)*Y^2;
(-a^12-3*b^4*a^8-3*b^8*a^4-b^12)*w^4*u^6+(-6*b^2*a^10-12*b^6*a^6-6*b^10*a^2)*w^4*u^5+(a^12-9*b^4*a^8-9*b^8*a^4+b^12)*w^4*u^4+((2*a^16+8*b^4*a^12+12*b^8*a^8+8*b^12*a^4+2*b^16)*w^4*v^2+(6*b^2*a^10+4*b^6*a^6+6*b^10*a^2)*w^4)*u^3+((12*b^2*a^14+36*b^6*a^10+36*b^10*a^6+12*b^14*a^2)*w^4*v^2+(12*b^4*a^8+12*b^8*a^4)*w^4)*u^2+((24*b^4*a^12+48*b^8*a^8+24*b^12*a^4)*w^4*v^2+8*b^6*a^6*w^4)*u+(16*b^6*a^10+16*b^10*a^6)*w^4*v^2
[11] fctr(G);
[[-1,1],[w,4],[(a^4+b^4)*u+2*b^2*a^2,3],[u^3-u+(-2*a^4-2*b^4)*v^2,1]]

最後に、楕円曲線E'_cが楕円曲線E_cにQ-isomorphicであることは容易に分かる。

■2⁴+3⁴=97より、楕円曲線C: x⁴+z⁴=97y²は楕円曲線
     E: y²=x³-37636x
にQ-isomorphicである。

楕円曲線Cから楕円曲線Eへの双有理変換φは、
     u = 34(4x²+97t_8z²)/(2x-3z)²,
     v = 37636(2x³+8xy+27z³)/(2x-3z)³
[逆変換φ^-1は、
     x = 388(16u+3v+6984)w,
     y = 150544(2u³+432u²-v²-3120v-5419584)w²,
     z = -776(27y-v+2328)w
ただし、wは0でない任意の有理数である。
]
である。

[pari/gpによる計算]

gp>  read("x3m37636x.gp")
time = 102 ms.
gp>  f2(f1([x,y,z]))
time = 46 ms.
%1 = [(776*x^2 + (18818*y + 1746*z^2))/(4*x^2 - 12*z*x + 9*z^2), (75272*x^3 + 301088*y*x + (1016172*z*y + 112908*z^3))/(8*x^3 - 36*z*x^2 + 54*z^2*x - 27*z^3)]
gp>  g1(g2([u,v]))
time = 9 ms.
%2 = [6208*u + (1164*v + 2709792), 301088*u^3 + 65035008*u^2 + (-150544*v^2 - 469697280*v - 815885853696), -20952*u + (776*v - 1806528)]

[asirによる計算]

[0] fctr(776*x^2 + (18818*y + 1746*z^2));
[[194,1],[4*x^2+97*y+9*z^2,1]]
[1] fctr(4*x^2 - 12*z*x + 9*z^2);
[[1,1],[2*x-3*z,2]]
[2] fctr(75272*x^3 + 301088*y*x + (1016172*z*y + 112908*z^3));
[[37636,1],[2*x^3+8*y*x+27*z*y+3*z^3,1]]
[3] fctr((8*x^3 - 36*z*x^2 + 54*z^2*x - 27*z^3));
[[1,1],[2*x-3*z,3]]
[4] fctr(6208*u + (1164*v + 2709792));
[[388,1],[16*u+3*v+6984,1]]
[5] fctr(301088*u^3 + 65035008*u^2 + (-150544*v^2 - 469697280*v - 815885853696));
[[150544,1],[2*u^3+432*u^2-v^2-3120*v-5419584,1]]
[6] fctr(-20952*u + (776*v - 1806528));         
[[-776,1],[27*u-v+2328,1]]

■mwrankによって、E(Q)のrankは2であり、その生成元は(10658/49, -497568/343),(1874137/4356, 2290047101/287496)であることが分かる。
これより、194=2*97は合同数である。

[mwrankによる計算]

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 saturate to obtain generating 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 15 2005 at 07:36:24 by GCC 3.3.3 (NetBSD nb3 20040520)
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,-37636,0]

Curve [0,0,0,-37636,0] :    
3 points of order 2:
[0 : 0 : 1], [194 : 0 : 1], [-194 : 0 : 1]

****************************
* Using 2-isogeny number 1 *
****************************

Using 2-isogenous curve [0,0,0,150544,0]
-------------------------------------------------------
First step, determining 1st descent Selmer groups
-------------------------------------------------------
After first local descent, rank bound = 2
rk(S^{phi}(E'))=      3
rk(S^{phi'}(E))=       1

-------------------------------------------------------
Second step, determining 2nd descent Selmer groups
-------------------------------------------------------
After second local descent, rank bound = 2
rk(phi'(S^{2}(E)))=  3
rk(phi(S^{2}(E')))=    1
rk(S^{2}(E))=  4
rk(S^{2}(E'))= 3

****************************
* Using 2-isogeny number 2 *
****************************

Using 2-isogenous curve [0,-1164,0,37636,0]
-------------------------------------------------------
First step, determining 1st descent Selmer groups
-------------------------------------------------------
After first local descent, rank bound = 2
rk(S^{phi}(E'))=        3
rk(S^{phi'}(E))=       1

-------------------------------------------------------
Second step, determining 2nd descent Selmer groups
-------------------------------------------------------
After second local descent, rank bound = 2
rk(phi'(S^{2}(E)))=  3
rk(phi(S^{2}(E')))=    1
rk(S^{2}(E))=  4
rk(S^{2}(E'))= 3

****************************
* Using 2-isogeny number 3 *
****************************

Using 2-isogenous curve [0,1164,0,37636,0]
-------------------------------------------------------
First step, determining 1st descent Selmer groups
-------------------------------------------------------
After first local descent, rank bound = 2
rk(S^{phi}(E'))= 2
rk(S^{phi'}(E))=       2

-------------------------------------------------------
Second step, determining 2nd descent Selmer groups
-------------------------------------------------------
After second local descent, rank bound = 2
rk(phi'(S^{2}(E)))=  2
rk(phi(S^{2}(E')))=    2
rk(S^{2}(E))=  4
rk(S^{2}(E'))= 3

After second local descent, combined upper bound on rank = 2
Third step, determining E(Q)/phi(E'(Q)) and E'(Q)/phi'(E(Q))
-------------------------------------------------------
1. E(Q)/phi(E'(Q))
-------------------------------------------------------
(c,d)  =(-582,75272)
(c',d')=(1164,37636)
First stage (no second descent yet)...
(97,0,-582,0,776):  (x:y:z) = (2:0:1)
    Curve E         Point [388 : 0 : 1], height = 0
After first global descent, this component of the rank = 2
-------------------------------------------------------
2. E'(Q)/phi'(E(Q))
-------------------------------------------------------
First stage (no second descent yet)...
(2,0,1164,0,18818):  (x:y:z) = (7:284:1)
        Curve E'        Point [98 : 3976 : 1], height = 4.27546911926715
        Curve E         Point [141148 : -497568 : 343], height = 8.55093823853429
(97,0,1164,0,388):  (x:y:z) = (11:1649:3)
      Curve E'        Point [35211 : 1759483 : 27], height = 5.02380690960883
 Curve E         Point [179467266 : 2290047101 : 287496], height = 10.0476138192177
After first global descent, this component of the rank = 2

-------------------------------------------------------
Summary of results:
-------------------------------------------------------
   rank(E) = 2
     #E(Q)/2E(Q) = 16

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,-37636,0]:

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


II. Points on phi(E') mod 2E
Point [74606 : -497568 : 343], height = 8.55093823853429
Point [123693042 : 2290047101 : 287496], height = 10.0476138192177
-------------------------------------------------------
Computing full set of 4 coset representatives for
2E(Q) in E(Q) (modulo torsion), and sorting into height order....done.

Rank = 2

Regulator (before saturation) = 38.6439672225803
Saturating...finished saturation (index was 1)
Regulator (after saturation) = 38.6439672225803

Generator 1 is [74606 : -497568 : 343]; height 8.55093823853429
Generator 2 is [123693042 : 2290047101 : 287496]; height 10.0476138192177

Regulator = 38.6439672225803

The rank and full Mordell-Weil basis have been determined unconditionally.
 (2.52 seconds)

■楕円曲線Eの有理点から、楕円曲線Cの有理点(x:y:z)をいくつか求める。

[pari/gpによる計算]

gp> read("x3m37636x.gp")
time = 94 ms.
gp> rpCC(2)
[13731978001341, -19493235973922211842773249, 6005859158666]
[-6005859158666, 19493235973922211842773249, -13731978001341]
[6005859158666, -19493235973922211842773249, -13731978001341]
[13731978001341, 19493235973922211842773249, -6005859158666]
[18388885479, -173523910661376094249, 40929505802]
[40929505802, 173523910661376094249, 18388885479]
[40929505802, -173523910661376094249, -18388885479]
[18388885479, 173523910661376094249, -40929505802]
[33745754578245579, -431663975437045045497682121766001, -64000334486210114]
[64000334486210114, 431663975437045045497682121766001, -33745754578245579]
[64000334486210114, -431663975437045045497682121766001, 33745754578245579]
[-33745754578245579, 431663975437045045497682121766001, -64000334486210114]
[56569757228031167875937254143633, -327658635516556991085437645060738623030879938023695035102969049, -20396137263685122406579335894674]
[20396137263685122406579335894674, 327658635516556991085437645060738623030879938023695035102969049, -56569757228031167875937254143633]
[20396137263685122406579335894674, -327658635516556991085437645060738623030879938023695035102969049, 56569757228031167875937254143633]
[56569757228031167875937254143633, 327658635516556991085437645060738623030879938023695035102969049, 20396137263685122406579335894674]
[24052825399614791904069864287243753481216207558017402451, -62918275086490740218345834579295104422806842856036937706810293064180139547079016269258629412414583434157284449, 14899975389758418095917019066232540644116785380338268806]
[-14899975389758418095917019066232540644116785380338268806, 62918275086490740218345834579295104422806842856036937706810293064180139547079016269258629412414583434157284449, -24052825399614791904069864287243753481216207558017402451]
[14899975389758418095917019066232540644116785380338268806, -62918275086490740218345834579295104422806842856036937706810293064180139547079016269258629412414583434157284449, -24052825399614791904069864287243753481216207558017402451]
[24052825399614791904069864287243753481216207558017402451, 62918275086490740218345834579295104422806842856036937706810293064180139547079016269258629412414583434157284449, -14899975389758418095917019066232540644116785380338268806]
[11619875966182, -25318076626587457417229929, -14478764558151]
[14478764558151, 25318076626587457417229929, -11619875966182]
[14478764558151, -25318076626587457417229929, 11619875966182]
[-11619875966182, 25318076626587457417229929, -14478764558151]
[54406, -300565249, -5949]
[5949, 300565249, -54406]
[0, 0, 0]
[54406, 300565249, 5949]
[74734, -737072569, 68097]
[-68097, 737072569, -74734]
[68097, -737072569, -74734]
[74734, 737072569, -68097]
[-1836778949566, -720486501472037686558646161, -84237570591099]
[84237570591099, 720486501472037686558646161, 1836778949566]
[84237570591099, -720486501472037686558646161, -1836778949566]
[1836778949566, 720486501472037686558646161, -84237570591099]
[31951294023088672460831208254858, -142242879984709938910764584129129745915970580820764300785264009, -30973759198706025026367910862409]
[30973759198706025026367910862409, 142242879984709938910764584129129745915970580820764300785264009, -31951294023088672460831208254858]
[30973759198706025026367910862409, -142242879984709938910764584129129745915970580820764300785264009, 31951294023088672460831208254858]
[-31951294023088672460831208254858, 142242879984709938910764584129129745915970580820764300785264009, -30973759198706025026367910862409]
[358122449396177141733, -25777209037795093896542436989745766453969, 468080583507025844042]
[-468080583507025844042, 25777209037795093896542436989745766453969, -358122449396177141733]
[468080583507025844042, -25777209037795093896542436989745766453969, -358122449396177141733]
[358122449396177141733, 25777209037795093896542436989745766453969, -468080583507025844042]
[418641, -340219290841, -1829258]
[1829258, 340219290841, -418641]
[1829258, -340219290841, 418641]
[-418641, 340219290841, -1829258]
[3, -1, -2]
[2, 1, -3]
[2, -1, 3]
[3, 1, 2]
[3177, -1029961, 1006]
[-1006, 1029961, -3177]
[1006, -1029961, -3177]
[3177, 1029961, -1006]
[887025399231723, -256226403773975592243907793329, 1548464267793862]
[1548464267793862, 256226403773975592243907793329, 887025399231723]
[1548464267793862, -256226403773975592243907793329, -887025399231723]
[887025399231723, 256226403773975592243907793329, -1548464267793862]
[445539889251174225869962977142272742, -20424405403352364368921333659808721007918872087493754099233468208065081, -180420965179935149446517071093851519]
[180420965179935149446517071093851519, 20424405403352364368921333659808721007918872087493754099233468208065081, -445539889251174225869962977142272742]
[180420965179935149446517071093851519, -20424405403352364368921333659808721007918872087493754099233468208065081, 445539889251174225869962977142272742]
[445539889251174225869962977142272742, 20424405403352364368921333659808721007918872087493754099233468208065081, 180420965179935149446517071093851519]
[1548464267793862, -256226403773975592243907793329, 887025399231723]
[-887025399231723, 256226403773975592243907793329, -1548464267793862]
[887025399231723, -256226403773975592243907793329, -1548464267793862]
[1548464267793862, 256226403773975592243907793329, -887025399231723]
[1006, -1029961, 3177]
[3177, 1029961, 1006]
[3177, -1029961, -1006]
[1006, 1029961, -3177]
[2, -1, -3]
[3, 1, -2]
[3, -1, 2]
[-2, 1, -3]
[1829258, -340219290841, -418641]
[418641, 340219290841, -1829258]
[418641, -340219290841, 1829258]
[1829258, 340219290841, 418641]
[-692806265698098549054195494786458406950237439726490308301, -11278897633514402472328948801409984782306934429312697836674636242259666994973496089361366248140168808059572211130529, -10539602553331836125484460337851563697672039740217449663626]
[10539602553331836125484460337851563697672039740217449663626, 11278897633514402472328948801409984782306934429312697836674636242259666994973496089361366248140168808059572211130529, 692806265698098549054195494786458406950237439726490308301]
[10539602553331836125484460337851563697672039740217449663626, -11278897633514402472328948801409984782306934429312697836674636242259666994973496089361366248140168808059572211130529, -692806265698098549054195494786458406950237439726490308301]
[692806265698098549054195494786458406950237439726490308301, 11278897633514402472328948801409984782306934429312697836674636242259666994973496089361366248140168808059572211130529, -10539602553331836125484460337851563697672039740217449663626]
[30973759198706025026367910862409, -142242879984709938910764584129129745915970580820764300785264009, -31951294023088672460831208254858]
[31951294023088672460831208254858, 142242879984709938910764584129129745915970580820764300785264009, -30973759198706025026367910862409]
[31951294023088672460831208254858, -142242879984709938910764584129129745915970580820764300785264009, 30973759198706025026367910862409]
[-30973759198706025026367910862409, 142242879984709938910764584129129745915970580820764300785264009, -31951294023088672460831208254858]
[84237570591099, -720486501472037686558646161, 1836778949566]
[-1836778949566, 720486501472037686558646161, -84237570591099]
[1836778949566, -720486501472037686558646161, -84237570591099]
[84237570591099, 720486501472037686558646161, -1836778949566]
[68097, -737072569, 74734]
[-74734, 737072569, -68097]
[74734, -737072569, -68097]
[68097, 737072569, -74734]
[5949, -300565249, -54406]
[54406, 300565249, -5949]
[54406, -300565249, 5949]
[-5949, 300565249, -54406]
time = 263 ms.

■不定方程式x⁴+z⁴=97y²の正整数解x,y,zで、primitiveなもの、つまり、gcd(x,y,z)=1となるものは、無数に存在する。

[参考文献]

[1]J. W. S. Cassels, E. V. Flynn, "Prolegomena to a Middlebrow Arithmetic of Curves of Genus 2", LMSLNS 230, Cambrdge University Press, 2005, ISBN0-521-48370-0.
[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}.

Last Update: 2005.08.21

H.Nakao