Rational Points on Elliptic Curve: x^4+z^4=17y^2

[2005.08.14]x^4+z^4=17y^2の有理点

x^4+z^4=17y^2, v^2=u^3-34^2*uの有理点を求めるプログラム(pari/gp)

Diophantus方程式
     C: x⁴+z⁴=17y²
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。

参考文献[1](p.188)によると、楕円曲線Cは、双有理変換φ
     u = (z²+4x²+17y)/(x-2z)², ------ (1)
     v = (z³+2x³+(z+8x)y)/(x-2z)³ ----- (2)
によって、楕円曲線
     E₁: 34v² = u³-u
に写される。

■双有理変換φの逆変換φ^-1を求める。
(1)×(z+8x)-(2)×17(x-2z)より、
     (z+8x)u-17(x-2z)v = -2(x+2z)
     (8u-17v+2)x+(u+34v+4)z = 0
を得る。これより、ある有理数wに対して、
     x = -(u+34v+4)w, ------ (3)
     z = (8u-17v+2)w ------ (4)
と表すことができる。
(3),(4)を(1)に代入して、yを求めると、
     y = (17u³+12u²-289v²-60v-4)w² ------ (5)
を得る。
よって、φ^-1: E₁→C, (u,v)→(x:y:z)は、
     x = -(u+34v+4)w,
     y = (17u³+12u²-289v²-60v-4)w²,
     z = (8u-17v+2)w
である。

■楕円曲線Cと楕円曲線E₁がQ-isomorphicであることを示す。

(u,v)が楕円曲線E₁の有理点であると仮定すると、
     u³-u-34v² = 0
である。このとき、(x:y:z)を(3),(4),(5)で定義されるものとすると、
     x⁴+z⁴-17y² = -w⁴(17u+8)³(u³-u-34v²) = 0
となるので、(x:y:z)は楕円曲線Cの有理点である。
よって、有理変換φ^-1によって、楕円曲線E₁は楕円曲線Cに写される。
[asirによる計算]

[0] X=-(u+34*v+4)*w;
-w*u-34*w*v-4*w
[1] Z=(8*u-17*v+2)*w;
8*w*u-17*w*v+2*w
[2] Y=(17*u^3+12*u^2-289*v^2-60*v-4)*w^2;
17*w^2*u^3+12*w^2*u^2-289*w^2*v^2-60*w^2*v-4*w^2
[3] F=X^4+Z^4-17*Y^2;
-4913*w^4*u^6-6936*w^4*u^5+1649*w^4*u^4+(167042*w^4*v^2+6424*w^4)*u^3+(235824*w^4*v^2+3264*w^4)*u^2+(110976*w^4*v^2+512*w^4)*u+17408*w^4*v^2
[4] fctr(F);
[[-1,1],[w,4],[17*u+8,3],[u^3-u-34*v^2,1]]

次に、(x:y:z)が楕円曲線Cの有理点であると仮定すると、
x⁴+z⁴-17y² = 0
である。
(u,v)を(1),(2)で定義されたものとすると、
u³-u-34v² = -(76x²-32xz+289y+49z²)(x⁴-17y²+z⁴)/(x-2z)⁶ = 0
となる。よって、(u,v)は楕円曲線E₁の有理点である。
双有理変換φ:C→E₁, (x:y:z)→(u,v)によって、楕円曲線Cは楕円曲線E₁に写される。

[asirによる計算]

[5] U=(z^2+4*x^2+17*y)/(x-2*z)^2;
(4*x^2+17*y+z^2)/(x^2-4*z*x+4*z^2)
[6] V=(z^3+2*x^3+(z+8*x)*y)/(x-2*z)^3;
(2*x^3+8*y*x+z*y+z^3)/(x^3-6*z*x^2+12*z^2*x-8*z^3)
[7] G=(U^3-U-34*V^2);
(-76*x^14+1248*z*x^13+(-289*y-9073*z^2)*x^12+(4624*z*y+38416*z^3)*x^11+(1292*y^2-32368*z^2*y-105020*z^4)*x^10+(-21216*z*y^2+129472*z^3*y+195232*z^5)*x^9+(4913*y^3+154241*z^2*y^2-323969*z^4*y-257489*z^6)*x^8+(-78608*z*y^3-653072*z^3*y^2+522512*z^5*y+261392*z^7)*x^7+(550256*z^2*y^3+1784048*z^4*y^2-550256*z^6*y-244976*z^8)*x^6+(-2201024*z^3*y^3-3297728*z^5*y^2+425408*z^7*y+252352*z^9)*x^5+(5502560*z^4*y^3+4223072*z^6*y^2-397664*z^8*y-260960*z^10)*x^4+(-8804096*z^5*y^3-3790592*z^7*y^2+517888*z^9*y+222976*z^11)*x^3+(8804096*z^6*y^3+2380544*z^8*y^2-517888*z^10*y-140032*z^12)*x^2+(-5030912*z^7*y^3-992256*z^9*y^2+295936*z^11*y+58368*z^13)*x+1257728*z^8*y^3+213248*z^10*y^2-73984*z^12*y-12544*z^14)/(x^14-28*z*x^13+364*z^2*x^12-2912*z^3*x^11+16016*z^4*x^10-64064*z^5*x^9+192192*z^6*x^8-439296*z^7*x^7+768768*z^8*x^6-1025024*z^9*x^5+1025024*z^10*x^4-745472*z^11*x^3+372736*z^12*x^2-114688*z^13*x+16384*z^14)
[8] G1=nm(G);
-76*x^14+1248*z*x^13+(-289*y-9073*z^2)*x^12+(4624*z*y+38416*z^3)*x^11+(1292*y^2-32368*z^2*y-105020*z^4)*x^10+(-21216*z*y^2+129472*z^3*y+195232*z^5)*x^9+(4913*y^3+154241*z^2*y^2-323969*z^4*y-257489*z^6)*x^8+(-78608*z*y^3-653072*z^3*y^2+522512*z^5*y+261392*z^7)*x^7+(550256*z^2*y^3+1784048*z^4*y^2-550256*z^6*y-244976*z^8)*x^6+(-2201024*z^3*y^3-3297728*z^5*y^2+425408*z^7*y+252352*z^9)*x^5+(5502560*z^4*y^3+4223072*z^6*y^2-397664*z^8*y-260960*z^10)*x^4+(-8804096*z^5*y^3-3790592*z^7*y^2+517888*z^9*y+222976*z^11)*x^3+(8804096*z^6*y^3+2380544*z^8*y^2-517888*z^10*y-140032*z^12)*x^2+(-5030912*z^7*y^3-992256*z^9*y^2+295936*z^11*y+58368*z^13)*x+1257728*z^8*y^3+213248*z^10*y^2-73984*z^12*y-12544*z^14
[9] fctr(G1);
[[-1,1],[x-2*z,8],[76*x^2-32*z*x+289*y+49*z^2,1],[x^4-17*y^2+z^4,1]]
[10] G2=dn(G);
x^14-28*z*x^13+364*z^2*x^12-2912*z^3*x^11+16016*z^4*x^10-64064*z^5*x^9+192192*z^6*x^8-439296*z^7*x^7+768768*z^8*x^6-1025024*z^9*x^5+1025024*z^10*x^4-745472*z^11*x^3+372736*z^12*x^2-114688*z^13*x+16384*z^14
[11] fctr(G2);
[[1,1],[x-2*z,14]]

■楕円曲線E₁は楕円曲線E:y^2=x(x^2-34^2)にQ-isomorphicである。
双有理変換ψ:E₁→E, (u,v)→(U,V)
     U = 34u,
     V = 34²v
[逆変換ψ^-1:E→E₁, (U,V)→(u,v)は
     u = U/34,
     v = V/34²
である。
]
によって、E₁とEは互いに写し合う。

■mwrankによって、E(Q)のrankは2であり、その生成元は(162,-2016),(153/4,-867/8)であることが分かる。
これより、34=2*17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,-1156,0]

Curve [0,0,0,-1156,0] :     
3 points of order 2:
[0 : 0 : 1], [34 : 0 : 1], [-34 : 0 : 1]

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

Using 2-isogenous curve [0,0,0,4624,0]
-------------------------------------------------------
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,-204,0,1156,0]
-------------------------------------------------------
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 [0,204,0,1156,0]
-------------------------------------------------------
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)  =(-102,2312)
(c',d')=(204,1156)
First stage (no second descent yet)...
(17,0,-102,0,136):  (x:y:z) = (2:0:1)
       Curve E         Point [68 : 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)...
(2,0,204,0,578):  (x:y:z) = (1:28:1)
     Curve E'        Point [2 : 56 : 1], height = 2.06859046682197
   Curve E         Point [196 : -2016 : 1], height = 4.13718093364394
(17,0,204,0,68):  (x:y:z) = (1:17:1)
  Curve E'        Point [17 : 289 : 1], height = 1.25648009037909
 Curve E         Point [578 : -867 : 8], height = 2.51296018075818
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,-1156,0]:

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


II. Points on phi(E') mod 2E
Point [162 : -2016 : 1], height = 4.13718093364394
Point [306 : -867 : 8], height = 2.51296018075818
-------------------------------------------------------
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) = 7.09967518240335
Saturating...finished saturation (index was 1)
Regulator (after saturation) = 7.09967518240335

Generator 1 is [162 : -2016 : 1]; height 4.13718093364394
Generator 2 is [306 : -867 : 8]; height 2.51296018075818

Regulator = 7.09967518240335

The rank and full Mordell-Weil basis have been determined unconditionally.
 (2.25 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("x3m1156x.gp")
time = 29 ms.
gp>  rpCC(2)
[-59780514843800714, -866837137379721943596367948910689, -7077947952036623]
[7077947952036623, 866837137379721943596367948910689, 59780514843800714]
[7077947952036623, -866837137379721943596367948910689, -59780514843800714]
[-59780514843800714, 866837137379721943596367948910689, 7077947952036623]
[-4899867526, -27710071709800439849, 10568845261]
[-10568845261, 27710071709800439849, 4899867526]
[-10568845261, -27710071709800439849, -4899867526]
[4899867526, 27710071709800439849, 10568845261]
[-262238, -20406039889, -220169]
[220169, 20406039889, 262238]
[-220169, -20406039889, 262238]
[-262238, 20406039889, 220169]
[-1186, -385241, 859]
[-859, 385241, 1186]
[-859, -385241, -1186]
[1186, 385241, 859]
[-314, -182209, -863]
[-863, 182209, -314]
[-863, -182209, 314]
[-314, 182209, 863]
[-1623999547, -665665180513001369, 871601578]
[-871601578, 665665180513001369, 1623999547]
[-871601578, -665665180513001369, -1623999547]
[1623999547, 665665180513001369, 871601578]
[-2297, -35732401, -12134]
[-12134, 35732401, -2297]
[-12134, -35732401, 2297]
[-2297, 35732401, 12134]
[-13, -41, -2]
[2, 41, 13]
[2, -41, -13]
[-13, 41, 2]
[-1, -1, 2]
[-2, 1, 1]
[-2, -1, -1]
[1, 1, 2]
[-43, -569, -38]
[38, 569, 43]
[-38, -569, 43]
[-43, 569, 38]
[-120854, -8344969681, 176503]
[-176503, 8344969681, 120854]
[-176503, -8344969681, -120854]
[120854, 8344969681, 176503]
[-38, -569, -43]
[43, 569, 38]
[-43, -569, 38]
[-38, 569, 43]
[-2, -1, 1]
[-1, 1, 2]
[-1, -1, -2]
[2, 1, 1]
[-2, -41, -13]
[-13, 41, -2]
[-13, -41, 2]
[-2, 41, 13]
[-12134, -35732401, -2297]
[2297, 35732401, 12134]
[2297, -35732401, -12134]
[-12134, 35732401, 2297]
[1789, -1921729289, -89014]
[-89014, 1921729289, 1789]
[-89014, -1921729289, -1789]
[1789, 1921729289, 89014]
[-863, -182209, -314]
[314, 182209, 863]
[0, 0, 0]
[-863, 182209, 314]
[-859, -385241, 1186]
[-1186, 385241, 859]
[-1186, -385241, -859]
[859, 385241, 1186]
[-220169, -20406039889, -262238]
[262238, 20406039889, 220169]
[-262238, -20406039889, 220169]
[-220169, 20406039889, 262238]
[-10568845261, -27710071709800439849, 4899867526]
[-4899867526, 27710071709800439849, 10568845261]
[-4899867526, -27710071709800439849, -10568845261]
[-10568845261, 27710071709800439849, -4899867526]
[-90196534, -5194443631595329, -140754913]
[140754913, 5194443631595329, 90196534]
[-140754913, -5194443631595329, 90196534]
[-90196534, 5194443631595329, 140754913]
[-57890234, -815586990800969, 16659299]
[-16659299, 815586990800969, 57890234]
[-16659299, -815586990800969, -57890234]
[-57890234, 815586990800969, -16659299]
[188556062, -2795522720147533489, -3395022919]
[-3395022919, 2795522720147533489, 188556062]
[-3395022919, -2795522720147533489, -188556062]
[188556062, 2795522720147533489, 3395022919]
[-28660544052254, -201750776049558579509482361, -11454586570699]
[11454586570699, 201750776049558579509482361, 28660544052254]
[-11454586570699, -201750776049558579509482361, 28660544052254]
[-28660544052254, 201750776049558579509482361, 11454586570699]
[-26144120221929475706, -328335108102132706400135115387234007969, 34183842481405005073]
[-34183842481405005073, 328335108102132706400135115387234007969, 26144120221929475706]
[-34183842481405005073, -328335108102132706400135115387234007969, -26144120221929475706]
[26144120221929475706, 328335108102132706400135115387234007969, 34183842481405005073]
time = 243 ms.

■不定方程式x⁴+z⁴=17y²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。

■ここで求めた双有理変換φおよびφ^-1は、a,bを正有理数の定数とするとき、楕円曲線C_a,b: x^4+z^4=(a^4+b^4)y^2と楕円曲線E_a,b: 2(a^4+b^4)v^2=u^3-uの間のQ-isomorphism φ_a,b,φ_a,b^-1に拡張できる。

[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