Rational Points on Elliptic Curves: u^3+v^3=u+v+1, y^2=x^3+12x^2-432

[2004.03.05]u^3+v^3=u+v+1, y^2=x^3+12x^2+432の有理点

u^3+v^3=u+v+a, y^2=x^3+12x^2-432a^2 (a!=0)の有理点を計算するプログラム(pari/gp)
u^3+v^3=u+v+1のグラフ(jpeg)
y^2=x^3+12x^2-432のグラフ(jpeg)

■Diophantus方程式
     C: u³+v³=u+v+1
で表される曲線の有理点(x,y)を求める。

■曲線Cは、整点を持たない。
なぜならば、u,vを整数とすると、
     u³≡u (mod 2),
     v³≡v (mod 2)
より、
     u³+v³≡u+v ≠ u+v+1 (mod 2)
を得るので、明らかである。

■曲線Cの無限遠点O_Cは、[1:-1:0]である。

■３次曲線Cは、楕円曲線である。
双有理変換ψ:(u,v)→((12/(u+v),36(u-v)/(u+v))
[逆変換は、φ:(x,y)→({36+y}/{6x},{36-y}/{6x})]
によって、曲線Cは、楕円曲線
     E: y² = x³+12x²-432
に写される。
ただし、ψ(O_C)=O, φ(O)=O_Cとする。ここで、O=[0:1:0]は、曲線Eの無限遠点である。

φとψは有理変換であり、しかも互いに逆変換であることは、容易に確かめられる。
[pari/gpによる計算]

gp>  read("./de12.gp") 
time = 23 ms.
gp>  cc(1,u,v)
time = 2 ms.
%1 = u^3 - u + (v^3 - v - 1)
gp>  cc(1,(36+y)/(6*x),(36-y)/(6*x))
time = 39 ms.
%2 = (-x^3 - 12*x^2 + (y^2 + 432))/x^3
gp>  cc(1,(36+y)/(6*x),(36-y)/(6*x))*x^3
time = 12 ms.
%3 = -x^3 - 12*x^2 + (y^2 + 432)
gp>  ee(1,x,y)
time = 0 ms.
%4 = x^3 + 12*x^2 + (-y^2 - 432)
gp>  ee(1,12/(u+v),36*(u-v)/(u+v))
time = 20 ms.
%5 = (-1728*u^3 + 1728*u + (-1728*v^3 + 1728*v + 1728))/(u^3 + 3*v*u^2 + 3*v^2*u + v^3)
gp>  ee(1,12/(u+v),36*(u-v)/(u+v))*(u+v)^3/1728
time = 14 ms.
%6 = -u^3 + u + (-v^3 + v + 1)

■楕円曲線Eのねじれ点群E_tors(Q)は、自明である。

pari/gpで計算すると、以下のようになる。
E_tors(Q) = { O }

[pari/gpでの計算結果]

gp>  e=ec(1)
time = 171 ms.
%7 = [0, 12, 0, 0, -432, 48, 0, -1728, -20736, 2304, 262656, -32845824, -4096/11, [5.035720531284966795311115387, -8.517860265642483397655557694 + 3.637744375243196215556053182*I, -8.517860265642483397655557694 - 3.637744375243196215556053182*I]~, 0.8422481664877393933750180083, 0.4211240832438696966875090041 + 0.9159729168547222548225389537*I, -2.003366971585723359916258856 + 3.37962245 E-29*I, -1.001683485792861679958129428 - 5.908736569693707298010327406*I, 0.7714765097733163823783361300]
gp>  elltors(e,1)
time = 53 ms.
%8 = [1, [], []]

■楕円曲線EのMordell-Weil群E(Q)をCremonaのmwrank3で計算すると、rankは0である。

E(Q) = {O}

つまり、Eの有理点は、無限遠点O=[0:1:0]に限る。

[mwrank3での計算結果]

bash-2.05a$ mwrank3 -b 15 -c 15
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, 12, 0, 0, -432]

Curve [0,12,0,0,-432] : Working with minimal curve [0,0,1,-3,-5]
        [u,r,s,t] = [2,-4,0,4]
No points of order 2
Basic pair: I=144, J=8208
disc=-55427328
2-adic index bound = 2
By Lemma 5.1(a), 2-adic index = 1
2-adic index = 1
One (I,J) pair
Looking for quartics with I = 144, J = 8208
Looking for Type 3 quartics:
Trying positive a from 1 up to 2 (square a first...)
Trying positive a from 1 up to 2 (...then non-square a)
Trying negative a from -1 down to -5
(-3,0,-6,12,-3) --trivial
(-3,0,12,12,0)  --trivial
Finished looking for Type 3 quartics.
Mordell rank contribution from B=im(eps) = 0
Selmer  rank contribution from B=im(eps) = 0
Sha     rank contribution from B=im(eps) = 0
Mordell rank contribution from A=ker(eps) = 0
Selmer  rank contribution from A=ker(eps) = 0
Sha     rank contribution from A=ker(eps) = 0
Rank = 0
After descent, rank of points found is 0


The rank and full Mordell-Weil basis have been determined unconditionally.
Regulator = 1

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

bash-2.05a$

■定数aを0でない整数とするとき、曲線C_a: u³+v³=u+v+a について、考察する。

曲線C_aは、双有理変換ψ:(u,v)→((12a/(u+v),36a(u-v)/(u+v))
[逆変換は、φ:(x,y)→({36a+y}/{6x},{36a-y}/{6x})]
によって、楕円曲線
E_a: y² = x³+12x²-432a²
に写される。

gp>  cc(a,u,v)
time = 1 ms.
%14 = -a + (u^3 - u + (v^3 - v))
gp>  cc(a,(36*a+y)/(6*x),(36*a-y)/(6*x))
time = 4 ms.
%15 = (-a*x^3 - 12*a*x^2 + (432*a^3 + y^2*a))/x^3
gp> cc(a,(36*a+y)/(6*x),(36*a-y)/(6*x))*x^3/a
time = 14 ms.
%16 = -x^3 - 12*x^2 + (432*a^2 + y^2)
gp> ee(a,x,y)
time = 0 ms.
%17 = x^3 + 12*x^2 + (-432*a^2 - y^2)
gp>  ee(a,12*a/(u+v),36*a*(u-v)/(u+v))
time = 14 ms.
%18 = 1728/(u^3 + 3*v*u^2 + 3*v^2*u + v^3)*a^3 + ((-1728*u^2 + 1728*v*u + (-1728*v^2 + 1728))/(u^2 + 2*v*u + v^2))*a^2
gp> ee(a,12*a/(u+v),36*a*(u-v)/(u+v))*(u+v)^3/(1728*a^2)
time = 15 ms.
%19 = a + (-u^3 + u + (-v^3 + v))

■a=2,3,...,10について、楕円曲線E_aのねじれ点群E_a(Q)_torsとMordell-Weil群E_a(Q)を求めると、以下のようになる。

     E_a(Q)_tors = {O} (a=2,3,...,10)

     E₂(Q) = E₃(Q) = E₅(Q) = E₇(Q) = E₈(Q) = {O}

     E₄(Q) = Z = { n(16,16) | n ∈ Z }

     E₆(Q) = Z² = { m(36,216)+n(504,11448) | m,n ∈ Z }

     E₉(Q) = Z = { n(189,2673) | n ∈ Z }

     E₁₀(Q) = Z = { n(40,200) | n ∈ Z }

よって、曲線C₂,C₃,C₅,C₇,C₈の有理点は、無限遠点に限る。

[pari/gpによる計算]

[mwrank3による計算結果]

■pari/gpで、楕円曲線C₄の有理点をいくつか計算すると、以下のようになる。

gp>  read("./de12.gp")
time = 32 ms.
gp>  rpCr1(4,[16,16],5)
[1, -1, 0]
[5/3, 4/3]
[4/3, 5/3]
[-5546/939, 5582/939]
[5582/939, -5546/939]
[179952964/98826543, 85579697/98826543]
[85579697/98826543, 179952964/98826543]
[-933453019488391/320426198265576, 983061263177863/320426198265576]
[983061263177863/320426198265576, -933453019488391/320426198265576]
[192957101295433209853529/105376465833855748105623, 37134986745105218337988/105376465833855748105623]
[37134986745105218337988/105376465833855748105623, 192957101295433209853529/105376465833855748105623]
time = 324 ms.
gp>  cc(4,37134986745105218337988/105376465833855748105623, 19295710129433209853529/105376465833855748105623)
time = 8 ms.
%1 = 0

■pari/gpで、楕円曲線C₆の有理点をいくつか計算すると、以下のようになる。

gp>  rpCr2(6,[36,216],[504,11448],10,10^10)
[1, -1, 0]
[1, 2]
[-1, 2]
[0, 2]
[2, 1]
[2, 0]
[2, -1]
[-26/7, 27/7]
[27/7, -26/7]
[-41/26, 57/26]
[57/26, -41/26]
[-67/19, 70/19]
[70/19, -67/19]
[-11/222, 443/222]
[-326/147, 380/147]
[380/147, -326/147]
[443/222, -11/222]
[656/441, 802/441]
[802/441, 656/441]
[-6018/6331, 12609/6331]
[7361/4820, 8639/4820]
[8639/4820, 7361/4820]
[6270/11563, 23521/11563]
[12609/6331, -6018/6331]
[14802/14105, 28073/14105]
[23521/11563, 6270/11563]
[28073/14105, 14802/14105]
[-54937/23824, 63129/23824]
[-61811/8945, 62186/8945]
[62186/8945, -61811/8945]
[63129/23824, -54937/23824]
[-96558/170581, 335079/170581]
[335079/170581, -96558/170581]
[-402242/5449, 402244/5449]
[402244/5449, -402242/5449]
[-516965/997451, 1959862/997451]
[1959862/997451, -516965/997451]
[-22539382/13756839, 30578740/13756839]
[30578740/13756839, -22539382/13756839]
[32707830/34425839, 69139733/34425839]
[-57468898/9100091, 57925431/9100091]
[57925431/9100091, -57468898/9100091]
[69139733/34425839, 32707830/34425839]
[74350613/163488108, 332244331/163488108]
[-241405818/229925831, 462085995/229925831]
[-273704198/33288151, 274690276/33288151]
[274690276/33288151, -273704198/33288151]
[-325481776/214532547, 464334838/214532547]
[332244331/163488108, 74350613/163488108]
[462085995/229925831, -241405818/229925831]
[464334838/214532547, -325481776/214532547]
[-412222386/850138001, 1671067972/850138001]
[543950573/1108452419, 2253773926/1108452419]
[1671067972/850138001, -412222386/850138001]
[2253773926/1108452419, 543950573/1108452419]
time = 366 ms.

■pari/gpで、楕円曲線C₉の有理点をいくつか計算すると、以下のようになる。

gp>  rpCr1(9,[189,2673],5)
[1, -1, 0]
[37/14, -29/14]
[-29/14, 37/14]
[180317953/80309196, 5979455/80309196]
[5979455/80309196, 180317953/80309196]
[911679550522682189/517158410106559594, 999403886519066507/517158410106559594]
[999403886519066507/517158410106559594, 911679550522682189/517158410106559594]
[-87559024075808338856825953064063/482481685439482695928863141665304, 1074717122217510506255989589980799/482481685439482695928863141665304]
[1074717122217510506255989589980799/482481685439482695928863141665304, -87559024075808338856825953064063/482481685439482695928863141665304]
[-1120225366831202779378484454344255215538267027181995/452959141845790295832995020131236410328837853649286, 1316647308051917105841077873485397133556710641579923/452959141845790295832995020131236410328837853649286]
[1316647308051917105841077873485397133556710641579923/452959141845790295832995020131236410328837853649286, -1120225366831202779378484454344255215538267027181995/452959141845790295832995020131236410328837853649286]
time = 32 ms.

■pari/gpで、楕円曲線C₁₀の有理点をいくつか計算すると、以下のようになる。

gp>  rpCr1(10,[40,200],5)
[1, -1, 0]
[7/3, 2/3]
[2/3, 7/3]
[-14456/10815, 25706/10815]
[25706/10815, -14456/10815]
[4868971082/1732183983, -3928587083/1732183983]
[-3928587083/1732183983, 4868971082/1732183983]
[283203439774114529/201352048108357830, 445580037455575471/201352048108357830]
[445580037455575471/201352048108357830, 283203439774114529/201352048108357830]
[-2488943469515908163629592977/375202885868437237002084657, 2517256161127537730891542798/375202885868437237002084657]
[2517256161127537730891542798/375202885868437237002084657, -2488943469515908163629592977/375202885868437237002084657]
time = 6 ms.

[参考文献]

[1]R.J.Stroeker, Benjamin M.M. de Weger, "Solving Elliptic Diophantine Equations: The General Cubic Case", 1991, p1-28.
[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: 2005.08.20

H.Nakao