Rational Points on Elliptic Curves: x^2-3y^4=1, v^2=u^3-12u
[2003.09.01]x^2-3y^4=1, v^2=u^3-12uの有理点
■Diophantus方程式
C: x2-3y4 = 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、自明な整点(±1,0)を持つ。
■双有理変換φ:(x,y)→(4(x+1)/y3,2(x+1)/y2)[逆変換は、φ-1:(u,v)→((2u3-v2)/v2,2u/v)]によって、曲線Cは、楕円曲線
E: v2 = u3-12u ----- (2)
に写される。
ただし、φ(-1,0)=(0,0), φ(1,0)=Oとする。ここで、O=[0:1:0]は、曲線Eの無限遠点である。
この双有理変換については、こちらを参照のこと。
楕円曲線Eは、自明な有理点(0,0)を持つ。
■楕円曲線Eのねじれ点群E(Q)torsは、 Z/2Zである。
pari/gpで計算すると、以下のようになる。
E(Q)tors = Z/2Z = { (0,0), O}
E(Q)torsの生成元(位数2)をTとする。
[pari/gpでの計算結果]
gp> e=ellinit([0,0,0,-12,0])
time = 91 ms.
%1 = [0, 0, 0, -12, 0, 0, -24, 0, -144, 576, 0, 110592, 1728, [3.464101615137754587054892682, 0.E-28, -3.464101615137754587054892683]~, 1.408792103676543125007999642, 1.408792103676543125007999642*I, -1.114995124330672325927969473, -3.344985372992016977783908419*I, 1.984695191381379833158134016]
gp> elltors(e,1)
time = 105 ms.
%2 = [2, [2], [[0, 0]]]
■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
P(-2,-4)
である。
E(Q) = Z/2Z×Z
[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: [0,0,0,-12,0]
Curve [0,0,0,-12,0] :
1 points of order 2:
[0 : 0 : 1]
Using 2-isogenous curve [0,0,0,48,0]
-------------------------------------------------------
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'))= 2
Third step, determining E(Q)/phi(E'(Q)) and E'(Q)/phi'(E(Q))
-------------------------------------------------------
1. E(Q)/phi(E'(Q))
-------------------------------------------------------
(c,d) =(0,-12)
(c',d')=(0,48)
First stage (no second descent yet)...
(-2,0,0,0,6): (x:y:z) = (1:2:1)
Curve E Point [-2 : -4 : 1], height = 0.250591196023589
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 = [0,0,0,-12,0]:
I. Points on E mod phi(E')
Point [-2 : -4 : 1], height = 0.250591196023589
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 : -4 : 1]; height 0.250591196023589
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) = 0.250591196023589
(5.7 seconds)
Enter curve: [0,0,0,0,0]
bash-2.05a$
■pari/gpで、楕円曲線E: v2 = u3-12uの有理点をいくつか計算すると、以下のようになる。
gp> read("de30.gp")
time = 36 ms.
gp> v=ratpointE(15,100);
[0]
[0, 0]
[-2, 4]
[-2, -4]
[6, -12]
[6, 12]
[-3, -3]
[-3, 3]
[4, -4]
[4, 4]
[-2/9, 44/27]
[-2/9, -44/27]
[54, 396]
[54, -396]
[49/4, -329/8]
[-48/49, 1128/343]
[49/4, 329/8]
[-48/49, -1128/343]
[2166/625, -2964/15625]
[2166/625, 2964/15625]
[-1250/361, -1300/6859]
[-1250/361, 1300/6859]
[-3267/3721, -712701/226981]
[14884/1089, -1756556/35937]
[-3267/3721, 712701/226981]
[14884/1089, 1756556/35937]
[-55778/201601, 164408828/90518849]
[-55778/201601, -164408828/90518849]
[1209606/27889, 1326099948/4657463]
[1209606/27889, -1326099948/4657463]
[6723649/1731856, -7871603809/2279122496]
[6723649/1731856, 7871603809/2279122496]
[-20782272/6723649, 47939979696/17434421857]
[-20782272/6723649, -47939979696/17434421857]
[3088744326/482285521, -144382816197972/10591472326681]
[-964571042/514790721, -46583381470076/11680086668769]
[3088744326/482285521, 144382816197972/10591472326681]
[-964571042/514790721, 46583381470076/11680086668769]
[-114391875/38179987609, 1414568011820925/7460255038835773]
[-114391875/38179987609, -1414568011820925/7460255038835773]
[152719950436/38130625, 59682017987751316/235456609375]
[152719950436/38130625, -59682017987751316/235456609375]
[68027689757814/12053832690769, 442953007035183973164/41849255727271332647]
[-24107665381538/11337948292969, 152241040356455297812/38176992098757408203]
[68027689757814/12053832690769, -442953007035183973164/41849255727271332647]
[-24107665381538/11337948292969, -152241040356455297812/38176992098757408203]
[3474056452030129/840024375114276, -111854308751402583911671/24346572628178331192024]
[3474056452030129/840024375114276, 111854308751402583911671/24346572628178331192024]
[-10080292501371312/3474056452030129, -660026651989526532174024/204764802544277241215383]
[-10080292501371312/3474056452030129, 660026651989526532174024/204764802544277241215383]
[-406947287285839682/2342998905050892169, -5171127074171700480844739284/3586395066685989433536489947]
[-406947287285839682/2342998905050892169, 5171127074171700480844739284/3586395066685989433536489947]
[14057993430305353014/203473643642919841, -52642732830738967299978248244/91782998812005768982875889]
[14057993430305353014/203473643642919841, 52642732830738967299978248244/91782998812005768982875889]
[-1021748999154070287747/941058173153198733361, 98952200388036929056131166343811/28868532422330691427620711977641]
[3764232692612794933444/340582999718023429249, 219311267529852206084715501445916/6285422116835922138309171391807]
[-1021748999154070287747/941058173153198733361, -98952200388036929056131166343811/28868532422330691427620711977641]
[3764232692612794933444/340582999718023429249, -219311267529852206084715501445916/6285422116835922138309171391807]
[12651458877340136796505734/3637974721409037441150625, -3962525677739209590409102021180284852/6938882214678612455828329644267359375]
[-7275949442818074882301250/2108576479556689466084289, -1734946129626160156488300042048967300/3061850887648929790852696173423146913]
[12651458877340136796505734/3637974721409037441150625, 3962525677739209590409102021180284852/6938882214678612455828329644267359375]
[-7275949442818074882301250/2108576479556689466084289, 1734946129626160156488300042048967300/3061850887648929790852696173423146913]
time = 784 ms.
■pari/gpで、楕円曲線C: x2-3y4 = 1の有理点をいくつか計算すると、以下のようになる。
gp> ratpointC(v)
[1, 0]
[-1, 0]
[2, -1]
[2, 1]
[-2, -1]
[-2, 1]
[-7, 2]
[7, 2]
[7, -2]
[-7, -2]
[-122/121, 3/11]
[-122/121, -3/11]
[122/121, 3/11]
[122/121, -3/11]
[2593/2209, -28/47]
[-2593/2209, 28/47]
[2593/2209, 28/47]
[-2593/2209, -28/47]
[-390794/169, -475/13]
[-390794/169, 475/13]
[390794/169, 475/13]
[390794/169, -475/13]
[58941127/51825601, 4026/7199]
[-58941127/51825601, -4026/7199]
[58941127/51825601, -4026/7199]
[-58941127/51825601, 4026/7199]
[61353342962/60575546641, -74983/246121]
[61353342962/60575546641, 74983/246121]
[-61353342962/60575546641, 74983/246121]
[-61353342962/60575546641, -74983/246121]
[81199358332033/9215553418369, -6824776/3035713]
[-81199358332033/9215553418369, 6824776/3035713]
[-81199358332033/9215553418369, -6824776/3035713]
[81199358332033/9215553418369, 6824776/3035713]
[-513813891524670482/281214567758429041, -498273129/530296679]
[513813891524670482/281214567758429041, 498273129/530296679]
[-513813891524670482/281214567758429041, 498273129/530296679]
[513813891524670482/281214567758429041, -498273129/530296679]
[5830850177127262819399/5830841453459885475649, -2413152950/76359946657]
[5830850177127262819399/5830841453459885475649, 2413152950/76359946657]
[-5830850177127262819399/5830841453459885475649, 2413152950/76359946657]
[-5830850177127262819399/5830841453459885475649, -2413152950/76359946657]
[-265471048509583706971158122/120176165972532276193346761, 11690411959381/10962489040931]
[265471048509583706971158122/120176165972532276193346761, -11690411959381/10962489040931]
[-265471048509583706971158122/120176165972532276193346761, -11690411959381/10962489040931]
[265471048509583706971158122/120176165972532276193346761, 11690411959381/10962489040931]
[-20536759641325726509302231806753/3601376822458609486068815706529, -3416601879194196/1897729385992273]
[-20536759641325726509302231806753/3601376822458609486068815706529, 3416601879194196/1897729385992273]
[20536759641325726509302231806753/3601376822458609486068815706529, 3416601879194196/1897729385992273]
[20536759641325726509302231806753/3601376822458609486068815706529, -3416601879194196/1897729385992273]
[8255166565433182393136033578274019482/8213765041775856459312186130322554201, 690462543706808627/2865966685391834149]
[8255166565433182393136033578274019482/8213765041775856459312186130322554201, -690462543706808627/2865966685391834149]
[-8255166565433182393136033578274019482/8213765041775856459312186130322554201, -690462543706808627/2865966685391834149]
[-8255166565433182393136033578274019482/8213765041775856459312186130322554201, 690462543706808627/2865966685391834149]
[3890352280124524518936819244595140223537287/3194371601942961635212808521880177971313281, -1132269253352186593166/1787280504549568916159]
[-3890352280124524518936819244595140223537287/3194371601942961635212808521880177971313281, 1132269253352186593166/1787280504549568916159]
[3890352280124524518936819244595140223537287/3194371601942961635212808521880177971313281, 1132269253352186593166/1787280504549568916159]
[-3890352280124524518936819244595140223537287/3194371601942961635212808521880177971313281, -1132269253352186593166/1787280504549568916159]
[-13286572192015104896274209510471259634271647898594/51712118403941313409900075506958624147710007969, -2769647618522056009700175/227402986796438787254063]
[13286572192015104896274209510471259634271647898594/51712118403941313409900075506958624147710007969, 2769647618522056009700175/227402986796438787254063]
[-13286572192015104896274209510471259634271647898594/51712118403941313409900075506958624147710007969, 2769647618522056009700175/227402986796438787254063]
[13286572192015104896274209510471259634271647898594/51712118403941313409900075506958624147710007969, -2769647618522056009700175/227402986796438787254063]
time = 50 ms.
[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]Michael A. Bennett, Gary Walsh, "The Diophantine Equation b^2X^4-dY^2=1", 1991, p1-10.
- [4]Michael A. Bennett, Gary Walsh, "Simultaneous quadratic quations with few or no solutions", 1999, p1-10.
- [5]J.H.E.Cohn, "The Diophantine Equation x^4+1=Dy^2",Math. of Comp, Vol.66(1997), No.219, p1347-1351.
- [6]Gary Walsh, "A note on a theorem of Ljunggren and the Diophantine equations x^2-kxy^2+y^4=1,4", Arch. Math. 73(1999), p119-125.
| Last Update: 2005.06.12 |
| H.Nakao |