Rational Points on Elliptic Curves: 3x^4-2y^2=1, v^2=u^3+3u

[2003.09.29]3x^4-2y^2=1, v^2=u^3+3uの有理点

y^2=6x^4-2とv^2=u^3+48uの間の双有理変換を計算するプログラム(asir)
3x^4-2y^2=1,v^2=u^3+3uの有理点を計算するプログラム(pari/gp)

■Diophantus方程式
     C: 3x⁴-2y² = 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、整点(±1,±1)を持つ。

■(1)を双有理変換(x,y)→(x,2y)[逆変換は(x,y)→(x,y/2)]で変換すると、
     C₁: y² = 6x⁴-2 ----- (2)
を得る。

曲線C₁は、整点(±1,±2)を持つ。

■双有理変換φ:(x,y)→(-2sqrt(6)y+12x²,24xy-24sqrt(6)x³) [逆変換は、φ^-1:(u,v)→(-v/{2sqrt(6)u},(v²-u³)/{4sqrt(6)u²})]によって、
曲線C₁は、楕円曲線
     E₁: v² = u³+48u ----- (3)
に写される。

■双有理変換φはQ(sqrt(6))-isomorphicであるが、Q-isomorphicではないので、
E₁の１つの有理点(1,2)を使って、双有理変換φ^を以下のように定義する。
     φ^(x,y)=φ(x,y)-φ(1,2)
このとき、
     φ^^-1(u,v)=φ^-1((x,y)+φ(1,2))
である。
asirを使って、φ(1,2)と(u,v)=φ^(x,y)と(x,y)=φ^^-1(u,v)をそれぞれ計算すると、このようになる。

     φ(1,2)=(12-4sqrt(6),48-24sqrt(6))

     u = 12(x+1)²/(3x²-y-1)
     v = 24(x+1)(6x³-3xy-y+2)/(3x²-y-1)²

     x = (u²-4v-48)/(u²-24u+48)
     y = 2(u⁴+24u³-12u²v+96uv-1152u-576v-2304)/(u²-24u+48)²

■双有理変換(u,v)→(u/4,v/8)[逆変換は、(u,v)→(4u,8v)]によって、曲線E₁は、楕円曲線
     E: v² = u³+3u ----- (4)
に写される。

楕円曲線Eは、自明な有理点(0,0)を持つ。

■以上の双有理変換を合成すると、楕円曲線C:3x⁴-2y²=1と楕円曲線E:v² = u³+3uの間の
双有理変換ψ:C→Eとψ^-1:E→Cが以下のように求められる。

     u = 3(x+1)²/(3x²-2y-1)
     v = 6(x+1)(3x³-3xy-y+1)/(3x²-2y-1)²

     x = (u²-2v-3)/(u²-6u+3)
     y = (u⁴+6u³-6u²v+12uv-18u-18v-9)/(u²-6u+3)²

■楕円曲線Eのねじれ点群E_tors(Q)は、 Z/2Zである。

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

E_tors(Q)の生成元(位数2)をTとする。
[pari/gpでの計算結果]

gp>  e=ellinit([0,0,0,3,0])
time = 86 ms.
%1 = [0, 0, 0, 3, 0, 0, 6, 0, -9, -144, 0, -1728, 1728, [0.E-28, 0.E-28 - 1.732050807568877293527446341*I, 0.E-28 + 1.732050807568877293527446341*I]~, 2.817584207353086250015999285, -1.408792103676543125007999642 + 1.408792103676543125007999642*I, -1.114995124330672325927969473 + 1.26740368 E-29*I, 0.5574975621653361629639847364 - 1.672492686496008488891954209*I, 3.969390382762759666316268033]
gp>  elltors(e,1)
time = 53 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(1,-2)
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: [0,0,0,3,0]

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

Using 2-isogenous curve [0,0,0,-12,0]
-------------------------------------------------------
First step, determining 1st descent Selmer groups
-------------------------------------------------------
After first local descent, rank bound = 1
rk(S^{phi}(E'))=    1
rk(S^{phi'}(E))=       2

-------------------------------------------------------
Second step, determining 2nd descent Selmer groups
-------------------------------------------------------
After second local descent, rank bound = 1
rk(phi'(S^{2}(E)))=  1
rk(phi(S^{2}(E')))=    2
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)  =(0,3)
(c',d')=(0,-12)
This component of the rank is 0
-------------------------------------------------------
2. E'(Q)/phi'(E(Q))
-------------------------------------------------------
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
 Curve E         Point [1 : -2 : 1], height = 0.501182392047178
After first global descent, this component of the rank = 2

-------------------------------------------------------
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 = [0,0,0,3,0]:

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


II. Points on phi(E') mod 2E
Point [1 : -2 : 1], height = 0.501182392047178
-------------------------------------------------------
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 [1 : -2 : 1]; height 0.501182392047178

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) = 0.501182392047178

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

bash-2.05a$

■pari/gpで、楕円曲線E: v² = u³+3uの有理点をいくつか計算すると、以下のようになる。

gp> read("./de3x4m2y2m1.gp")
time = 21 ms.
gp> v=ratpointE(50,100);
[0]
[0, 0]
[1, -2]
[1, 2]
[3, 6]
[3, -6]
[1/4, 7/8]
[12, 42]
[1/4, -7/8]
[12, -42]
[27/121, -1098/1331]
[121/9, -1342/27]
[121/9, 1342/27]
[27/121, 1098/1331]
[2209/784, -121871/21952]
[2209/784, 121871/21952]
[2352/2209, -217812/103823]
[2352/2209, 217812/103823]
[676875/169, 556881450/2197]
[676875/169, -556881450/2197]
[169/225625, -5080322/107171875]
[169/225625, 5080322/107171875]
[51825601/16208676, -424317173273/65256129576]
[51825601/16208676, 424317173273/65256129576]
[48626028/51825601, 711890931906/373092501599]
[48626028/51825601, -711890931906/373092501599]
[16867350867/60575546641, -13801373145958938/14908914114829561]
[60575546641/5622450289, 15100346123150402/421588190020087]
[16867350867/60575546641, 13801373145958938/14908914114829561]
[60575546641/5622450289, -15100346123150402/421588190020087]
[9215553418369/46577567450176, -246497947680210894529/317881464472342360576]
[9215553418369/46577567450176, 246497947680210894529/317881464472342360576]
[139732702350528/9215553418369, -1662502295879576548824/27975775314337212097]
[139732702350528/9215553418369, 1662502295879576548824/27975775314337212097]
[281214567758429041/248276111083450641, -272473800299599003173929278/123709314725502531008125689]
[281214567758429041/248276111083450641, 272473800299599003173929278/123709314725502531008125689]
[744828333250351923/281214567758429041, 768058966360992425280234534/149127151368715394699454839]
[744828333250351923/281214567758429041, -768058966360992425280234534/149127151368715394699454839]
[5830841453459885475649/5823307160093702500, 445243408490396790389727064799143/14052530852136240464297375000]
[5830841453459885475649/5823307160093702500, -445243408490396790389727064799143/14052530852136240464297375000]
[17469921480281107500/5830841453459885475649, -42212199917828030394174042231150/445242742351621203009886712455393]
[17469921480281107500/5830841453459885475649, 42212199917828030394174042231150/445242742351621203009886712455393]
[120176165972532276193346761/136665731780038311593903161, 2910223459970773268796314939722131091582/1597678705238915879143728461559579503341]
[120176165972532276193346761/136665731780038311593903161, -2910223459970773268796314939722131091582/1597678705238915879143728461559579503341]
[409997195340114934781709483/120176165972532276193346761, -9310397761097552890707989909594376727446/1317429902454990029336244337518005274491]
[409997195340114934781709483/120176165972532276193346761, 9310397761097552890707989909594376727446/1317429902454990029336244337518005274491]
[3601376822458609486068815706529/11673168400913311478026280086416, -38973212264403963652753523766321662998487219569/39882569094710727922720617190415979463325641536]
[3601376822458609486068815706529/11673168400913311478026280086416, 38973212264403963652753523766321662998487219569/39882569094710727922720617190415979463325641536]
[35019505202739934434078840259248/3601376822458609486068815706529, 210497794749338999454087020865138962690493616764/6834438626011180151704011508999581622229650417]
[35019505202739934434078840259248/3601376822458609486068815706529, -210497794749338999454087020865138962690493616764/6834438626011180151704011508999581622229650417]
[8213765041775856459312186130322554201/476738524262076613525095357201625129, -23659032358892029498276208747750920794831382482538890818/329170094145023518807050094025409729769798266597187883]
[8213765041775856459312186130322554201/476738524262076613525095357201625129, 23659032358892029498276208747750920794831382482538890818/329170094145023518807050094025409729769798266597187883]
[1430215572786229840575286071604875387/8213765041775856459312186130322554201, 17099649916477181872552994390764569480177058397531013642/23540376971365671484960975628504252221877334805855209949]
[1430215572786229840575286071604875387/8213765041775856459312186130322554201, -17099649916477181872552994390764569480177058397531013642/23540376971365671484960975628504252221877334805855209949]
[3846100986260154331936196539205620793710668/3194371601942961635212808521880177971313281, -13214778814480716119368239194531271391467375190272779382301141926/5709238088439431189724694566643921020862586652134291233012207679]
[3194371601942961635212808521880177971313281/1282033662086718110645398846401873597903556, 6953150786096526051030485141064230117925439303560659797713320633/1451607297343297804136489209548506413706224097928684347476698296]
[3846100986260154331936196539205620793710668/3194371601942961635212808521880177971313281, 13214778814480716119368239194531271391467375190272779382301141926/5709238088439431189724694566643921020862586652134291233012207679]
[3194371601942961635212808521880177971313281/1282033662086718110645398846401873597903556, -6953150786096526051030485141064230117925439303560659797713320633/1451607297343297804136489209548506413706224097928684347476698296]
[23012843792354688875189674929391107641080185091875/51712118403941313409900075506958624147710007969, 110397369089808026271428143379757357482805854775421343961934114256232161850/11759490178627345706146650741336046746470940433149891935430341357628047]
[51712118403941313409900075506958624147710007969/7670947930784896291729891643130369213693395030625, -3021406200750741653888043137557088545180348979621135318993262045538487422/21245822668305081352130104651770854854443514228529616524796293141192859375]
[23012843792354688875189674929391107641080185091875/51712118403941313409900075506958624147710007969, -110397369089808026271428143379757357482805854775421343961934114256232161850/11759490178627345706146650741336046746470940433149891935430341357628047]
[51712118403941313409900075506958624147710007969/7670947930784896291729891643130369213693395030625, 3021406200750741653888043137557088545180348979621135318993262045538487422/21245822668305081352130104651770854854443514228529616524796293141192859375]
[41261133130834642410031653822433208622530017500335436289/11320442684433531154473197019712800072005807303642616064, 293444103919076629865758547718013029606906263817221925288891662618891060391286609151/38088609249911937440416396198659029663519102020889535194837263026742874227123392512]
[41261133130834642410031653822433208622530017500335436289/11320442684433531154473197019712800072005807303642616064, -293444103919076629865758547718013029606906263817221925288891662618891060391286609151/38088609249911937440416396198659029663519102020889535194837263026742874227123392512]
[33961328053300593463419591059138400216017421910927848192/41261133130834642410031653822433208622530017500335436289, 461113431095447461283566689627320901837794175207003877260288296030718811737623140272/265040184912739972506539377093728376663222479059078700947663672406061419891085841663]
[33961328053300593463419591059138400216017421910927848192/41261133130834642410031653822433208622530017500335436289, -461113431095447461283566689627320901837794175207003877260288296030718811737623140272/265040184912739972506539377093728376663222479059078700947663672406061419891085841663]
[350587027507885830030843460546564642981195388104133028131207907/1030457769177959751286825531039553422595449346652936287098193761, -34057252721441294798932998873714601801232985892319165791893466227983091088104755338563992008122/33078461112576200483014146492155218305837931504428365854941868619504242743474300255914015483759]
[1030457769177959751286825531039553422595449346652936287098193761/116862342502628610010281153515521547660398462701377676043735969, 33710575173029333615697258964682753130727858066473807912720578618780273973672162735800441120638/1263315664531943868294852935056994559350609155187288119779032155859429147966424472580635794703]
[350587027507885830030843460546564642981195388104133028131207907/1030457769177959751286825531039553422595449346652936287098193761, 34057252721441294798932998873714601801232985892319165791893466227983091088104755338563992008122/33078461112576200483014146492155218305837931504428365854941868619504242743474300255914015483759]
[1030457769177959751286825531039553422595449346652936287098193761/116862342502628610010281153515521547660398462701377676043735969, -33710575173029333615697258964682753130727858066473807912720578618780273973672162735800441120638/1263315664531943868294852935056994559350609155187288119779032155859429147966424472580635794703]
[2800603075498149678331774357422064337675923647877249931672318114943361/18432508050011962188745199978247371817874981488735513281147530080223044, 1696037337652482602261038588056405669782389529183916149377308631718708117261863578592965006751561231270087/2502514764087678580227284312446876956202960262758251515895048807084560413466036150144398032362073108143928]
[2800603075498149678331774357422064337675923647877249931672318114943361/18432508050011962188745199978247371817874981488735513281147530080223044, -1696037337652482602261038588056405669782389529183916149377308631718708117261863578592965006751561231270087/2502514764087678580227284312446876956202960262758251515895048807084560413466036150144398032362073108143928]
[55297524150035886566235599934742115453624944466206539843442590240669132/2800603075498149678331774357422064337675923647877249931672318114943361, -13053383545479468935947852113400666508531653701530618998965064375177239280457180066973710698775468049340422/148209943630703447354280217458116745455568663558068775308018201157817437155855062444480462491901659992641]
[55297524150035886566235599934742115453624944466206539843442590240669132/2800603075498149678331774357422064337675923647877249931672318114943361, 13053383545479468935947852113400666508531653701530618998965064375177239280457180066973710698775468049340422/148209943630703447354280217458116745455568663558068775308018201157817437155855062444480462491901659992641]
[2674468381663777477305482447996545779184961116642670166750257404430125458281881/2090976852759333612231029399961807714821922928221480934385793978518278805743081, -7362723147979423064448252929769349107889331859837422565681319405504704010788529479757915489516496329773609235907816258/3023596526595326647662739653492950152242372362669780697308993728408734208343104920075583413824556739682220902212906229]
[2674468381663777477305482447996545779184961116642670166750257404430125458281881/2090976852759333612231029399961807714821922928221480934385793978518278805743081, 7362723147979423064448252929769349107889331859837422565681319405504704010788529479757915489516496329773609235907816258/3023596526595326647662739653492950152242372362669780697308993728408734208343104920075583413824556739682220902212906229]
[6272930558278000836693088199885423144465768784664442803157381935554836417229243/2674468381663777477305482447996545779184961116642670166750257404430125458281881, 19530600827426551061662195362029623990801764101690399595170950496732906795210402882670397035209442129915897781623501174/4373772623089619737983714885920580635688939557471598122979852821471502028341768622453152430950192539353934099215940029]
[6272930558278000836693088199885423144465768784664442803157381935554836417229243/2674468381663777477305482447996545779184961116642670166750257404430125458281881, -19530600827426551061662195362029623990801764101690399595170950496732906795210402882670397035209442129915897781623501174/4373772623089619737983714885920580635688939557471598122979852821471502028341768622453152430950192539353934099215940029]
time = 2,736 ms.

■pari/gpで、楕円曲線C: 3x⁴-2y² = 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, 1]
[-1, -1]
[1, 1]
[1, -1]
[-3, -11]
[3, -11]
[3, 11]
[-3, 11]
[-19/25, -13/625]
[19/25, 13/625]
[19/25, -13/625]
[-19/25, 13/625]
[-449/167, 246121/27889]
[449/167, 246121/27889]
[-449/167, -246121/27889]
[449/167, -246121/27889]
[-22689/21961, -530296679/482285521]
[22689/21961, 530296679/482285521]
[22689/21961, -530296679/482285521]
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time = 88 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