Rational Points on Genus 2 Curve: y^2=278271081x^2(x^2-9)^2-229833600(x^2-1)^2

[2004.04.25]y^2=278271081x^2(x^2-9)^2-229833600(x^2-1)^2の有理点

y^2=278271081x^2(x^2-9)^2-229833600(x^2-1)^2の有理点を計算するプログラム(pari/gp)
y^2=278271081x^2(x^2-9)^2-229833600(x^2-1)^2のグラフ(jpeg)

■参考文献[1](p130,Theorem 6.2.3[Kuller-Kulesz]),[2](p13, Miscellaneous examples of genus 2 curves)に記述されているgenus 2の超楕円曲線
C₂: y² = 278271081x²(x²-9)²-229833600(x²-1)²
の有理点は、588個以上であることが示されている。
ここでは、実際に、超楕円曲線C₂の有理点をいくつか求める。

■Machael Stollのratpoints-1.4.cを使って、超楕円曲線C₂の有理点のx座標を求めると、以下のようになる。
[pari/gpによる計算]

gp>  g2(x)=278271081*x^2*(x^2-9)^2-229833600*(x^2-1)^2
time = 4 ms.
gp>  g2(x)
time = 8 ms.
%1 = 278271081*x^6 - 5238713058*x^4 + 22999624761*x^2 - 229833600

[ratpoints-1.4による計算]

bash-2.05a$ ./ratpoints '-229833600 0 22999624761 0 -5238713058 0 278271081' 1000000

This is ratpoints-1.4 by Michael Stoll (1998-11-10).

Please acknowledge use of the program in published work.


y^2 = 278271081 x^6 - 5238713058 x^4 + 22999624761 x^2 - 229833600

max. Height = 1000000
Search region:
  [-1000000.500000, 1000000.500000]
Using speed ratios 1000.000000 and 9.000000
12 primes used for first stage of sieving,
43 primes used for both stages of sieving together.
Sieving primes:
 First stage: 223, 163, 139, 173, 233, 157, 251, 149, 191, 197, 199, 179
 Second stage: 113, 181, 241, 229, 211, 239, 131, 193, 127, 227, 107, 167, 73, 79, 151, 109, 137, 103, 53, 97, 83, 47, 101, 67, 29, 43, 19, 37, 41, 13, 31
Probabilities: Min(223) = 0.457399, Cut1(179) = 0.603352, Cut2(31) = 0.870968, Max(7) = 1.000000

Forbidden divisors of the denominator:
  23, 29, 31, 41, 67, 73, 79, 83, 103, 109, 127, 131, 137, 163, 173, 179, 197, 211, 223, 227, 229, 239, 241, 251

(-109 : 1)
(-10 : 1)
(10 : 1)
(109 : 1)
(-97 : 3)
(-52 : 3)
(-13 : 3)
(13 : 3)
(52 : 3)
(97 : 3)
(-85 : 4)
(-1 : 4)
(1 : 4)
(85 : 4)
(-11 : 5)
(11 : 5)
(-50 : 7)
(-37 : 7)
(-25 : 7)
(25 : 7)
(37 : 7)
(50 : 7)
(-29 : 8)
(-1 : 8)
(1 : 8)
(29 : 8)
(-35 : 9)
(-23 : 9)
(-13 : 9)
(13 : 9)
(23 : 9)
(35 : 9)
(-125 : 11)
(-41 : 11)
(-7 : 11)
(-4 : 11)
(-2 : 11)
(2 : 11)
(4 : 11)
(7 : 11)
(41 : 11)
(125 : 11)
(-239 : 13)
(-31 : 13)
(-2 : 13)
(2 : 13)
(31 : 13)
(239 : 13)
(-173 : 14)
(173 : 14)
(-37 : 15)
(-29 : 15)
(29 : 15)
(37 : 15)
(-142 : 17)
(142 : 17)
(-100 : 19)
(100 : 19)
(-53 : 21)
(53 : 21)
(-197 : 25)
(-22 : 25)
(22 : 25)
(197 : 25)
(-28 : 27)
(28 : 27)
(-119 : 33)
(119 : 33)
(-23 : 34)
(23 : 34)
(-5 : 37)
(5 : 37)
(-5 : 38)
(5 : 38)
(-109 : 43)
(-71 : 43)
(-68 : 43)
(68 : 43)
(71 : 43)
(109 : 43)
(-53 : 47)
(53 : 47)
(-61 : 49)
(61 : 49)
(-53 : 55)
(-43 : 55)
(43 : 55)
(53 : 55)
(-79 : 57)
(-29 : 57)
(29 : 57)
(79 : 57)
(-50 : 63)
(50 : 63)
(-167 : 66)
(167 : 66)
(-157 : 81)
(-97 : 81)
(97 : 81)
(157 : 81)
(-73 : 89)
(73 : 89)
(-364 : 95)
(364 : 95)
(-365 : 101)
(365 : 101)
(-61 : 111)
(61 : 111)
(-139 : 113)
(139 : 113)
(-43 : 119)
(43 : 119)
(-193 : 125)
(193 : 125)
(-911 : 159)
(-215 : 159)
(-91 : 159)
(91 : 159)
(215 : 159)
(911 : 159)
(-131 : 187)
(131 : 187)
(-347 : 188)
(347 : 188)
(-500 : 193)
(500 : 193)
(-1361 : 195)
(1361 : 195)
(-31 : 233)
(31 : 233)
(-2359 : 247)
(2359 : 247)
(-649 : 269)
(649 : 269)
(-1079 : 307)
(1079 : 307)
(-1388 : 321)
(1388 : 321)
(-2324 : 379)
(2324 : 379)
(-194 : 389)
(194 : 389)
(-79 : 459)
(79 : 459)
(-775 : 528)
(775 : 528)
(-217 : 535)
(217 : 535)
(-1858 : 539)
(1858 : 539)
(-2020 : 573)
(2020 : 573)
(-973 : 583)
(973 : 583)
(-4913 : 601)
(4913 : 601)
(-79 : 693)
(79 : 693)
(-2626 : 745)
(2626 : 745)
(-2351 : 1067)
(2351 : 1067)
(-1679 : 1078)
(1679 : 1078)
(-809 : 1303)
(809 : 1303)
(-3475 : 1319)
(3475 : 1319)
(-5371 : 1409)
(5371 : 1409)
(-3739 : 1447)
(3739 : 1447)
(-5146 : 1493)
(5146 : 1493)
(-6017 : 1511)
(6017 : 1511)
(-5899 : 1617)
(5899 : 1617)
(-286 : 1695)
(286 : 1695)
(-425 : 1709)
(425 : 1709)
(-8995 : 1812)
(8995 : 1812)
(-262 : 1879)
(262 : 1879)
(-4861 : 1881)
(4861 : 1881)
(-371 : 1882)
(371 : 1882)
(-6599 : 1893)
(6599 : 1893)
(-3461 : 1945)
(3461 : 1945)
(-4799 : 1981)
(4799 : 1981)
(-230 : 2123)
(230 : 2123)
(-5375 : 2141)
(5375 : 2141)
(-5275 : 2253)
(5275 : 2253)
(-6139 : 2353)
(6139 : 2353)
(-241 : 2397)
(241 : 2397)
(-22388 : 2545)
(22388 : 2545)
(-14812 : 2557)
(14812 : 2557)
(-301 : 2593)
(301 : 2593)
(-1187 : 2703)
(1187 : 2703)
(-1555 : 2757)
(1555 : 2757)
(-391 : 3371)
(391 : 3371)
(-9625 : 3653)
(9625 : 3653)
(-13675 : 3913)
(13675 : 3913)
(-484 : 4397)
(484 : 4397)
(-12707 : 4881)
(12707 : 4881)
(-667 : 6639)
(667 : 6639)
(-14431 : 7183)
(14431 : 7183)
(-3559 : 10807)
(3559 : 10807)
(-22483 : 12255)
(22483 : 12255)
(-52718 : 14131)
(52718 : 14131)
(-7141 : 17369)
(7141 : 17369)
(-175708 : 17645)
(175708 : 17645)
(-40588 : 18579)
(40588 : 18579)
(-30023 : 19843)
(30023 : 19843)
(-96325 : 22009)
(96325 : 22009)
(-14753 : 24933)
(14753 : 24933)
(-95111 : 38587)
(95111 : 38587)
(-172294 : 48663)
(172294 : 48663)
(-344923 : 54182)
(344923 : 54182)
(-329183 : 58789)
(329183 : 58789)
(-15149 : 59167)
(15149 : 59167)
(-234127 : 63521)
(234127 : 63521)
(-10325 : 66849)
(10325 : 66849)
(-10891 : 74412)
(10891 : 74412)
(-212345 : 85303)
(212345 : 85303)
(-427775 : 94857)
(427775 : 94857)
(-38204 : 96993)
(38204 : 96993)
(-318283 : 123631)
(318283 : 123631)
(-35801 : 130658)
(35801 : 130658)
(-252775 : 135197)
(252775 : 135197)
(-228643 : 158063)
(228643 : 158063)
(-356173 : 166459)
(356173 : 166459)
(-122773 : 193353)
(122773 : 193353)
(-26305 : 220957)
(26305 : 220957)
(-507469 : 290741)
(507469 : 290741)
(-182377 : 399105)
(182377 : 399105)

1234705650 candidates survived the first stage,
4172 candidates survived the second stage.

288 rational point pairs found.
bash-2.05a$

よって、超楕円曲線C₂の有理点のx座標が288個求まった。

■超楕円曲線C₂の有理点(x,y)をいくつか求めると。以下のようになる。

[pari/gpによる計算]

gp> read("kk.gp")
time = 63 ms.
gp> rpc2(c2x)
[-109, 21585857568]
[-109, -21585857568]
[-10, 15105750]
[-10, -15105750]
[10, 15105750]
[10, -15105750]
[109, 21585857568]
[109, -21585857568]
[-97/3, 5029198448/9]
[-97/3, -5029198448/9]
[-52/3, 757326812/9]
[-52/3, -757326812/9]
[-13/3, 5880568/9]
[-13/3, -5880568/9]
[13/3, 5880568/9]
[13/3, -5880568/9]
[52/3, 757326812/9]
[52/3, -757326812/9]
[97/3, 5029198448/9]
[97/3, -5029198448/9]
[-85/4, 10030798425/64]
[-85/4, -10030798425/64]
[-1/4, 2205213/64]
[-1/4, -2205213/64]
[1/4, 2205213/64]
[1/4, -2205213/64]
[85/4, 10030798425/64]
[85/4, -10030798425/64]
[-11/5, 17641704/125]
[-11/5, -17641704/125]
[11/5, 17641704/125]
[11/5, -17641704/125]
[-50/7, 1697544690/343]
[-50/7, -1697544690/343]
[-37/7, 555381024/343]
[-37/7, -555381024/343]
[-25/7, 46387560/343]
[-25/7, -46387560/343]
[25/7, 46387560/343]
[25/7, -46387560/343]
[37/7, 555381024/343]
[37/7, -555381024/343]
[50/7, 1697544690/343]
[50/7, -1697544690/343]
[-29/8, 86913825/512]
[-29/8, -86913825/512]
[-1/8, 5798445/512]
[-1/8, -5798445/512]
[1/8, 5798445/512]
[1/8, -5798445/512]
[29/8, 86913825/512]
[29/8, -86913825/512]
[-35/9, 81307600/243]
[-35/9, -81307600/243]
[-23/9, 15462520/243]
[-23/9, -15462520/243]
[-13/9, 40282000/243]
[-13/9, -40282000/243]
[13/9, 40282000/243]
[13/9, -40282000/243]
[23/9, 15462520/243]
[23/9, -15462520/243]
[35/9, 81307600/243]
[35/9, -81307600/243]
[-125/11, 2745430680/121]
[-125/11, -2745430680/121]
[-41/11, 2564112/11]
[-41/11, -2564112/11]
[-7/11, 10986000/121]
[-7/11, -10986000/121]
[-4/11, 6311148/121]
[-4/11, -6311148/121]
[-2/11, 2771850/121]
[-2/11, -2771850/121]
[2/11, 2771850/121]
[2/11, -2771850/121]
[4/11, 6311148/121]
[4/11, -6311148/121]
[7/11, 10986000/121]
[7/11, -10986000/121]
[41/11, 2564112/11]
[41/11, -2564112/11]
[125/11, 2745430680/121]
[125/11, -2745430680/121]
[-239/13, 221385547920/2197]
[-239/13, -221385547920/2197]
[-31/13, 243922800/2197]
[-31/13, -243922800/2197]
[-2/13, 38782194/2197]
[-2/13, -38782194/2197]
[2/13, 38782194/2197]
[2/13, -38782194/2197]
[31/13, 243922800/2197]
[31/13, -243922800/2197]
[239/13, 221385547920/2197]
[239/13, -221385547920/2197]
[-173/14, 81035812275/2744]
[-173/14, -81035812275/2744]
[173/14, 81035812275/2744]
[173/14, -81035812275/2744]
[-37/15, 103419184/1125]
[-37/15, -103419184/1125]
[-29/15, 185127008/1125]
[-29/15, -185127008/1125]
[29/15, 185127008/1125]
[29/15, -185127008/1125]
[37/15, 103419184/1125]
[37/15, -103419184/1125]
[-142/17, 41286117486/4913]
[-142/17, -41286117486/4913]
[142/17, 41286117486/4913]
[142/17, -41286117486/4913]
[-100/19, 10914030780/6859]
[-100/19, -10914030780/6859]
[100/19, 10914030780/6859]
[100/19, -10914030780/6859]
[-53/21, 231770200/3087]
[-53/21, -231770200/3087]
[53/21, 231770200/3087]
[53/21, -231770200/3087]
[-197/25, 108086291232/15625]
[-197/25, -108086291232/15625]
[-22/25, 1885949418/15625]
[-22/25, -1885949418/15625]
[22/25, 1885949418/15625]
[22/25, -1885949418/15625]
[197/25, 108086291232/15625]
[197/25, -108086291232/15625]
[-28/27, 899410732/6561]
[-28/27, -899410732/6561]
[28/27, 899410732/6561]
[28/27, -899410732/6561]
[-119/33, 1419800/9]
[-119/33, -1419800/9]
[119/33, 1419800/9]
[119/33, -1419800/9]
[-23/34, 3774967185/39304]
[-23/34, -3774967185/39304]
[23/34, 3774967185/39304]
[23/34, -3774967185/39304]
[-5/37, 695310600/50653]
[-5/37, -695310600/50653]
[5/37, 695310600/50653]
[5/37, -695310600/50653]
[-5/38, 708657675/54872]
[-5/38, -708657675/54872]
[5/38, 708657675/54872]
[5/38, -708657675/54872]
[-109/43, 5669261400/79507]
[-109/43, -5669261400/79507]
[-71/43, 13580357520/79507]
[-71/43, -13580357520/79507]
[-68/43, 13510786404/79507]
[-68/43, -13510786404/79507]
[68/43, 13510786404/79507]
[68/43, -13510786404/79507]
[71/43, 13580357520/79507]
[71/43, -13580357520/79507]
[109/43, 5669261400/79507]
[109/43, -5669261400/79507]
[-53/47, 15087595344/103823]
[-53/47, -15087595344/103823]
[53/47, 15087595344/103823]
[53/47, -15087595344/103823]
[-61/49, 18175843488/117649]
[-61/49, -18175843488/117649]
[61/49, 18175843488/117649]
[61/49, -18175843488/117649]
[-53/55, 1962350688/15125]
[-53/55, -1962350688/15125]
[-43/55, 1652349408/15125]
[-43/55, -1652349408/15125]
[43/55, 1652349408/15125]
[43/55, -1652349408/15125]
[53/55, 1962350688/15125]
[53/55, -1962350688/15125]
[-79/57, 10066579160/61731]
[-79/57, -10066579160/61731]
[-29/57, 4526785840/61731]
[-29/57, -4526785840/61731]
[29/57, 4526785840/61731]
[29/57, -4526785840/61731]
[79/57, 10066579160/61731]
[79/57, -10066579160/61731]
[-50/63, 9224397830/83349]
[-50/63, -9224397830/83349]
[50/63, 9224397830/83349]
[50/63, -9224397830/83349]
[-167/66, 635030975/8712]
[-167/66, -635030975/8712]
[167/66, 635030975/8712]
[167/66, -635030975/8712]
[-157/81, 29104082080/177147]
[-157/81, -29104082080/177147]
[-97/81, 26748795800/177147]
[-97/81, -26748795800/177147]
[97/81, 26748795800/177147]
[97/81, -26748795800/177147]
[157/81, 29104082080/177147]
[157/81, -29104082080/177147]
[-73/89, 80246387400/704969]
[-73/89, -80246387400/704969]
[73/89, 80246387400/704969]
[73/89, -80246387400/704969]
[-364/95, 255534986796/857375]
[-364/95, -255534986796/857375]
[364/95, 255534986796/857375]
[364/95, -255534986796/857375]
[-365/101, 167648177400/1030301]
[-365/101, -167648177400/1030301]
[365/101, 167648177400/1030301]
[365/101, -167648177400/1030301]
[-61/111, 36028763744/455877]
[-61/111, -36028763744/455877]
[61/111, 36028763744/455877]
[61/111, -36028763744/455877]
[-139/113, 221385547920/1442897]
[-139/113, -221385547920/1442897]
[139/113, 221385547920/1442897]
[139/113, -221385547920/1442897]
[-43/119, 87312246240/1685159]
[-43/119, -87312246240/1685159]
[43/119, 87312246240/1685159]
[43/119, -87312246240/1685159]
[-193/125, 330288939888/1953125]
[-193/125, -330288939888/1953125]
[193/125, 330288939888/1953125]
[193/125, -330288939888/1953125]
[-911/159, 2982202255336/1339893]
[-911/159, -2982202255336/1339893]
[-215/159, 216095499400/1339893]
[-215/159, -216095499400/1339893]
[-91/159, 110096313296/1339893]
[-91/159, -110096313296/1339893]
[91/159, 110096313296/1339893]
[91/159, -110096313296/1339893]
[215/159, 216095499400/1339893]
[215/159, -216095499400/1339893]
[911/159, 2982202255336/1339893]
[911/159, -2982202255336/1339893]
[-131/187, 58935136200/594473]
[-131/187, -58935136200/594473]
[131/187, 58935136200/594473]
[131/187, -58935136200/594473]
[-347/188, 1118325845751/6644672]
[-347/188, -1118325845751/6644672]
[347/188, 1118325845751/6644672]
[347/188, -1118325845751/6644672]
[-500/193, 343484619060/7189057]
[-500/193, -343484619060/7189057]
[500/193, 343484619060/7189057]
[500/193, -343484619060/7189057]
[-1361/195, 11287425105968/2471625]
[-1361/195, -11287425105968/2471625]
[1361/195, 11287425105968/2471625]
[1361/195, -11287425105968/2471625]
[-31/233, 167648177400/12649337]
[-31/233, -167648177400/12649337]
[31/233, 167648177400/12649337]
[31/233, -167648177400/12649337]
[-2359/247, 196300617914280/15069223]
[-2359/247, -196300617914280/15069223]
[2359/247, 196300617914280/15069223]
[2359/247, -196300617914280/15069223]
[-649/269, 2044279894368/19465109]
[-649/269, -2044279894368/19465109]
[649/269, 2044279894368/19465109]
[649/269, -2044279894368/19465109]
[-1079/307, 2747876952480/28934443]
[-1079/307, -2747876952480/28934443]
[1079/307, 2747876952480/28934443]
[1079/307, -2747876952480/28934443]
[-1388/321, 7121718113180/11025387]
[-1388/321, -7121718113180/11025387]
[1388/321, 7121718113180/11025387]
[1388/321, -7121718113180/11025387]
[-2324/379, 156374901086388/54439939]
[-2324/379, -156374901086388/54439939]
[2324/379, 156374901086388/54439939]
[2324/379, -156374901086388/54439939]
[-194/389, 4232784414738/58863869]
[-194/389, -4232784414738/58863869]
[194/389, 4232784414738/58863869]
[194/389, -4232784414738/58863869]
[-79/459, 681426631456/32234193]
[-79/459, -681426631456/32234193]
[79/459, 681426631456/32234193]
[79/459, -681426631456/32234193]
[-775/528, 743562946645/4460544]
[-775/528, -743562946645/4460544]
[775/528, 743562946645/4460544]
[775/528, -743562946645/4460544]
[-217/535, 8946606766008/153130375]
[-217/535, -8946606766008/153130375]
[217/535, 8946606766008/153130375]
[217/535, -8946606766008/153130375]
[-1858/539, 228360540810/14235529]
[-1858/539, -228360540810/14235529]
[1858/539, 228360540810/14235529]
[1858/539, -228360540810/14235529]
[-2020/573, 6462212334700/62710839]
[-2020/573, -6462212334700/62710839]
[2020/573, 6462212334700/62710839]
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[-228643/158063, 654970595786640507936/3949032077556047]
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[-356173/166459, 683586284854299661560/4612345627840579]
[-356173/166459, -683586284854299661560/4612345627840579]
[356173/166459, 683586284854299661560/4612345627840579]
[356173/166459, -683586284854299661560/4612345627840579]
[-122773/193353, 218323531928880169312/2409525294532659]
[-122773/193353, -218323531928880169312/2409525294532659]
[122773/193353, 218323531928880169312/2409525294532659]
[122773/193353, -218323531928880169312/2409525294532659]
[-26305/220957, 9562986201008569200/980687430618863]
[-26305/220957, -9562986201008569200/980687430618863]
[26305/220957, 9562986201008569200/980687430618863]
[26305/220957, -9562986201008569200/980687430618863]
[-507469/290741, 34639370112764846568/203111011630901]
[-507469/290741, -34639370112764846568/203111011630901]
[507469/290741, 34639370112764846568/203111011630901]
[507469/290741, -34639370112764846568/203111011630901]
[-182377/399105, 1397121261214848811576/21190453504360875]
[-182377/399105, -1397121261214848811576/21190453504360875]
[182377/399105, 1397121261214848811576/21190453504360875]
[182377/399105, -1397121261214848811576/21190453504360875]
at least 576 rational points on C_2: y^2=278271081*x^2*(x^2-9)^2-229833600*(x^2-1)^2
time = 209 ms.

超楕円曲線C₂は少なくとも288*2=576個の有理点を持つことが分かる。

[2004.04.28追記]
ratpointsでmax_heightを10⁸で指定することにより、C₂の新たな有理点のx座標が2個見つかった。

(-4207979 : 680553)
(4207979 : 680553)

よって、新たに見つかったC₂の有理点は、以下の4個である。

[-4207979/680553, 311183882506716810456800/105066581873157459]
[-4207979/680553, -311183882506716810456800/105066581873157459]
[4207979/680553, 311183882506716810456800/105066581873157459]
[4207979/680553, -311183882506716810456800/105066581873157459]

超楕円曲線C₂は580個以上の有理点を持つことが分かった。

[2004.04.29追記]
■超楕円曲線C₂の有理点(x,y)を求めるには、楕円曲線
     E₂: Y² = 278271081X(X-9)²-229833600(X-1)²
の有理点(X,Y)で、X=□となるものを求めれば十分である。

楕円曲線E₂は、双有理変換
     x = 278271081X,
     y = 278271081Y
[逆変換は、
     X = x/278271081,
     Y = y/278271081
]
によって、楕円曲線
     E: y² = x³ - 5238713058x² + 6400130444837836641x - 17797117590000689845449600
に写される。

Cremonaのmwrank3では、必要な計算時間が大き過ぎるためか、楕円曲線EのMordell-Weil群の基底とrankを求めることができなかった。

[参考文献]

[1]Alice Silverberg, "Open Questions in Arithmetic Algebraic Geometry", pp84-142 :
Brain Conrad, Karl Rubin(Ed), "Arithmetic Algebraic Geometry", American Mathematical Society, Ω IAS/PARK CITY Mathematics Series Volume 9, 2001, ISBN0-8218-2173-3.
[2]Bjorn Poonen, "Computaional aspect of curves of genus at least 2", April 10, 1996, p1-21.

Last Update: 2005.08.21

H.Nakao