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Rational Points on Genus 3 Curve: y^2=7920000(x^2+1)^4-136782591x^2(x^2-1)^2

[2004.04.24]y^2=7920000(x^2+1)^4-136782591x^2(x^2-1)^2の有理点

■参考文献[1](p130,Theorem[Kuller-Kulesz] 6.2.3)に記述されているgenus 3の超楕円曲線
     C1: y2 = 7920000(x2+1)4-136782591x2(x2-1)2
の有理点は、176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C1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C1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> g3(x)=7920000*(x^2+1)^4-136782591*x^2*(x^2-1)^2
time = 2 ms.
gp> g3(x)
time = 9 ms.
%1 = 7920000*x^8 - 105102591*x^6 + 321085182*x^4 - 105102591*x^2 + 7920000
[ratpoints-1.4による計算]
bash-2.05a$ ./ratpoints '7920000 0 -105102591 0 321085182 0 -105102591 0 7920000' 100000

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

Please acknowledge use of the program in published work.


y^2 = 7920000 x^8 - 105102591 x^6 + 321085182 x^4 - 105102591 x^2 + 7920000

max. Height = 100000
Search region:
  [-100000.500000, 100000.500000]
Using speed ratios 1000.000000 and 12.000000
12 primes used for first stage of sieving,
44 primes used for both stages of sieving together.
Sieving primes:
 First stage: 131, 179, 181, 197, 229, 211, 173, 73, 151, 199, 239, 233
 Second stage: 191, 113, 227, 193, 223, 97, 251, 139, 167, 89, 67, 241, 37, 79, 157, 71, 149, 47, 109, 137, 41, 103, 61, 43, 101, 107, 53, 83, 17, 19, 23, 31
Probabilities: Min(131) = 0.488550, Cut1(233) = 0.583691, Cut2(31) = 0.903226, Max(7) = 1.000000

Forbidden divisors of the denominator:
  17, 19, 23, 31, 37, 41, 43, 47, 53, 71, 73, 83, 103, 107, 131, 139, 157, 181, 191, 193, 199, 211, 223, 227, 229, 233, 241

(-59 : 1)
(-25 : 1)
(-9 : 1)
(-5 : 1)
(-4 : 1)
(-3 : 1)
(-2 : 1)
(2 : 1)
(3 : 1)
(4 : 1)
(5 : 1)
(9 : 1)
(25 : 1)
(59 : 1)
(-3 : 2)
(-1 : 2)
(1 : 2)
(3 : 2)
(-5 : 3)
(-2 : 3)
(-1 : 3)
(1 : 3)
(2 : 3)
(5 : 3)
(-25 : 4)
(-5 : 4)
(-1 : 4)
(1 : 4)
(5 : 4)
(25 : 4)
(-4 : 5)
(-3 : 5)
(-1 : 5)
(1 : 5)
(3 : 5)
(4 : 5)
(-1 : 9)
(1 : 9)
(-13 : 12)
(13 : 12)
(-12 : 13)
(12 : 13)
(-29 : 21)
(29 : 21)
(-79 : 25)
(-4 : 25)
(-1 : 25)
(1 : 25)
(4 : 25)
(79 : 25)
(-52 : 27)
(52 : 27)
(-101 : 29)
(-30 : 29)
(-21 : 29)
(21 : 29)
(30 : 29)
(101 : 29)
(-97 : 30)
(-29 : 30)
(29 : 30)
(97 : 30)
(-65 : 36)
(65 : 36)
(-27 : 52)
(27 : 52)
(-1 : 59)
(1 : 59)
(-197 : 60)
(197 : 60)
(-36 : 65)
(36 : 65)
(-127 : 67)
(127 : 67)
(-25 : 79)
(25 : 79)
(-30 : 97)
(30 : 97)
(-29 : 101)
(29 : 101)
(-67 : 127)
(67 : 127)
(-257 : 137)
(257 : 137)
(-60 : 197)
(60 : 197)
(-137 : 257)
(137 : 257)

12382132 candidates survived the first stage,
132 candidates survived the second stage.

88 rational point pairs found.
bash-2.05a$
よって、超楕円曲線C1の有理点のx座標が88個求まった。

■超楕円曲線C1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 = 44 ms.
gp> rpc1(c1x)
[-59, 34036269960]
[-59, -34036269960]
[-25, 1087640400]
[-25, -1087640400]
[-9, 16946160]
[-9, -16946160]
[-5, 1284360]
[-5, -1284360]
[-4, 411180]
[-4, -411180]
[-3, 20328]
[-3, -20328]
[-2, 5082]
[-2, -5082]
[2, 5082]
[2, -5082]
[3, 20328]
[3, -20328]
[4, 411180]
[4, -411180]
[5, 1284360]
[5, -1284360]
[9, 16946160]
[9, -16946160]
[25, 1087640400]
[25, -1087640400]
[59, 34036269960]
[59, -34036269960]
[-3/2, 160545/8]
[-3/2, -160545/8]
[-1/2, 2541/8]
[-1/2, -2541/8]
[1/2, 2541/8]
[1/2, -2541/8]
[3/2, 160545/8]
[3/2, -160545/8]
[-5/3, 548240/27]
[-5/3, -548240/27]
[-2/3, 107030/27]
[-2/3, -107030/27]
[-1/3, 6776/27]
[-1/3, -6776/27]
[1/3, 6776/27]
[1/3, -6776/27]
[2/3, 107030/27]
[2/3, -107030/27]
[5/3, 548240/27]
[5/3, -548240/27]
[-25/4, 227730525/64]
[-25/4, -227730525/64]
[-5/4, 1059135/64]
[-5/4, -1059135/64]
[-1/4, 102795/64]
[-1/4, -102795/64]
[1/4, 102795/64]
[1/4, -102795/64]
[5/4, 1059135/64]
[5/4, -1059135/64]
[25/4, 227730525/64]
[25/4, -227730525/64]
[-4/5, 847308/125]
[-4/5, -847308/125]
[-3/5, 328944/125]
[-3/5, -328944/125]
[-1/5, 256872/125]
[-1/5, -256872/125]
[1/5, 256872/125]
[1/5, -256872/125]
[3/5, 328944/125]
[3/5, -328944/125]
[4/5, 847308/125]
[4/5, -847308/125]
[-1/9, 5648720/2187]
[-1/9, -5648720/2187]
[1/9, 5648720/2187]
[1/9, -5648720/2187]
[-13/12, 22659175/1728]
[-13/12, -22659175/1728]
[13/12, 22659175/1728]
[13/12, -22659175/1728]
[-12/13, 271910100/28561]
[-12/13, -271910100/28561]
[12/13, 271910100/28561]
[12/13, -271910100/28561]
[-29/21, 1214562800/64827]
[-29/21, -1214562800/64827]
[29/21, 1214562800/64827]
[29/21, -1214562800/64827]
[-79/25, 1112573616/15625]
[-79/25, -1112573616/15625]
[-4/25, 36436884/15625]
[-4/25, -36436884/15625]
[-1/25, 43505616/15625]
[-1/25, -43505616/15625]
[1/25, 43505616/15625]
[1/25, -43505616/15625]
[4/25, 36436884/15625]
[4/25, -36436884/15625]
[79/25, 1112573616/15625]
[79/25, -1112573616/15625]
[-52/27, 2317861700/177147]
[-52/27, -2317861700/177147]
[52/27, 2317861700/177147]
[52/27, -2317861700/177147]
[-101/29, 122193289680/707281]
[-101/29, -122193289680/707281]
[-30/29, 8509067490/707281]
[-30/29, -8509067490/707281]
[-21/29, 3643688400/707281]
[-21/29, -3643688400/707281]
[21/29, 3643688400/707281]
[21/29, -3643688400/707281]
[30/29, 8509067490/707281]
[30/29, -8509067490/707281]
[101/29, 122193289680/707281]
[101/29, -122193289680/707281]
[-97/30, 2491897331/27000]
[-97/30, -2491897331/27000]
[-29/30, 283635583/27000]
[-29/30, -283635583/27000]
[29/30, 283635583/27000]
[29/30, -283635583/27000]
[97/30, 2491897331/27000]
[97/30, -2491897331/27000]
[-65/36, 2545693535/139968]
[-65/36, -2545693535/139968]
[65/36, 2545693535/139968]
[65/36, -2545693535/139968]
[-27/52, 1738396275/1827904]
[-27/52, -1738396275/1827904]
[27/52, 1738396275/1827904]
[27/52, -1738396275/1827904]
[-1/59, 34036269960/12117361]
[-1/59, -34036269960/12117361]
[1/59, 34036269960/12117361]
[1/59, -34036269960/12117361]
[-197/60, 23143701119/216000]
[-197/60, -23143701119/216000]
[197/60, 23143701119/216000]
[197/60, -23143701119/216000]
[-36/65, 6109664484/3570125]
[-36/65, -6109664484/3570125]
[36/65, 6109664484/3570125]
[36/65, -6109664484/3570125]
[-127/67, 299027679720/20151121]
[-127/67, -299027679720/20151121]
[127/67, 299027679720/20151121]
[127/67, -299027679720/20151121]
[-25/79, 27814340400/38950081]
[-25/79, -27814340400/38950081]
[25/79, 27814340400/38950081]
[25/79, -27814340400/38950081]
[-30/97, 74756919930/88529281]
[-30/97, -74756919930/88529281]
[30/97, 74756919930/88529281]
[30/97, -74756919930/88529281]
[-29/101, 122193289680/104060401]
[-29/101, -122193289680/104060401]
[29/101, 122193289680/104060401]
[29/101, -122193289680/104060401]
[-67/127, 299027679720/260144641]
[-67/127, -299027679720/260144641]
[67/127, 299027679720/260144641]
[67/127, -299027679720/260144641]
[-257/137, 5554488268560/352275361]
[-257/137, -5554488268560/352275361]
[257/137, 5554488268560/352275361]
[257/137, -5554488268560/352275361]
[-60/197, 1388622067140/1506138481]
[-60/197, -1388622067140/1506138481]
[60/197, 1388622067140/1506138481]
[60/197, -1388622067140/1506138481]
[-137/257, 5554488268560/4362470401]
[-137/257, -5554488268560/4362470401]
[137/257, 5554488268560/4362470401]
[137/257, -5554488268560/4362470401]
at least 176 rational points on C_1: y^2=7920000*(x^2+1)^4-136782591*x^2*(x^2-1)^2
time = 135 ms.
超楕円曲線C1は少なくとも88*2=176個の有理点を持つことが分かる。


[参考文献]

Last Update: 2008.05.25
H.Nakao

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