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when I Find the diophantine-equation rational points $$2y^2=x^6-x^2+2$$

I using Faltings's theorem showed that there are only finitely many solutions,if we assmue that $(x_{i},y_{i}),i=1,2,\cdots,N$ is solution,and $x_{i}=\dfrac{p_{i}}{q_{i}},i=1,2,\cdots,N$,can we estimate upper bound of $|p_{i}|$ or $|q_{i}|?$

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    $\begingroup$ This is a very tough sort of problem. You might take a look at Bjorn Poonen's article here that talks about the challenges of finding rational points on curves. The Jacobian of your hyperelliptic curve has rank 3, and so the method of Chabauty is out. There is an etale double cover of your curve that maps to an elliptic curve over a quartic number field, and one might be able to get elliptic curve Chabauty to work in this context. $\endgroup$ – Jeremy Rouse Jun 27 '15 at 17:32
  • $\begingroup$ Correction - the elliptic curve is defined over a cubic number field. $\endgroup$ – Jeremy Rouse Jun 27 '15 at 17:51
  • $\begingroup$ math110, exactly where did you get this problem? $\endgroup$ – Will Jagy Jun 27 '15 at 17:58
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    $\begingroup$ @WillJagy,when I deal this problem get it:mathoverflow.net/questions/208662/… $\endgroup$ – math110 Jun 28 '15 at 3:03

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