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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
    Commented Jun 27, 2015 at 17:32
  • $\begingroup$ Correction - the elliptic curve is defined over a cubic number field. $\endgroup$
    – Jeremy Rouse
    Commented Jun 27, 2015 at 17:51
  • $\begingroup$ math110, exactly where did you get this problem? $\endgroup$
    – Will Jagy
    Commented Jun 27, 2015 at 17:58
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    $\begingroup$ @WillJagy,when I deal this problem get it:mathoverflow.net/questions/208662/… $\endgroup$
    – math110
    Commented Jun 28, 2015 at 3:03

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