Let $k$ be any number field, and suppose we want to study the $k$-rational points on $$y^2 x = f(x),$$ where $f$ is a polynomial of degree greater or equal than 3. In other words, $y^2 x = f(x)$ is a sort of twisted elliptic or hyperelliptic curve. Question: What are the techniques available to tackle equations like $y^2 x = f(x)$?
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5$\begingroup$ Substitute $z=yx$, then the rational points with $x\ne 0$ correspond to rational points of $z^2=xf(x)$. $\endgroup$– Michael ZieveCommented Oct 3, 2013 at 6:58
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1$\begingroup$ I should have thought of that. Thanks! $\endgroup$– user40810Commented Oct 3, 2013 at 7:06
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