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I'm trying to classify all of the rational points on the hyperelliptic curve $y^2 = x^5 - 6x^3 + \frac{2}{9} x^2 -3x$.

This is a genus 2 curves and MAGMA gives a RankBound on this curve's Jacobian of 2, so I cannot yet use the method of Chabauty. How can I find all of the rational points?

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    $\begingroup$ Why this curve? That is, how does this genus-2 curve arise? $\endgroup$
    – Noam D. Elkies
    Commented Dec 9, 2016 at 23:49
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    $\begingroup$ There are two rational Weierstrass points, so there a "descent" that might work more easily than what one must usually resort to: there are only a few possible choices of $x$ mod squares, each of which yields a hyperelliptic genus-$3$ curve; if in each case the genus-$1$ quotient happens to have finitely many rational points, you're done. $\endgroup$
    – Noam D. Elkies
    Commented Dec 9, 2016 at 23:52
  • $\begingroup$ This curves is a modular curve that parametrizes elliptic curves with a particular torsion structure I'm interested in. Could you elaborate on the "few possible choices of $x$ mod squares" statement? $\endgroup$
    – M C
    Commented Dec 10, 2016 at 0:33
  • $\begingroup$ Are you sure about that? There's bad reduction at $3,5,7,37$ and it seems unlikely that anything involving $3885$-torsion would give a curve of genus $2$ or anywhere near that low. ("Few possible choices": write $x=m/n$ in lowest terms and get square = $mnP(m,n)$ for some homogeneous octic $P$, and consider possible common factors. $\endgroup$
    – Noam D. Elkies
    Commented Dec 10, 2016 at 3:31

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