Rational points on genus 3 curves defined by short equations

Question

(a) Find all pairs of rational numbers $(x,y)$ such that $$ y^3-y=x^4-x. $$ (b) Find all pairs of rational numbers $(x,y)$ such that $$ y^3+y=x^4+x. $$ If not a complete answer, I would be happy to receive some comments on the difficulty of these equations. Are they any easier than general genus $3$ equations?

Answer 1

I have looked a bit at the first equation. It has (at least) seven rational points (as a projective curve). The differences of these points generate a free abelian group of rank three in the Mordell-Weil group of the Jacobian $J$ of the curve; the MW group has no torsion. So even if one could determine that the known subgroup is of finite index (this would require computing the 2-Selmer group or (assuming BSD for the Jacobian) an upper bound for the order of vanishing of the L-function), standard Chabauty would not be sufficient. Also, the endomorphism ring of $J$ over $\mathbb Q$ is only $\mathbb Z$, so there are no obvious additional structures one could hope to exploit. So, in a way, at least the first curve looks like it will be on the upper end of the range of difficulty for "small" plane quartic curves.

Now for the second curve. It has four small rational points, whose differences generate a free abelian group of rank two; again the MW group has no torsion and the endomorphism ring of the Jacobian is only $\mathbb Z$. If one could verify that the Mordell-Weil rank is indeed two, this case would be amenable to Chabauty's method, so I would rank this one as probably within reach.