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Q:
Is a three-dimensional elliptic curve the only elliptic curve with infinitely many rational points?
In 3.5 of Silverman’s Rational Points on Elliptic Curves, he shows that the three-dimensional elliptic curve defined by $y^2+y=x^3+x$ has infinitely many rational points (over $\mathbb Q$).
My question is: what if the degree of the polynomial $y^2+y$ were greater than three? (Note that in the proof he does not actually make use of the degree being three.) Are there counterexamples?
A:
This is the only family of curves for which all nontrivial questions are completely understood.
EDIT: In the most recent paper where this is proved we also get an explicit parametrization of the counterexamples. The section Theorem 1.9(1) in the arXiv version says:
Let $\mathbb Q (a_1,\ldots,a_{2g+2})$ be the field generated by the rational functions $a_1,\ldots,a_{2g+2}$ and let $k=\mathbb{Q}(a_1,\ldots,a_{2g+2}) (\sqrt[k]{x})$. Then $k(\sqrt[k]{x})$ is the smallest intermediate field containing $\mathbb{Q}(\sqrt{x})$ and $k(\sqrt[k]{x})(\sqrt{x})$. If $c\in k(\sqrt[k
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