Suppose $f(x,y)$ is a polynomial over $\mathbb Q$ of degree $d>3$ , irreducible and smooth(for its projective closure). The Mordell conjecture states that $f(x,y)=0$ has finitely many solutions. I wonder if there is an example where the number of solutions is larger than $d^2$?
The example with the greatest number of solutions that I can find is something like
$$\prod_{n=1}^d(y-n^2)-\prod_{n=1}^d(x-n)=0.$$
(Here, I include the $y-n^2$ terms to make the function irreducible.) I believe it may be the case that this equation has exactly $d^2$ rational solutions.