Do you know explicit examples of superelliptic curves $y^{\ell} = g(x)$ (for some prime $\ell > 3$) covering some elliptic curves?

Question

For every elliptic curve $E$ Icart in $\S 2$ of the paper explicitly constructs a superelliptic curve $S\!: y^3 = f(x)$ and a cover $\varphi\!: S \to E$. Do you know explicit examples of superelliptic curves $T\!: y^{\ell} = g(x)$ (for some prime $\ell > 3$) and explicit covers $\psi\!: T \to E$ at least onto some elliptic curves $E$ over $\overline{\mathbb{Q}}$ with complex multiplication?

Thanks in advance.

Answer 1

Interestingly, I only found the Klein quartic $K = x^3y + y^3z + z^3x$ and the Fermat curve $F = x^7 + y^7 + z^7$. Both of them cover (at least over $\overline{\mathbb{Q}}$) the elliptic curve of $j$-invariant $-3^3 5^3$ with complex multiplication by $\mathbb{Z}[(1 + \sqrt{-7})/2]$. As is known, $F$ covers $K$ and $K$ is birationally isomorphic to the superelliptic curve $y^7 - x(1-x)^2$.