Quotient of algebraic curve

Question

Say we have an algebraic curve $C:s^6 = t^6 + 1$ defined over $\mathbb{F}_p$, and the automorphism $\phi :(s,t) \mapsto (s,\xi_6t)$ (we assume this is defined over the base field). I am trying to construct equations for the quotient curve $C/<\phi>$ and the corresponding quotient map. Magma seems do be able to do it, but I can't find any literature on the topic. Anyone knows how to do this or has a reference? Very much appreciated!