Let $X$ be an algebraic curve defined over a number field $K$ and has genus $g \geq 2$. Let $P$ be a $K$-point of $X$. We say that $X_P$ is a Parshin cover of $X$ if $X_P$ is defined over a finite field extension $K'$ of $K$ and such that there is a morphism $X_P \rightarrow X$ defined over $K$ which is only ramified at $P$.
Mazur gave a very short and easy-to-understand construction here, which is supposedly similar to Parshin's original construction. However the construction seems to pass through Riemann surface theory and so it is not clear to me how to produce explicit algebraic equations from it.
Are there any examples, say when $X$ is a fairly simple curve in $\mathbb{P}^2$ say with a straightforward defining equation, such that one can construct a Parshin cover explicitly? For example, take $X$ to be the Fermat curve $x^n + y^n = z^n$ with $n \geq 5$. Can one give an explicit Parshin cover for it?
