I have an explicit genus one curve $E$ with two points $p_1$ and $p_2$ on it and am looking for an explicit degree seven cover $X\to E$ with ramification precisely over $p_i$, with a single preimage point over each (by Hurwitz formula, $g(X)=7$). This should be a non-Galois extension, and the splitting field should have the Galois group ${\rm PSL}(2,7)$.

Literature search did not yield anything useful: there is a fair amount of work on covers of genus one curves with one ramification point (called "origami") and even that seems nontrivial. Similarly, there does not appear to be a simple general theory of ${\rm PSL}(2,7)$ Galois covers. Any suggestions will be appreciated.

I have an explicit genus one curve $E$ with two points $p_1$ and $p_2$ on it and am looking for an explicit degree seven cover $X\to E$ with ramification precisely over $p_i$, with a single preimage point over each (by Hurwitz formula, $g(X)=7$). This should be a non-Galois extension, and the splitting field should have the Galois group ${\rm PSL}(2,7)$.

Literature search did not yield anything useful: there is a fair amount of work on covers of genus one curves with one ramification point (called "origami") and even that seems nontrivial. Similarly, there does not appear to be a simple general theory of ${\rm PSL}(2,7)$ Galois covers. Any suggestions will be appreciated.

Added for clarity. My question is what algorithms/tricks can be used to explicitly compute the cover $X$, given explicit formulas for $E$, $p_1$ and $p_2$.