Let $C$ be a curve of genus 3 and suppose that it admits a branched cover $\varphi:C\rightarrow E$ with $E$ elliptic and such that $\varphi$ does not factor through any \' etale cover. Then the degree of the ramification divisor $R$ is $4$ and the branch divisor $B\subseteq E$ is supported in a set of cardinality between 1 and 4. I wonder if all 4 hypothesis are possible and given a branch locus, which is the minimun degree for which it can be achieved.
Obviously if $B$ is supported on 4 point I can take a double cover ramified on 4 point.
Let us suppose that $B$ is supported on 3 point. Then if I understood well the theory, the existence of a branched covering of degree $n$ such as mine is equivalent to the existence of a primitive, transitive subgroup of $S\_n$ generated by 5 element $a,b,g\_1,g\_2,g\_3$ such that $[a,b]g\_1g\_2g\_3=\mathrm{id}$. Is this correct? Does this groupo exist? for which $n$?
If the above is correct then I have also to find a primitive transitive subgroup of $S\_n$ 
generated by $a,b,g\_1, g\_2$ satisfying $[a,b]g\_1g\_2=\mathrm{id}$. And finally I would need a primitive transitive subgroup of $S\_n$ generated by three element with relation $[ab]g=\mathrm{id}$.
Thank you very much for your help!



Configuration of the branch locus of  a branched covering of an elliptic curve
According to Sofia's and my comments to the original question, we look for instance for elements $a,b\in S\_n$ which generate a primitive subgroup of $S\_n$ and where the commutator $[a,b]=a^{-1}b^{-1}ab$ is a $5$-cycle.


This is possible for each $n\ge5$: Set $a=(1,2\,3,\cdots,n)$, $b=(1,2,\cdots,n-4,n-3,n-1,n-2)$. Then $[a,b]=(1,2,n-2,n-1,n)$. Note that in this computation, I use the right action of the symmetric group.


Note that $a$ is an $n$--cycle, so $G=\langle a,b\rangle$ is transitive. Furthermore, $b$ is an $(n-1)$-cycle, so $G$ is doubly transitive and hence primitive. Actually, it is not difficult to show that even $G=S\_n$.


So this is an example for a branch locus of size $1$. The size $2$ can be obtained by replacing the $5$-cycle $g\_1$ by two $3$-cycles $h\_1$, $h\_2$ with $g\_1=h\_1h\_2$. The branch locus of size $3$ or $4$ will be obtained by further splitting up $h\_1$ or $h\_1$ and $h\_2$ into a product of two transpositions.


The cases of $2$, $3$ and $4$ branch points however can be realized with smaller degrees, namely $n=3$ for $2$ and $3$ branch points, and $n=2$ for $4$ branch points. These are the minimal degrees.


