The cohomology of an $S_{3}$ cover of an elliptic curve ramified in one point Let $E/\mathbb{C}$ be an elliptic curve. Let $C \to E$ be a Galois cover with group $G = S_{3}$ (symmetric group on $3$ elements), ramified in one point. (To clarify: there is a unique point in $E$ over which the cover is branched. Thanks Jason.) Then $C$ is a curve of genus $3$.
I want to understand $H^{1}(C, \mathbb{Q})$. The $G$-invariants are $2$-dimensional, $H^{1}(C, \mathbb{Q})^{G} = H^{1}(E, \mathbb{Q})$. Using Chevalley–Weil one can compute the representation of $G$ on the complement $H^{1}(C, \mathbb{Q})^{\perp}$. It consists of twice the $2$-dimensional irrep $V$ of $G$.
Consequently there is an action of $M_{2}(\mathbb{Q})$ on $H^{1}(C, \mathbb{Q})^{\perp}$, and in the category of $\mathbb{Q}$-Hodge structures, we obtain $H^{1}(C, \mathbb{Q})^{\perp} = H \otimes V$, where $H$ is a $2$-dimensional Hodge structure of type $(1,0) + (0,1)$, and $V$ is the $2$-dimensional irrep of $G$.
Therefore, $H$ “is” an elliptic curve (up to isogeny).

Q. Can we explicitly understand the isogeny class of $H$? Is it the same as $E$?


One actually does not have to use the Hodge theory. The cover $C \to E$ gives rise to a map $\mathrm{Jac}(C) \to \mathrm{Jac}(E)$. The kernel $A$ is an abelian variety of dimension $2$, with an action of $M_{2}(\mathbb{Z})$. Up to isogeny, we get $A \cong \smash{E'}^{2}$. Note that $H^{1}(E', \mathbb{Q}) \cong H$ (the Hodge structure introduced above).
So, equivalently, the question is: how does $E'$ relate to $E$?
 A: I am going to take a shot at partially answering this question myself. (Parts of these arguments are due to Ben Moonen, my supervisor. All errors are mine.) Any feedback is welcome!
The exact relation between $E'$ and $E$ is possibly difficult to describe more explicitly than the tautological: “given by some correspondence on $\mathcal{A}_{1} \times \mathcal{A}_{1}$”.
What I can show:


*

*$E$ and $E'$ are not isogenous in general.

*$E'$ does depend on $E$ (i.e. is not constant in $\mathcal{A}_{1}$).



Proof of claim 1
Note that the cover $C \to E$ factors as $C \to E_{1} \to E$, where


*

*$C \to E_{1}$ is the quotient of $C_{3} \subset S_{3}$, and

*$E_{1} \to E$ is induced $C_{2}$-cover, which is explicitly given by modding out some $2$-torsion point $P \in E_{1}[2]$.


The study of the cover $C \to E$ may hence be replaced by the study of $C_{3}$-covers $C \to E_{1}$ ramified above the origin and a $2$-torsion points such that the corresponding monodromies are inverse to each other.
If we compose $C \to E_{1}$ with the hyperelliptic quotient map $E_{1} \to \mathbb{P}^{1}$, we obtain a $C_{6}$-cover $C \to \mathbb{P}^{1}$ ramified above $4$ points. More precisely, above two of these points lies $1$ point, while above the other two there lie $3$ points. We thus have a family of cyclic covers of $\mathbb{P}^{1}$.
Ben Moonen (my supervisor) has studied [M] the Jacobian of these covers, to see if they give rise to special subvarieties of $\mathcal{A}_{g}$. In his notation, this family corresponds to ramification data $(1,5,3,3)$, which is not in his list (p.10 (p.508 in print)). By the André–Oort conjecture (in low dimension a theorem) it follows that the Jacobian of $C$ is CM only finitely many times. Since $\mathrm{Jac}(C) \sim E \times \smash{E'}^{2}$, this implies that $E \not\sim E'$, proving the first claim.

Proof of claim 2
I am not very well versed in the language of Shimura varieties, so the moduli space of elliptic curves with a marked point of order $2$ might have a standard notation. I'll just call it $\mathcal{M}$. Over this, we have a universal family $\mathcal{E}$, and over that a $C_{3}$-cover $\mathcal{C}$. We can compactify everything and look at the boundary. Obviously $\mathcal{M} \subset \bar{\mathcal{M}}_{1,2}$.
One point on the boundary of $\mathcal{M}$ can be described as follows:


*

*mark $\mathbb{P}^{1}$ with the points $\{0,\infty\}$,

*attach a nodal curve to the $\mathbb{P}^{1}$ at $\{1\}$.


The dual graph of this curve consists of two vertices joined by an edge; one vertex has a loop, and the other has two half-edges (corresponding to the marked points). This has a $C_{3}$-cover:


*

*take a $\mathbb{P}^{1}$ and attach three nodal curves to the $3$rd roots of unity.


The map on $\mathbb{P}^{1}$ given by $z \mapsto z^{3}$ is ramified at $\{0,\infty\}$, and can be extended to the nodal curves.
The dual graph contains $3$ loops (from the nodal curves), and hence the toric rank of the Jacobian is $3$. The procedure of assigning $E'$ to $E_{1}$ (described partly in the question, and partly in the proof of claim 1) gives a map $\mathcal{M} \to \mathcal{A}_{1}, E_{1} \mapsto E'$. The above example shows that if we pass to the closures $\bar{\mathcal{M}} \to \bar{\mathcal{A}}_{1}$, the map goes through the boundary of $\mathcal{A}_{1}$, and is in particular not constant. This proves the second claim.

References
[M]: B. Moonen — Special subvarieties arising from families of cyclic covers of the projective line. Documenta Math. 15 (2010), 793-819. 
