Presentation for a Finite Etale Cover of an (Affine) Elliptic Curve

Question

I will write the question first, then try and explain myself more clearly after:

Question: How can one find a presentation for a finite etale cover of an affine piece of an elliptic curve?

If $E/\mathbb{C}$ is an elliptic curve, then it is homeomorphic to a torus. For this reason we know $\pi_{1}(E) \cong \mathbb{Z}\times \mathbb{Z}$. If we puncture $E/\mathbb{C}$, then we obtain an affine curve, let us denote it $X$. Moreover, $X$ can be thought of as the elliptic curve minus the point at infinity. Since this is affine, we can give a presentation for it, say, $$X:= \text{Spec}(\mathbb{C}[x,y]/\langle y^{2} - f(x) \rangle)$$ for $f(x) = x(x-1)(x-2)$.

We know $\pi_{1}^{et}(X) \cong \hat{\pi_{1}(X(\mathbb{C})^{an})}$ and $X(\mathbb{C})^{an}$ is homeomorphic to a punctured torus. The fundamental group of a punctured torus is the free group on two generators $F_{2}$. The profinite completion of which is non-trivial. So, there are algebraic covers of such an affine curve. How can we get our hands on them? Is there a presentation which in some sense is in terms of $f(x)$? Does anyone have a reference for a discussion on such a construction?

Thanks in advance :)

Answer 1

Question: How can one find a presentation for a finite etale cover of an affine piece of an elliptic curve?

If $E/\mathbb{C}$ is an elliptic curve, then it is homeomorphic to a torus. For this reason we know $\pi_{1}(E) \cong \mathbb{Z}\times \mathbb{Z}$. If we puncture $E/\mathbb{C}$, then we obtain an affine curve, let us denote it $X$. Moreover, $X$ can be thought of as the elliptic curve minus the point at infinity. Since this is affine, we can give a presentation for it, say, $$X:= \text{Spec}(\mathbb{C}[x,y]/\langle y^{2} - f(x) \rangle)$$ for $f(x) = x(x-1)(x-2)$.

In general, $X$ admits such a presentation only if one allows $f(x)$ to be a more general monic polynomial of degree $3$, i.e. not necessarily $x(x - 1)(x - 2)$.

We know $\pi_{1}^{et}(X) \cong \hat{\pi_{1}(X(\mathbb{C})^{an})}$ and $X(\mathbb{C})^{an}$ is homeomorphic to a punctured torus. The fundamental group of a punctured torus is the free group on two generators $F_{2}$. The profinite completion of which is non-trivial. So, there are algebraic covers of such an affine curve. How can we get our hands on them? Is there a presentation which in some sense is in terms of $f(x)$? Does anyone have a reference for a discussion on such a construction?

I do not, nor do the experts, know of explicit polynomial presentations of the sort given above---i.e. $y^2 - f(x)$---for general algebraic covers of $X$. On the other hand, most if not all of the significant research concerning punctured elliptic curves that has been carried out until now---such as, for instance, research concerning anabelian geometry---may be understood without any consideration whatsoever of such explicit polynomial presentations.