Equations for double etale covers of the hyperelliptic curve $y^2 = x^5+1$ Let $X$ be the (smooth projective model) of the hyperelliptic curve $y^2=x^5+1$ over $\mathbf C$.
Can we "easily" write down equations for all double unramified covers of $X$?
Topologically, these covers correspond to (normal) subgroups of index two in the fundamental group of $X$. So there's only a finite number of them. The function field of $X$ is $K=\mathbf C(x)[y]/(y^2-x^5-1)$. A double etale cover of $X$ corresponds to  (a certain) quadratic extension $L/K$. I guess every quadratic extension of $K$ is given by taking the root of some element $D$ in $K$, or by adjoining $(1+\sqrt{D})/2$ to $K$ for some $D$.
I didn't get much further than this though.

As my reputation is still too low, I could not add a comment to your answer Cantlog. What I wanted to say is the following: shouldn't it be true that the divisor of the rational function $f_{i}$ you define equals $2(w_{1} - w_{i})$, as the order of vanishing of the rational function $x$ at the point $(0,0)$ is $2$? Then indeed the class of the divisor $w_{1} - w_{i}$ is a 2-torsion point on the Jacobian of $X$, as twice that divisor becomes rationally equivalent to the class of the zero divisor. 
