fiber product of the $s$ pairs $(E_{1},\pi_{1}),\ldots,(E_{s},\pi_{s})$. Such a fiber product is given by the curve $\widehat{C}$ formed of the tuples $(x,y_{1},\ldots,y_{s})$ so that $(x,y_{j}) \in E_{j}$, for $j=1,\ldots,s$. This curve is reducible and contains two irreducible components, both of them being isomorphic. The curve $\widehat{C}$ admits the group of automorphisms $N=\langle f_{1},\ldots,f_{s}\rangle \cong {\mathbb Z}_{2}^{s}$, where $$f_{j}(x,y_{1},\ldots,y_{s})=(x,y_{1},\ldots,y_{j-1},-y_{j},y_{j+1},\ldots,y_{s}).$$

The two irreducible factors are permuted by some elements of $N$ and each one is invariant under a subgroup isomorphic to ${\mathbb Z}_{2}^{s-1}$.

In what follows we will construct a Riemann surface $S$ which is isomorphic to the irreducible components of $\widehat{C}$.

Let us now consider the (affine) generalized Humbert curve (see [@C-G-H-R] for details) $$D: \left\{ \begin{array}{ccc}
z_{1}^{2}+z_{2}^{2}+z_{3}^{2}&=&0\\
\lambda z_{1}^{2}+z_{2}^{2}+z_{4}^{2}&=&0\\
\mu_{1,1}z_{1}^{2}+z_{2}^{2}+z_{5}^{2}&=&0\\
\mu_{1,2}z_{1}^{2}+z_{2}^{2}+z_{6}^{2}&=&0\\
\vdots & \vdots& \vdots\\
\mu_{k,1}z_{1}^{2}+z_{2}^{2}+z_{2k+3}^{2}&=&0\\
\mu_{k,2}z_{1}^{2}+z_{2}^{2}+z_{2k+4}^{2}&=&0\\
\vdots & \vdots& \vdots\\
\mu_{s-2,1}z_{1}^{2}+z_{2}^{2}+z_{2s-1}^{2}&=&0\\
\mu_{s-2,2}z_{1}^{2}+z_{2}^{2}+1&=&0
\end{array}
\right\}.$$

The conditions on the parameters ensure that $D$ is a non-singular algebraic curve, that is, a closed Riemann surface. On $D$ we have the abelian group $\langle b_{1},\ldots,b_{2s-1}\rangle=H_{0} \cong {\mathbb Z}_{2}^{2s-1}$ of conformal automorphisms, where $$b_{j}(z_{1},\ldots,z_{2s-1})=(z_{1},\ldots,z_{j-1}, - z_{j},z_{j+1},\ldots,z_{2s-1}), \; j=1,...,2s-1.$$

Inside the group $H_{0}$, the only non-trivial elements acting with fixed points are $b_{1},\ldots, b_{2s-1}$ and $b_{2s}=b_{1}b_{2}\cdots b_{2s-1}$. The degree $2^{2s-1}$ holomorphic map $$P:D \to 