a_{s-1}a_{s},a_{1},\ldots,a_{s-2}\rangle$$ $$H_{s,1}=\langle a_{s}a_{1},a_{2},\ldots,a_{s-1}\rangle$$

The quotient orbifold $C/H_{j,j+1}$ has underlying Riemann surface structure $E_{j} \cong E_{\lambda_{j}}$, for $j=1,\ldots,s-1$, and $C/H_{s,1}$ has underlying Riemann surface structure $E_{s} \cong E_{\lambda_{s}}$.

\[lema02\] With the above notations, the following hold for the above defined subgroups.

1.  Any two such subgroups $H_{i_{1},i_{2},\ldots,i_{k}}$ and $H_{j_{1},j_{2},\ldots,j_{l}}$ commute.

2.  The quotient $C/H_{i_{1},i_{2},\ldots,i_{k}}$ has genus $k-1$ and its underlying Riemann surface is given by the (elliptic) hyperelliptic curve $$\nu^{2}=(\upsilon-\rho_{i_{1},1})(\upsilon-\rho_{i_{1},2})\cdots(\upsilon-\rho_{i_{k},1})(\upsilon-\rho_{i_{k},2}),$$ where $$\rho_{i_{j},1}=\left\{\begin{array}{cl}
    \infty, & i_{j}=1\\
    1, & i_{j}=2\\
    \mu_{r-2,1}, & i_{j}=r \geq 3
    \end{array}
    \right. \quad 
    \rho_{i_{j},2}=\left\{\begin{array}{cl}
    0, & i_{j}=1\\
    \lambda, & i_{j}=2\\
    \mu_{r-2,2}, & i_{j}=r\geq 3
    \end{array}
    \right.$$ In the case that $\rho_{i_{j},1}=\infty$, then the factor $(u-\rho_{i_{j},1})$ is deleted from the above expression.

3.  The group generated by any two different such subgroups is $H$.

Property (1) holds trivially as $H$ is an abelian group. Property (2) follows from Riemann-Hurwitz formula and Lemma \[lema01\]. Property (3) is clear as in the product we obtain all the generators.

The next result states that the sum of the genera appearing in all quotients of the form $C/H_{i_{1},i_{2},\ldots,i_{k}}$ is equal to the genus of $C$ (that is, the genus of $S$). Recall that we are considering $k$ even and $2 \leq k \leq 