closed Riemann surface $S$ whose jacobian variety is isogenous to the product $E_{1} \times E_{2} \times E_{3}$. We may use the constructed Riemann surface in Theorem \[construccion\], for $s=3$, to construct explicitly equations for such surface $S$.

\[coro1\] Let $E_{1}$, $E_{2}$ and $E_{3}$ be three elliptic curves. Choose $(\lambda_{1},\lambda_{2},\lambda_{3}) \in \Delta_{3}$ so that $E_{j}$ is isogenous to $E_{\lambda_{j}}$, for $j=1,2,3$, and let $\mu$ be a root of $$\lambda_{2}\lambda_{3} \mu^{2}-(\lambda_{1}\lambda_{2}+\lambda_{2}\lambda_{3}+\lambda_{1}\lambda_{3}-\lambda_{1}-\lambda_{3}+1) \mu +\lambda_{1}\lambda_{2}=0.$$

If $S$ is the genus three Riemann surface defined by the curve $$\left\{ \begin{array}{lcl}
w_{1}^{2}&=&\mu(\lambda_{3} \mu -1)(\lambda_{3} \mu -\lambda_{1})(\lambda_{3} - 1)
z \left( z-\dfrac{1}{\lambda_{1}-\lambda_{3}\mu} \right) \left(z-\dfrac{1}{\mu (1-\lambda_{3})} \right)\\
w_{2}^{2}&=&-\lambda_{3}\mu^{2}(\lambda_{3}-1)z\left(z+\dfrac{1}{\lambda_{3} \mu}\right)  \left(z-\dfrac{1}{1-\lambda_{3}\mu} \right)\\
w_{3}^{2}&=&-\lambda_{3}\mu^{2}(\lambda_{3}\mu-1)(\lambda_{3}-1)z^{2}\left(z+\dfrac{1}{\lambda_{3} \mu}\right) \left(z-\dfrac{1}{\mu(1-\lambda_{3})}\right) \\
\end{array}
\right\},$$ then $JS$ is isogenous to the product $E_{1} \times E_{2} \times E_{3}$.

Upper bounds for $e(r)$, $r \geq 4$
-----------------------------------

Another direct consequence of the construction provided by Theorem \[construccion\] is the following upper bound for $e(r)$.

\[coro2\] $$e(r) \leq \left\{ \begin{array}{ll}
1+2^{(r-2)/2} r, & r \geq 4 \quad \mbox{even}\\
1+2^{(r-3)/2}(r-1), & r \geq 5 \quad \mbox{odd}
\end{array}
\right.$$

We conjecture that the above inequalities are in fact equalities.

Another fiber product construction
----------------------------------

In Section \[Sec:otraconstruccion\] we describe another similar construction which permits to obtain the following construction.

\[construccion2\] If 