product of $g$ elliptic curves (Theorem \[coro3\]).

We also provide a similar construction, a fiber product of $r$ given elliptic curves $E_{1}, \ldots, E_{r}$, which turns out to be irreducible. Such a fiber product is a closed Riemann surface of genus $g=1+2^{r-2}(r-1)$ whose jacobian variety is isogenous to the product $E_{1} \times \cdots \times E_{r} \times A_{g-r}$, where $A_{g-r}$ is again the product of jacobian varieties of elliptic/hyperelliptic Riemann surfaces. This construction also permits to obtain a $2$-dimensional family of Riemann surfaces of genus $g=5$ and a $1$-dimensional family of Riemann surfaces of genus $g=13$ whose jacobian varieties are isogenous to the product of elliptic curves (see Corollaries \[g=5\] and \[g=13\]).

Main results
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Before to state the main results, we need to recall some facts on elliptic curves. Set $\Delta_{1}={\mathbb C}-\{0,1\}$ and, for $s \geq 2$, set $\Delta_{s}=\{(\lambda_{1},\ldots,\lambda_{s}) \in {\mathbb C}^{s}: \lambda_{j} \in \Delta_{1};\; \lambda_{i} \neq \lambda_{j},\; i \neq j\}$. If $\lambda \in \Delta_{1}$, then we set the elliptic curve $$E_{\lambda}: y^{2}=x(x-1)(x-\lambda).$$

It is known that $E_{\lambda}$ and $E_{\mu}$ are isomorphic if and only if there is some $T \in 
{\mathbb G}=\langle u(\lambda)=1/\lambda, V(\lambda)=1-\lambda\rangle \cong {\mathfrak S}_{3}$ so that $\mu=T(\lambda)$. It is also well known that for every $\lambda 