\alpha \rangle$ is isomorphic to $E_1$ and $S/\langle \alpha \iota \rangle$ is isomorphic to $E_2$, where $\iota$ is the hyperelliptic involution. But, there are also Riemann surfaces of genus two with no extra automorphisms (with the exception of the hyperelliptic involution) whose jacobian variety is also isogenous to $E_1 \times E_2$; these can be obtained by considering non-constant holomorphic maps $h:S \to E$, where $S$ is a closed Riemann surface of genus two and $E$ being some elliptic curve.

A general construction
----------------------

Next, given $2s-3$ elliptic curves, for $s \geq 3$, we make an explicit construction of a closed Riemann surface of genus $g=1+2^{s-2}(s-2)$, whose jacobian variety is isogenous to the product of at least $s(s-1)/2$ elliptic curves and jacobian varieties of some elliptic/hyperelliptic Riemann surfaces. Let us consider the set of cardinality $2^{s-1}-1$ defined as $$V_{s}=\{\alpha=(\alpha_{1},\ldots,\alpha_{s}) \in \{0,1\}^{s}-\{(0,\ldots,0)\}: \alpha_{1}+\cdots+\alpha_{s} \; \mbox{is even}\}.$$

As previously noted, given $2s-3$ elliptic curves, $E_{1},\ldots, E_{2s-3}$, we may find a tuple $(\lambda_{1},\ldots,\lambda_{2s-3}) \in \Delta_{2s-3}$ so that $E_{j}$ is isogenous to the elliptic curve $y^{2}=x(x-1)(x-\lambda_{j})$.

\[construccion\] Let $s \geq 3$ and $(\lambda,\mu_{1,1},\mu_{1,2},\mu_{2,1},\mu_{2,2},\ldots,\mu_{s-2,1},\mu_{s-2,2}) \in \Delta_{2s-3}$. Set

$$\eta_{0}=\frac{-1}{\mu_{s-2,2}}, \; \eta_{1}=\frac{1}{1-\mu_{s-2,2}}, \; \eta_{2}=\frac{1}{\lambda-\mu_{s-2,2}}, \; \eta_{3}=\frac{1}{\mu_{s-2,1}-\mu_{s-2,2}},$$ $$\eta_{j,t}=\frac{1}{\mu_{j,t}-\mu_{s-2,2}}, \quad t=1,2, \quad j=1,\ldots,s-3.$$

For each $\alpha=(\alpha_{1},\ldots,\alpha_{s}) \in V_{s}$, set $K_{\alpha} 