lattice in ${\mathbb C}^{g}$. The quotient torus ${\mathbb C}^{g}/\Omega$ is isomorphic to $JS$.

A decomposition result of the jacobian variety
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Next, we recall the following decomposition result due to Kani-Rosen [@K-R] which will be used in our constructions.

\[coroKR\] Let $S$ be a closed Riemann surface of genus $g \geq 1$ and let $H_{1},\ldots,H_{s}<{\rm Aut}(S)$ such that:

1.  $H_{i} H_{j}=H_{j} H_{i}$, for all $i,j =1,\ldots,s$;

2.  $g_{H_{i}H_{j}}=0$, for $1 \leq i < j \leq s$

3.  $g=\sum_{j=1}^{s} g_{H_{j}}$.

Then $$JS \cong_{isog.}  \prod_{j=1}^{s} J(S_{H_{j}}).$$

Proof of Theorem \[r=2\] {#Sec:r=2}
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Let us consider the fiber product $C$ of $(E_{\lambda_{1}},\pi_{1})$ and $(E_{\lambda_{2}},\pi_{2})$, where $\pi_{j}(x,y)=x$, that is, $$C: \left\{ (x,y_{1},y_{2}): y_{1}=x(x-1)(x-\lambda_{1}), \quad y_{2}=x(x-1)(x-\lambda_{2}) \right\}.$$

We consider the projection $\pi:C \to \widehat{\mathbb C}$, where $\pi(x,y_{1},y_{2})=x$. On $C$ we have the automorphisms $$A_{1}(x,y_{1},y_{2})=(x,-y_{1},y_{2}), \quad A_{2}(x,y_{1},y_{2})=(x,y_{1},-y_{2}),$$ with $H=\langle A_{1}, A_{2} \rangle \cong {\mathbb Z}_{2}^{2}$, $H$ being the deck group of $\pi$, $C/H$ being of signature $(0;2,2,2,2,2)$ and branch values being $\infty$, $0$, $1$, $\lambda_{1}$ and $\lambda_{2}$. The Riemann surface $S$ defined by $C$ has genus two

If $H_{j}=\langle A_{j} \rangle$, it can be seen that the orbifold $C/H_{j}$ has underlying Riemann surface $E_{\lambda_{3-j}}$. We may apply Proposition \[coroKR\], using $H_{1}$ and $H_{2}$, to obtain that 