that $A_{1}$ and $A_{2}$ are [*isogenous*]{}.

An abelian variety $A$ is called [*decomposable*]{} if it is isogenous to the product of abelian varieties of smaller dimensions. It is called [*completely decompossable*]{} if it is the product of elliptic curves (varieties of dimension $1$). If the abelian variety is not isogenous to a product of lowest dimensional abelian varieties, then we say that it is [*simple*]{}.

If $A$ is an abelian variety, then there exist simple polarized abelian varieties $A_{1}, \ldots, A_{s}$ and positive integers $n_{1},\ldots,n_{s}$ such that its jacobian variety $JS$ is isogenous to the product $A_{1}^{n_{1}} \times \cdots A_{s}^{n_{s}}$. Moreover, the factors $A_{j}$ and the integers $n_{j}$ are unique up to isogeny and permutation of the factors.

In general, to describe these simple factors of an abelian variety seems to be a very difficult problem. When the abelian variety $A$ admits a non-trivial group $G$ of automorphisms, then there is a method to compute factors (non-necessarilly simple ones) by using the rational representations of $G$ (the isotipical decomposition) [@CR; @LR; @Anita].

The Jacobian variety
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Let $S$ be a closed Riemann surface of genus $g \geq 1$. The first homology group $H_{1}(S,{\mathbb Z})$ is isomorphic to ${\mathbb Z}^{2g}$ and 