  A closed Riemann surface $S$ is called a generalized Fermat curve of type
$(p,n)$, where $p,n \geq 2$ are integers, if it admits a group $H \cong
{\mathbb Z}_{p}^{n}$ of conformal automorphisms so that $S/H$ is an orbifold of
genus zero with exactly $n+1$ cone points, each one of order $p$. It is known
that $S$ is a fiber product of $(n-1)$ classical Fermat curves of degree $p$
and, for $(p-1)(n-1)>2$, that it is a non-hyperelliptic Riemann surface. In
this paper, assuming $p$ to be a prime integer, we provide a decomposition, up
to isogeny, of the Jacobian variety $JS$ as a product of Jacobian varieties of
certain cyclic $p$-gonal curves. Explicit equations for these $p$-gonal curves
are provided in terms of the equations for $S$. As a consequence of this
decomposition, we are able to provide explicit positive-dimensional families of
closed Riemann surfaces whose Jacobian variety is isogenous to the product of
elliptic curves.
