Symmetric correspondences with decomposable minimal equation

We study symmetric correspondences with completely decomposable minimal equation on smooth projective curves $C$. The Jacobian of $C$ then decomposes correspondingly. For all positive integers $g$ and $\ell$, we give series of examples of smooth curves $C$ of genus $n^\ell (g-1) +1$ with correspondences satisfying minimal equations of degree $\ell+1$ such that the Jacobian of $C$ has at least $2^\ell$ isogeny components.