  Motivated by a conjecture of Xiao, we study families of coverings of elliptic
curves and their corresponding Prym map $\Phi$. More precisely, we describe the
codifferential of the period map $P$ associated to $\Phi$ in terms of the
residue of meromorphic $1$-forms and then we use it to give a characterization
for the coverings for which the dimension of $\ker(dP)$ is the least possibile.
This is useful in order to exclude the existence of non isotrivial fibrations
with maximal relative irregularity and thus also in order to give
counterexamples to the Xiao's conjecture mentioned above. The first
counterexample to the original conjecture, due to Pirola, is then analysed in
our framework.
