Let $C\rightarrow C'$ be a double cover of curves, the restriction of the polarisation of $J_C$ to the Prym varieties $P$ attached to this double cover, gives a polarization on $P$,

Does this polarization have a square root in general?

I khow that, in some special cases, this is true an a square root gives a principal polarization. thanks

Prym varieties I.$\endgroup$algebraicclass (because $\mathrm{Pic}^0(A)$ is a divisible group). So it should not depend on the translate chosen. $\endgroup$