Prym varieties of \'etale covers of hyperelliptic curves

It is well known that the Prym variety of an \'etale cyclic covering of a hyperelliptic curve is isogenous to the product of two Jacobians. Moreover, if the degree of the covering is odd or congruent to 2 mod 4, then the canonical isogeny is an isomorphism. We compute the degree of this isogeny in the remaining cases and show that only in the case of coverings of degree 4 it is an isomorphism.