  The Prym map assigns to each covering of curves a polarized abelian variety.
In the case of unramified cyclic covers of curves of genus two, we show that
the Prym map is ramified precisely on the locus of bielliptic covers. The key
observation is that we can naturally associate to such a cover an abelian
surface with a cyclic polarization, and then the codifferential of the Prym map
can be interpreted in terms of multiplication of sections on the abelian
surface. Furthermore, we give a different proof of a result by Ramanan that a
genus two cyclic cover of degree sufficiently high is never hyperelliptic.
