
joint with Nils Bruin
Let C be a 4cover of an elliptic curve E, written as a quadric intersection in P^{3}. Let E' be another elliptic curve with 4torsion isomorphic to that of E. We show how to write down the 4cover C' of E' with the property that C and C' are represented by the same cohomology class on the 4torsion. In fact we give equations for C' as a curve of degree 8 in P^{5}.
We also study the K3surfaces fibred by the curves C' as we vary E'. In particular we show how to write down models for these surfaces as complete intersections of quadrics in P^{5} with exactly 16 singular points. This allows us to give examples of elliptic curves over Q that have elements of order 4 in their TateShafarevich group that are not visible in a principally polarized abelian surface.