joint with Nils Bruin
Let C be a 4-cover of an elliptic curve E, written as a quadric intersection in P3. Let E' be another elliptic curve with 4-torsion isomorphic to that of E. We show how to write down the 4-cover C' of E' with the property that C and C' are represented by the same cohomology class on the 4-torsion. In fact we give equations for C' as a curve of degree 8 in P5.
We also study the K3-surfaces 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 P5 with exactly 16 singular points. This allows us to give examples of elliptic curves over Q that have elements of order 4 in their Tate-Shafarevich group that are not visible in a principally polarized abelian surface.