Let C be a 4-cover of an elliptic curve E , written as a quadric intersection in $${mathbb P}^3$$ P 3 . Let $$E'$$ E ′ be another elliptic curve with 4-torsion isomorphic to that of E . We show how to write down the 4-cover $$C'$$ C ′ of $$E'$$ E ′ with the property that C and $$C'$$ C ′ are represented by the same cohomology class on the 4-torsion. In fact we give equations for $$C'$$ C ′ as a curve of degree 8 in $${mathbb P}^5$$ P 5 . We also study the K3-surfaces fibred by the curves $$C'$$ C ′ as we vary $$E'$$ E ′ . In particular we show how to write down models for these surfaces as complete intersections of quadrics in $${mathbb P}^5$$ P 5 with exactly 16 singular points. This allows us to give examples of elliptic curves over $${mathbb Q}$$ Q that have elements of order 4 in their Tate–Shafarevich group that are not visible in a principally polarized abelian surface.
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