Visibility of 4-covers of elliptic curves

Let $C$ be a $4$-cover of an elliptic curve $E$, written as a quadric intersection in $\mathbb{P}^3$. 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 $\mathbb{P}^5$. 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 $\mathbb{P}^5$ with exactly $16$ singular points. This allows us to give examples of elliptic curves over $\mathbb{Q}$ that have elements of order $4$ in their Tate-Shafarevich group that are not visible in a principally polarized abelian surface.