Let p be a prime and let K be a field containing the p'th roots of unity. Let E be an elliptic curve over K. We consider the the moduli problem $Y_E(p)$, which sends L to set of elliptic curves F/L, and symplectic isomorphisms $\phi:E[p] \rightarrow F[p]$. We know that this moduli problem is representable by a curve over $K$, and we let the compactification of this curve be $X_E(p)$. We know $X_E(p)$ is a twist of $X(p)$. Similarly, we can construct $X_E(p^2)$, and we see that $X_E(p^2)$ is a normal cover of $X_E(p)$. I think the Galois group of $X_E(p^2)/X_E(p)$ is $(Z/pZ)^3$. If that is the case, then given any K point of $X_E(p)$, we can look at fiber over this point. This fiber is defined over K, hence it will define a field extension of K, with Galois group a subset of $(Z/pZ)^3$.
This means, if we have E and F defined over K, with $E[p] \equiv F[p]$, then we should be able to construct a cyclic extension of K of order p. What is that extension?