What can we say if E^sigma is isogeneous to a twist of E?

Let $K$ be a quadratic number field, and let $E_1$ and $E_2$ be two isogeneous elliptic curves over $K$. Assume we know that $j(E_1)^\sigma=j(E_2)$ where $\sigma$ is the generator of the Galois group of $K/Q$. Can we then say that some twist of $E_1$ is a $Q$-curve? If so, is there a good way of describing the necessary twist?