Suppose $E_1$ and $E_2$ are elliptic curves defined over $\mathbb{Q}$. Now we know that both curves are isomorphic over $\mathbb{C}$ iff they have the same $j$-invariant.

But $E_1$ and $E_2$ could also be isomorphic over a subfield of $\mathbb{C}$. As is the case for $E$ and its quadratic twist $E_d$. Now the question general is.

$E_1$ and $E_2$ defined over $\mathbb{Q}$ and isomorphic over $\mathbb{C}$. Let $K$ the smallest subfield of $\mathbb{C}$ such that $E_1$ and $E_2$ become isomorphic over $K$. What can be said about $K$. Is it always a finite extension of $\mathbb{Q}$. If so, what can be said about the extension $K|\mathbb{Q}$.

My second question is something goes something like in the opposite direction. I start again with quadratic twists. Let $E$ be an elliptic curve over $\mathbb{Q}$ and consider the quadratic extension $\mathbb{Q}|\mathbb{Q}(\sqrt{d})$. Describe the curves over $\mathbb{Q}$(or isomorphism classes over $\mathbb{Q}$) which become isomorphic to $E$ over $\mathbb{Q}(\sqrt{d})$. I think the answer is $E$ and $E_d$. Again I would like to know what happens if we take a larger extension.

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $K|\mathbb{Q}$ a finite extension. Describe the isomorphism classes of elliptic curves over $\mathbb{Q}$ which become isomorphic to $E$ over K.

I have no idea what is the right context to answer such questions.