Strongly modular models of $\mathbb Q$-curves

Let $E$ be a $\mathbb Q$-curve without complex multiplication. We address the problem of deciding whether $E$ is geometrically isomorphic to a strongly modular $\mathbb Q$-curve. We show that the question has a positive answer if and only if $E$ has a model that is completely defined over an abelian number field. Next, if $E$ is completely defined over a quadratic or biquadratic number field $L$, we classify all strongly modular twists of $E$ over $L$ in terms of the arithmetic of $L$. Moreover, we show how to determine which of these twists come, up to isogeny, from a subfield of $L$.