Strongly modular models of Q-curves

Let E be a Q-curve without complex multiplication. We address the problem of deciding whether E is geometrically isomorphic to a strongly modular 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.