A ℚ-curve is an elliptic curve over a number field K which is geometrically isogenous to each of its Galois conjugates. K. Ribet  asked whether every ℚ-curve is modular, and he showed that a positive answer would follow from J.-P. Serre's conjecture on mod p Galois representations. We answer Ribet's question in the affirmative, subject to certain local conditions at 3.
"On the modularity of ℚ-curves." Duke Math. J. 109 (1) 97 - 122, 15 July 2001. https://doi.org/10.1215/S0012-7094-01-10914-9