$\mathbb{Q}$-curves (reviewed)

An elliptic curve $E$ defined over a number field $K$ is a $\mathbb{Q}$-curve if it is isogenous over $\overline{K}$ to each of its Galois conjugates. Note that the isogenies need not be defined over $K$ itself.

An elliptic curve which is the base change of a curve defined over $\mathbb{Q}$ is a $\mathbb{Q}$-curve, but not all $\mathbb{Q}$-curves are base-change curves.

Elliptic curves with CM are all $\mathbb{Q}$-curves, as are all those whose $j$-invariant is in $\mathbb{Q}$.