Base change (reviewed)

If $E$ is an elliptic curve defined over a field $K$, and $L$ is an extension field of $K$, then the same equation defining $E$ as an elliptic curve over $K$ also defines a curve over $L$ called the base change of $E$ from $K$ to $L$. Any curve defined over $L$ which is isomorphic to $E$ over $L$ is called a base-change curve from $K$ to $L$. A sufficient but not necessary condition for a curve to be a base change is that the coefficients of its Weierstrass equation lie in $K$.

When $K=\Q$ and $L$ is a number field, elliptic curves over $L$ which are base-changes of curves over $\Q$ may simply be called base-change curves. A necessary, but not sufficient, condition for this is that the $j$-invariant of $E$ should be in $\Q$.