Isogeny classes of elliptic curves

The isogeny class (over a field $K$) of an elliptic curve $E$ defined over $K$ is the set of all isomorphism classes of elliptic curves defined over $K$ that are isogenous to $E$ over $K$. Over a number field $K$ this is always a finite set; over $\Q$, it has at most 8 elements by a theorem of Kenku [MR:0675184, 10.1016/0022-314X(82)90025-7].