Isomorphism of elliptic curves (reviewed)

An isomorphism between two elliptic curves $E$, $E'$ defined over a field $K$ is an isogeny $f:E\to E'$ such that there exist an isogeny $g:E'\to E$ with the compositions $g\circ f$ and $f\circ g$ being the identity maps. Equivalently, an isomorphism $E\to E'$ is an isogeny of degree $1$.

Isomorphism is an equivalence relation, the equivalnce classes being called isomorphism classes.

When $E$ and $E'$ are defined by Weierstrass models, such an isomorphism is uniquely represented as a Weierstrass isomorphism between these models.