Suppose one has an elliptic curve $E = (C,O)$ over a field $k$ where $C$ is a non-singular genus one curve over $k$ and $O$ is a $k$-rational point on $C$. By moving $O$ on $C$ one gets a family of elliptic curves. Is this family trivial in the sense that all the curves are mutually isomorphic? If not (as I suspect), what are the isomorphism classes of elliptic curves one gets this way and how are they related to the original curve $E$?
Special cases of $k = \mathbb{C}$ and $\mathbb{Q}$ would be specially helpful to know about, for $\mathbb{C}$ in terms of lattices, and for $\mathbb{Q}$ in terms of the associated modular forms after Wiles et al.
(Depending on the answer) should CM curves be distinguished?
Thanks!
