$K$ a number field, $G_K$ its Galois group, $E_1, E_2$ two elliptic curves defined over $K$. The isogeny theorem says that if for some prime number $\ell$, The Tate modules (tensored with $\mathbb{Q}$) $V_{\ell}(E_1)$ is isomorphic to $V_{\ell}(E_2)$ as Galois modules. Then these two elliptic curves are isogenous.

My question is, when are these two curves isomorphic? Namely, what more invariants are needed to fully characterize the elliptic curve (besides the Tate modules). Thanks!