Let $E_1$ be an elliptic curve over $\Bbb Q$ and $S \subset E_1(\Bbb Q)$ be a subgroup.
Is there an elliptic curve $E_2$ with an algebraic group morphism $E_1 \to E_2$, and such that $E_2(\Bbb Q) \cong E_1(\Bbb Q) / S$ ? Or such that $E_2(\overline{\Bbb Q}) \cong E_1(\overline{\Bbb Q}) / S$ (seeing $E_1(\Bbb Q)$ as a subgroup $E_1(\overline{\Bbb Q})$)?
According to Silverman's books on elliptic curves, this holds if $S$ is finite, but what if $S$ has rank $1$, for instance?
Thank you!