I'm trying to prove question 2.12.25 from "Elliptic Curves, Modular Forms, and Their L-Functions by élvaro Lozano-Robledo".
Let $E$ be an elliptic curve, such that the image of the points $P_1,...,P_n \in E(\mathbb{Q})$ generate the group $E(\mathbb{Q}) / 2E(\mathbb{Q})$ (i.e. the weak mordell weil group). Let $G$ be the subgroup of $E(\mathbb{Q})$ generated by $P_1,...,P_n$.
Show that the index $[E(\mathbb{Q}): G]$ is finite, and that it can be arbitrarly large (depending only on the choice of the generators $P_1,...,P_n$).
Using Weak Mordell Weil theorem it is easy to see that the index is indeed finite, however I cannot figure out why it can be aritrarly large?
Isn't it true that $E(\mathbb{Q}) / G \cong {E(\mathbb{Q})/2E(\mathbb{Q})} / {G / 2E(\mathbb{Q})}$ which is not depandent of the generators? what am i missing?