For $E$ an elliptic over $\mathbb{Q}$, I want to see the structure of $E_\text{tor}(\mathbb{Q})$.
First of all, $E(\bar{\mathbb{Q}})[m]=\mathbb{Z}/m\mathbb{Z}\oplus\mathbb{Z}/m\mathbb{Z}$, thus $E_{\text{tor}}(\bar{\mathbb{Q}})\cong\lim\mathbb{Z}/m\mathbb{Z}\oplus\mathbb{Z}/m\mathbb{Z}\cong\mathbb{Q}/\mathbb{Z}\oplus\mathbb{Q}/\mathbb{Z}$. Also by using weil pairing, I can see that for any $m>2$, $E(\bar{\mathbb{Q}})[m]$ is not contained in $E_{\text{tor}}(\mathbb{Q})$.
I want to ask how to prove $E_\text{tor}(\mathbb{Q})$ is either isomorphic to a subgroup of $\mathbb{Q}/\mathbb{Z}$ or isomorphic to the direct sum of $\mathbb{Z}/2\mathbb{Z}$ and a subgroup of $\mathbb{Q}/\mathbb{Z}$.
Thanks!