Exploring the Possibilities

As $E$ varies over all elliptic curves over  $\mathbb{Q}$, what are the possibilities for $E(\mathbb{Q})$? What finitely generated abelian groups occur? Mordell's theorem implies that

$$E(\mathbb{Q}) \approx \mathbb{Z}^r \oplus E(\mathbb{Q})_{\tor},$$

where $E(\mathbb{Q})_{\tor}$ is the set of points of finite order in $E(\mathbb{Q})$ and $\mathbb{Z}^r\approx E(\mathbb{Q})/E(\mathbb{Q})_{\tor}$. The number $r$ is called the rank of $E$.