Torsion of rational elliptic curves over the maximal abelian extension of Q

Let $E$ be an elliptic curve defined over $\mathbb{Q}$, and let ${\mathbb{Q}}^{ab}$ be the maximal abelian extension of $\mathbb{Q}$. In this article we classify the groups that can arise as $E({\mathbb{Q}}^{ab}{)}_{\text{tors}}$ up to isomorphism. The method illustrates techniques for finding explicit models of modular curves of mixed level structure. Moreover we provide an explicit algorithm to compute $E({\mathbb{Q}}^{ab}{)}_{\text{tors}}$ for any elliptic curve $E\mathrm{/}\mathbb{Q}$.