Torsion of rational elliptic curves over different types of cubic fields

Let $E$ be an elliptic curve defined over $\text{Q}$, and let $G$ be the torsion group $E(K{)}_{tors}$ for some cubic field $K$ which does not occur over $\text{Q}$. In this paper, we determine over which types of cubic number fields (cyclic cubic, non-Galois totally real cubic, complex cubic or pure cubic) $G$ can occur, and if so, whether it can occur infinitely often or not. Moreover, if it occurs, we provide elliptic curves $E\mathrm{/}\text{Q}$ together with cubic fields $K$ so that $G= E(K)_{tors}$.