Torsion groups of Mordell curves over cubic and sextic fields

Let $E$ be a Mordell curve defined over a number field $K$ by the equation $y^2 = x^3 + c$, $c \in K$. Let $E(L)$ denote the set of $L$-rational points of $E$, where $L$ is a number field containing $K$. We classify the possible torsion subgroups of $E(L)$ when $L$ is a cubic or sextic field, and $E$ is an elliptic curve over $L$ or $\mathbb{Q}$. We also describe the conditions on $c$ under which $E$ has a certain torsion group from the set of all torsion subgroups of $E(L)$ in the following cases: (i) $c \in \mathbb{Q}$, $L$ is cubic or sextic; (ii) $c \in L$, $L$ is cubic.