Sporadic cubic torsion

Let $K$ be a number field, and let $E/K$ an elliptic curve over $K$. The Mordell--Weil theorem asserts that the $K$-rational points $E(K)$ of $E$ form finitely generated abelian group. In this work, we complete classification finite groups which appear as torsion subgroup for cubic field. To do so, determine on modular curves $X_1(N)$ \[N = 21, 22, 24, 25, 26, 28, 30, 32, 33, 35, 36, 39, 45, 65, 121.\] As part our analysis, list $N$ $J_0(N)$ (resp., $J_1(N)$, resp., $J_1(2,2N)$) has rank 0. We also provide evidence to generalized version conjecture Conrad, Edixhoven, Stein by proving $J_1(N)(\mathbb{Q})$ is $\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$-orbits cusps $X_1(N)_{\bar{\mathbb{Q}}}$ $N\leq 55$, $N \neq 54$.