  Let $E$ be an elliptic defined over a number field $K$. Then its Mordell-Weil
group $E(K)$ is finitely generated: $E(K)\cong E(K)_{tor}\times\mathbb{Z}^r$.
In this paper, we discuss the cyclic torsion subgroup of elliptic curves over
cubic number fields. For $N=169,143,91,65,77$ or $55$, we show that
$\mathbb{Z}/N\mathbb{Z}$ is not a subgroup of $E(K)_{tor}$ for any elliptic
curve $E$ over a cubic number field $K$.
