On the cyclic torsion of elliptic curves over cubic number fields

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 $\mathbb{Z}\mathrm{/}N\mathbb{Z}$ is not a subgroup of $E(K{)}_{tor}$ for any elliptic curve $E$ over a cubic number field $K$.