On the small cyclic torsion of elliptic curves over cubic number fields

Merel’s result on the strong uniform boundedness conjecture made it meaningful to classify the torsion part of the Mordell-Weil groups of all elliptic curves defined over number fields of fixed degree $d$. In this paper, we discuss the cyclic torsion subgroup of elliptic curves over cubic number fields. For $N=49,40,25$ or $22$, 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 $K$.