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