  Let $\Phi^\infty(d)$ denote the set of finite abelian groups that occur
infinitely often as the torsion subgroup of an elliptic curve over a number
field of degree $d$. The sets $\Phi^\infty(d)$ are known for $d\le 4$. In this
article we determine $\Phi^\infty(5)$ and $\Phi^\infty(6)$.
