Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 125 / 2 / b / i / Solution

For good reduction at $p$, the reduction of an elliptic curve gives a mapwhose kernel is its formal group of an elliptic curve, and the torsion in that kernel is $p$-primary. Hence the prime-to-$p$ part of any torsion subgroup injects into $\tilde{E}({\mathbb{F}}_p)$.

At $p=2$, the odd part of $E(\mathbb{Q}{)}_{\mathrm{tors}}$ must divide $4$, so it is trivial. The torsion group is therefore a $2$-group. At $p=3$, all of this group has order prime to $3$ and injects into a group of order $7$, so it too is trivial. Thus