Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 24 / 3 / d / Solution

At $2$ there is good reduction, and the surjectivity of good reduction over a local field gives an exact sequenceThe quotient has order $5$ and is consequently a cyclic group. For every odd integer $n$, multiplication by $n$ is an automorphism of $E_1({\mathbb{Q}}_2)$: its formal group law has multiplication series $nt+O(t^2)$ with unit linear coefficient, whose inverse formal power series has integral coefficients and converges on $2{\mathbb{Z}}_2$. This is the prime-to-residue-characteristic multiplication on a formal group.

Thus the odd-primary torsion subgroup injects into the order-five quotient. It also realizes that quotient: choose a lift $P$ of any reduced point, so $[5]P\in E_1$. There is a unique $Q\in E_1$ with $[5]Q=[5]P$. Then $P-Q$ is killed by five and has the prescribed reduction. This proves the prime-to-p torsion lifts uniquely at good reduction directly here. Therefore, for an odd prime $p$,The five-primary subgroup consists of the unique five-torsion lifts of the five reduced points. These lifts need not be rational over $\mathbb{Q}$; the preceding answer about rational torsion points of an elliptic curve does not contradict their existence over the p-adic field.