= Solution
Let $E_r(\mathbb Q_2)$ be the $r$th term in the <filtration of elliptic-curve points over a local field>. Reduction gives
$$
E(\mathbb Q_2)/E_1(\mathbb Q_2)\simeq\widetilde E(\mathbb F_2),
$$
of order four, while
$$
E_1(\mathbb Q_2)/E_2(\mathbb Q_2)\simeq(\mathbb F_2,+)
$$
has order two. The <formal logarithm> is injective on $E_2(\mathbb Q_2)$ and identifies it with an additive subgroup of $\mathbb Q_2$, so $E_2(\mathbb Q_2)$ is torsion-free. A finite subgroup of $E(\mathbb Q_2)$ therefore injects into $E(\mathbb Q_2)/E_2(\mathbb Q_2)$, whose order is $4\cdot2=8$. Thus $|E(\mathbb Q_2)_{\rm tors}|$ divides $8$.
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