= Torsion exclusion from good reduction counts
For an <elliptic curve> over $\mathbb Q$, reduction at a good prime $p$ is injective on the $\ell$-primary <torsion subgroup> when $\ell\ne p$: its kernel is a <formal group>, where multiplication by $\ell$ is bijective. Hence a rational point of order $\ell$ forces $\ell\mid\#\widetilde E(\mathbb F_p)$. If for every possible prime $\ell$ some good $p\ne\ell$ has a point count not divisible by $\ell$, rational torsion is trivial. In particular two distinct good primes $p,r$ with coprime counts, $p\nmid\#\widetilde E(\mathbb F_r)$ and $r\nmid\#\widetilde E(\mathbb F_p)$ suffice: the coprime counts exclude other primes, and the two cross conditions exclude $p,r$. This argument does not incorrectly apply prime-to-$p$ injectivity to $p$-power torsion.
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