 Two-prime bound on rational elliptic torsion

Let an elliptic curve over $\mathbb{Q}$ have good reduction at distinct primes $p,r$, with point counts $N_p,N_r$. For a torsion point, its $\ell$-primary part injects into every reduction whose residue characteristic differs from $\ell$. Consequently its $\ell$-power order has exponent at most $\min (v_{\ell }(N_p),v_{\ell }(N_r))$ when $\ell \ne p,r$, at most $v_p(N_r)$ when $\ell =p$, and at most $v_r(N_p)$ when $\ell =r$. This precise form avoids assuming injectivity on residue-characteristic torsion. Counts five at two and seven at three force trivial rational torsion.