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Let $K$ be a number field. We consider a local-global principle for elliptic curves $E/K$ that admit (or do not admit) a rational isogeny of prime degree $\ell$. For suitable $K$ (including $K=\mathbb{Q}$), we prove that this principle holds for all $\ell \equiv 1mod4$, and for $\ell \<7$, but find a counterexample when $\ell =7$ for an elliptic curve with $j$-invariant $2268945/128$. For $K=\mathbb{Q}$ we show that, up to isomorphism, this is the only counterexample.

A well known theorem of Mestre and Schoof implies that the order of an elliptic curve $E$ over a prime field ${𝔽}_q$ can be uniquely determined by computing the orders of a few points on $E$ and its quadratic twist, provided that $q\>229$. We extend this result to all finite fields with $q\>49$, and all prime fields with $q\>29$.