= Two-prime formal-kernel test for nontorsion
A nonidentity rational point in the <formal kernel of a minimal Weierstrass equation> at two distinct <primes> has infinite <order of a group element>. Indeed, the <prime-to-residue-characteristic multiplication on a formal group> implies that a finite order in the kernel at $p$ must be a power of $p$. Membership in the kernel at a different prime $q$ also forces a power of $q$. The only common possibility is order one, contradicting nonidentity. For an integral model, negative <valuations> of the affine coordinates provide a convenient kernel-membership test.
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