We give an elementary way to distinguish between the twists of an ordinary elliptic curve E over Fₚ in order to identify the one with $p+1-2U$ points, when p=U²+dV² with 2U, 2V∈ Z and E is constructed using the CM method for finding elliptic curves with a prescribed number of points. Our algorithms consist in most cases of reading off simple congruence conditions on U and V modulo $4$.
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Rubin et al. (2009) studied this question.
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