Abstract
By using the relations between supersingular elliptic curves defined over a finite field of characteristic p>3 and the p-th division polynomial, we present a property about the reduction modulo a prime p>3 of the p-th division polynomial based on a heuristic argument and numerical evidence. We prove this property determining the p-th division polynomial of supersingular elliptic curves. As a consequence of this result, we present a criterion to discard supersingular elliptic curves.
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