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The Frobenius endomorphism satisfies a quadratic polynomial which is linked to the cardinality of <math>E(\mathbb{F}_{q})</math> by the following theorem:
'''Theorem:''' The Frobenius endomorphism given by <math>\phi</math> satisfies the characteristic equation
<math> \phi ^{2} - t\phi + q = 0 </math> where <math> t = q + 1 - \sharp E(\mathbb{F}_{q}) </math>
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<math> (x^{q^{2}}, y^{q^{2}}) + \bar{q}(x, y) \equiv \bar{t}(x^{q}, y^{q}) \pmod l </math>
==Computation modulo primes==
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