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Given the elliptic curve <math>E</math> defined over <math>\mathbb{F}_{q}</math> we consider points on <math>E</math> over <math>\bar{\mathbb{F}}_{q}</math>, the [[algebraic closure]] of <math>\mathbb{F}_{q}</math>; i.e. we allow points with coordinates in <math>\bar{\mathbb{F}}_{q}</math>. The [[Frobenius endomorphism]] of <math>\bar{\mathbb{F}}_{q}</math> over <math>\mathbb{F}_q</math> extends to the elliptic curve by <math> \phi : (x, y) \mapsto (x^{q}, y^{q})</math>.
This map is the identity on <math>E(\mathbb{F}_{q})</math> and one can extend it to the point at infinity <math>O</math>, making it a [[group morphism]] from <math>E(\bar{\mathbb{F}}_{q
The Frobenius endomorphism satisfies a quadratic polynomial which is linked to the cardinality of <math>E(\mathbb{F}_{q})</math> by the following theorem:
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