Elliptic curve

In mathematics, an elliptic curve is a plane curve defined by an equation of the form

Related Topics:
Mathematics - Plane curve

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:y2 = x3 + a x + b,

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which is non-singular; that is, its graph has no cusps or self-intersections. One finds that elliptic curves correspond to embeddings of the torus into the complex projective plane; such embeddings generalize to arbitrary fields, and so it is said that elliptic curves are non-singular projective algebraic curves of genus 1 over a field K, together with a distinguished point defined over K.

Related Topics:
Cusps - Torus - Complex projective plane - Field - Non-singular - Projective - Algebraic curve - Genus

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Elliptic curves are especially important in number theory, and constitute a major area of current research; for example, they were used in the proof of Fermat's last theorem. They also find applications in cryptography (see the article elliptic curve cryptography) and integer factorization.

Related Topics:
Number theory - Fermat's last theorem - Cryptography - Elliptic curve cryptography - Integer factorization

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An elliptic curve is not the same as an ellipse: see elliptic integral for the origin of the term.

Related Topics:
Ellipse - Elliptic integral

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The natural group structure of a torus manifests itself as a curious geometric way on an elliptic curve; the set of points of the curve form an abelian group.

Related Topics:
Group - Abelian group

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