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# Topics: Number Theory - Elliptic Curves

### Elliptic Curves

In mathematics, an elliptic curve is a plane algebraic curve defined by an equation of the form$${\displaystyle y^{2}=x^{3}+ax+b} y^{2}=x^{3}+ax+b$$

that is non-singular; that is, it has no cusps or self-intersections. (When the characteristic of the coefficient field is equal to 2 or 3, the above equation is not quite general enough to comprise all non-singular cubic curves; see below for a more precise definition.)

Formally, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety – that is, it has a multiplication defined algebraically, with respect to which it is an abelian group – and O serves as the identity element. Often the curve itself, without O specified, is called an elliptic curve. The point O is actually the "point at infinity" in the projective plane.

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