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In mathematics, 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. Any elliptic curve can be written as a plane algebraic curve defined by an equation of the form which is non-singular; that is, its graph 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.) The point O is actually the "point at infinity" in the projective plane. If y2 = P(x), where P is any polynomial of degree three in x with no repeated roots, then we obtain a nonsingular plane curve of genus one, which is thus also an elliptic curve. If P has degree four and is squarefree this equation again describes a plane curve of genus one; however, it has no natural choice of identity element. More generally, any algebraic curve of genus one, for example from the intersection of two three-dimensional quadric surfaces, is called an elliptic curve, provided that it has at least one rational point.
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