13 Facts About Elliptic curve

In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point.

An elliptic curve is defined over a field and describes points in, the Cartesian product of with itself.

The Elliptic curve is required to be non-singular, which means that the Elliptic curve has no cusps or self-intersections.

An elliptic curve is an abelian variety – that is, it has a group law defined algebraically, with respect to which it is an abelian group – and serves as the identity element.

An elliptic curve is not an ellipse in the sense of a projective conic, which has genus zero: see elliptic integral for the origin of the term.

Topologically, a complex elliptic curve is a torus, while a complex ellipse is a sphere.

Real graph of a non-singular Elliptic curve has two components if its discriminant is positive, and one component if it is negative.

In Weierstrass normal form, such a Elliptic curve will have an additional point at infinity, which serves as the identity of the group.

Since the Elliptic curve is symmetrical about the -axis, given any point, we can take to be the point opposite it.

In 1999 this was shown to be a consequence of the proof of the Shimura–Taniyama–Weil conjecture, which asserts that every elliptic curve over Q is a modular curve, which implies that its L-function is the L-function of a modular form whose analytic continuation is known.

In other words, the number of points on the Elliptic curve grows proportionally to the number of elements in the field.

One typically takes the curve to be the set of all points which satisfy the above equation and such that both x and y are elements of the algebraic closure of K Points of the curve whose coordinates both belong to K are called K-rational points.

Many of the preceding results remain valid when the field of definition of E is a number field K, that is to say, a finite field extension of Q In particular, the group E of K-rational points of an elliptic curve E defined over K is finitely generated, which generalizes the Mordell–Weil theorem above.