Elliptic Curve

Informally, an elliptic curve is a type of cubic curve whose solutions are confined to a region of space that is topologically equivalent to a torus. The Weierstrass elliptic function P(z;g_2,g_3) describes how to get from this torus to the algebraic form of an elliptic curve.

Formally, an elliptic curve over a field K is a nonsingular cubic curve in two variables, f(X,Y)=0, with a K-rational point (which may be a point at infinity). The field K is usually taken to be the complex numbers C, reals R, rationals Q, algebraic extensions of Q, p-adic numbers Q_p, or a finite field.

By an appropriate change of variables, a general elliptic curve over a field with field characteristic !=2,3, a general cubic curve


where A, B, ..., are elements of K, can be written in the form


where the right side of (2) has no repeated factors. Any elliptic curve not of characteristic 2 or 3 can also be written in Legendre normal form


(Hartshorne 1999).


Elliptic curves are illustrated above for various values of a and b.

If K has field characteristic three, then the best that can be done is to transform the curve into


(the x^2 term cannot be eliminated). If K has field characteristic two, then the situation is even worse. A general form into which an elliptic curve over any K can be transformed is called the Weierstrass form, and is given by


where a, b, c, d, and e are elements of K. Luckily, Q, R, and C all have field characteristic zero.

An elliptic curve of the form y^2=x^3+n for n an integer is known as a Mordell curve.

Whereas conic sections can be parameterized by the rational functions, elliptic curves cannot. The simplest parameterization functions are elliptic functions. Abelian varieties can be viewed as generalizations of elliptic curves.


If the underlying field of an elliptic curve is algebraically closed, then a straight line cuts an elliptic curve at three points (counting multiple roots at points of tangency). If two are known, it is possible to compute the third. If two of the intersection points are K-rational, then so is the third. Mazur and Tate (1973/74) proved that there is no elliptic curve over Q having a rational point of order 13.

Let (x_1,y_1) and (x_2,y_2) be two points on an elliptic curve E with elliptic discriminant




A related quantity known as the j-invariant of E is defined as


Now define

 lambda={(y_1-y_2)/(x_1-x_2)   for x_1!=x_2; (3x_1^2+a)/(2y_1)   for x_1=x_2.

Then the coordinates of the third point are


For elliptic curves over Q, Mordell proved that there are a finite number of integral solutions. The Mordell-Weil theorem says that the group of rational points of an elliptic curve over Q is finitely generated. Let the roots of y^2 be r_1, r_2, and r_3. The discriminant is then


The amazing Taniyama-Shimura conjecture states that all rational elliptic curves are also modular. This fact is far from obvious, and despite the fact that the conjecture was proposed in 1955, it was not even partially proved until 1995. Even so, Wiles' proof for the semistable case surprised most mathematicians, who had believed the conjecture unassailable. As a side benefit, Wiles' proof of the Taniyama-Shimura conjecture also laid to rest the famous and thorny problem which had baffled mathematicians for hundreds of years, Fermat's last theorem.

Curves with small j-conductors are listed in Swinnerton-Dyer (1975) and Cremona (1997). Methods for computing integral points (points with integral coordinates) are given in Gebel et al. and Stroeker and Tzanakis (1994). The Schoof-Elkies-Atkin algorithm can be used to determine the order of an elliptic curve E/F_p over the finite field F_p.

See also

Cubic Curve, Elliptic Curve Factorization Method, Elliptic Curve Group Law, Fermat's Last Theorem, Frey Curve, j-Invariant, Legendre Normal Form, Minimal Discriminant, Mordell Curve, Mordell-Weil Theorem, Ochoa Curve, Ribet's Theorem, Schoof-Elkies-Atkin Algorithm, Siegel's Theorem, Swinnerton-Dyer Conjecture, Taniyama-Shimura Conjecture, Weierstrass Elliptic Function, Weierstrass Form Explore this topic in the MathWorld classroom

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Elliptic Curve

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