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 describes how to get from this torus to the algebraic form of an elliptic curve.
Formally, an elliptic curve over a field is a nonsingular cubic curve in two variables, , with a rational point (which may be a point at infinity). The field is usually taken to be the complex numbers , reals , rationals , algebraic extensions of , padic numbers , or a finite field.
By an appropriate change of variables, a general elliptic curve over a field with field characteristic , a general cubic curve
(1)

where , , ..., are elements of , can be written in the form
(2)

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
(3)

(Hartshorne 1999).
Elliptic curves are illustrated above for various values of and .
If has field characteristic three, then the best that can be done is to transform the curve into
(4)

(the term cannot be eliminated). If has field characteristic two, then the situation is even worse. A general form into which an elliptic curve over any can be transformed is called the Weierstrass form, and is given by
(5)

where , , , , and are elements of . Luckily, , , and all have field characteristic zero.
An elliptic curve of the form for 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 rational, then so is the third. Mazur and Tate (1973/74) proved that there is no elliptic curve over having a rational point of order 13.
Let and be two points on an elliptic curve with elliptic discriminant
(6)

satisfying
(7)

A related quantity known as the jinvariant of is defined as
(8)

Now define
(9)

Then the coordinates of the third point are
(10)
 
(11)

For elliptic curves over , Mordell proved that there are a finite number of integral solutions. The MordellWeil theorem says that the group of rational points of an elliptic curve over is finitely generated. Let the roots of be , , and . The discriminant is then
(12)

The amazing TaniyamaShimura 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 TaniyamaShimura 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 jconductors are listed in SwinnertonDyer (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 SchoofElkiesAtkin algorithm can be used to determine the order of an elliptic curve over the finite field .