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 
, p-adic 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 j-invariant 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 Mordell-Weil 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 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 
over the finite field 
.
