A 
-rational point is a point 
on an algebraic curve 
, where 
and 
are in a field 
. For example, rational point in the field 
of ordinary rational numbers is a point
satisfying the given equation such
that both 
and 
are rational numbers.
The rational point may also be a point at infinity. For example, take the elliptic curve
 |
(1)
|
and homogenize it by introducing a third variable 
so that each term has degree 3 as follows:
 |
(2)
|
Now, find the points at infinity by setting 
, obtaining
 |
(3)
|
Solving gives 
,
equal to any value, and (by definition)
. Despite freedom in the choice of
, there is only a single point
at infinity because the two triples (
, 
, 
), (
, 
, 
) are considered to be equivalent (or identified) only if
one is a scalar multiple of the other. Here, (0, 0, 0) is not considered to be a
valid point. The triples (
, 
,
1) correspond to the ordinary points (
, 
),
and the triples (
,
, 0) correspond to the points
at infinity, usually called the line at infinity.
The rational points on elliptic curves over the finite field GF(
) are 5, 7, 9, 10, 13, 14, 16, ... (OEIS A005523).
