A rational point over a field 
, also called a 
-rational point, is a point of an
algebraic variety defined over 
whose coordinates can be chosen
in 
(Milne 2024, p. 125). For an affine
plane curve 
defined over 
, a 
-rational point is a point 
with 
satisfying the equation.
For example, over the field 
, both coordinates of an affine
rational point are rational numbers.
The rational point may also be a point at infinity. For example, take the elliptic curve over 
 |
(1)
|
and homogenize it by introducing a third variable 
to obtain a homogeneous
polynomial of total degree 3 as follows:
 |
(2)
|
Now, find the points at infinity by setting 
, obtaining
 |
(3)
|
Solving gives 
,
, and 
. Despite freedom in the choice of 
, there is only a single point
at infinity because the two triples 
and 
are considered to be equivalent
(or identified) iff one is a nonzero scalar multiple of the other. Here, 
is not considered to be a valid point.
In homogeneous coordinates, the triples 
correspond to the affine points 
, and the triples 
with 
correspond to the points
at infinity, which lie on the line at infinity
(Milne 2024, pp. 130-131).
The maximum numbers of rational points on elliptic curves over the finite field GF(
), including the point at infinity,
are 5, 7, 9, 10, 13, 14, 16, ... (OEIS A005523),
as 
runs through the prime powers 2, 3, 4, 5, 7, 8, 9,
... (Waterhouse 1969, p. 536).
