The group of an elliptic curve which has been transformed to the form
 |
is the set of 
-rational points, including the single point
at infinity. The group law (addition) is defined as follows: Take 2 
-rational points 
and 
. Now 'draw' a straight line through them and compute the third
point of intersection 
(also a 
-rational point). Then
 |
gives the identity point at infinity. Now find the inverse of 
,
which can be done by setting 
giving 
.
This remarkable result is only a special case of a more general procedure. Essentially, the reason is that this type of elliptic curve has a single point at infinity which is an inflection point (the line at infinity meets the curve at a single point at infinity, so it must be an intersection of multiplicity three).
