The elliptic curve group law is the addition operation on the 
-rational points, including
the single point at infinity, of an elliptic
curve transformed to the form
 |
. To add two 
-rational points 
and 
, 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).
