Elliptic Curve Group Law

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).