Elliptic Curve Group Law

The Group of an Elliptic Curve which has been transformed to the form

$$y^2 = x^3 + ax + b$$

is the set of $K$-Rational Points, including the single Point at Infinity. The group law (addition) is defined as follows: Take 2 $K$-Rational Points $P$ and $Q$. Now `draw' a straight line through them and compute the third point of intersection $R$ (also a $K$-Rational Point). Then

$$P + Q + R = 0$$

gives the identity point at infinity. Now find the inverse of $R$, which can be done by setting $R = (a, b)$ giving $-R = (a, -b)$.

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).