Elliptic Curves Over the Reals

An elliptic curve over the reals is given by an equation

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

where

$$-16(4a^3+27b^2)\ne 0$$

The condition on the discriminant says the curve is nonsingular, so it has no cusps or self-intersections.

Point at Infinity

We adjoin one extra point, denoted $O$, called the point at infinity.

Group Law

The points on the curve, together with $O$, form an abelian group.

Geometric Addition

If $P$ and $Q$ are points on the curve:

• Draw the line through $P$ and $Q$ if $P\ne Q$.
• If $P=Q$, draw the tangent line at $P$.
• This line meets the curve at a third point.
• Reflect that third point across the $x$-axis.

The reflected point is defined to be $P+Q$.

Identity and Inverses

• The identity element is $O$.
• If $P=(x,y)$, then

$$-P=(x,-y)$$

so $P+(-P)=O$.

Order of a Point

If $P$ is a point on an elliptic curve $E$, the order of $P$ is the smallest positive integer $r$ such that

$$rP=O$$

if such an $r$ exists. Otherwise, the order is $\infty$.

Over the real numbers, most points have infinite order.