Here is a first example of an elliptic curve over a finite field where you can work everything out by hand.
Consider the elliptic curve defined by the equation
over the field
It’s easy to find all the solutions
Thus the points on our elliptic curve are
The point
The nice thing about working over an unextended finite field like
Fortunately, the familiar geometric description of the group operation on elliptic curves in terms of line intersections still works (why?). That is, any two points can be added by drawing a line through them, finding the third point of intersection, and reflecting through the line
For example, it is immediate from this rule that
As in the case over
- If a vertical line passes through two distinct affine points such as
and , then (since it also intersects with in the projective plane) these points are inverses of one another w.r.t. the group operation. (We’ve labelled to reflect this.) - If a vertical line hits a single affine point (e.g. the line
) then this point is its own inverse.
Thus
Amusingly, the geometric rule for point doubling using tangents still works, as well. The slope of the tangent at a point
These slopes are depicted on our diagram with dashed blue lines. Following these tangents, you can immediately verify that
and so
The orders of our group elements are enough to conclude that our group (call it
is an isomorphism of groups