Elliptic Curves

New (2026): the Tate Pairing Visualizer colours the points of a curve over F_p by the pairing with a chosen n-torsion point — plane waves on the group torus, noise in the (x, y) plane — and E[n] by the self-pairing t(P, P). Also the Elliptic Curve Playground draws the dynamics of P ↦ aP and P ↦ P + Q on an elliptic curve mod n, as cycles and trees or as points in the plane, with the failures of the group law over composite n shown as sinks labelled by the factor they reveal.

Some images relating to elliptic curves.

A family of elliptic curves

A large elliptic curve over a finite field

Elliptic Curve Group Law