A point of order 8

TL;DR

This paper presents a formula for a point of order 8 on an elliptic curve using roots of the cubic polynomial, and explores its doubling to obtain a new point of order 4, expanding understanding of elliptic curve points.

Contribution

It introduces a novel explicit formula for points of order 8 on elliptic curves in terms of polynomial roots, and distinguishes the resulting points from known order 4 points.

Findings

Derived a formula for points of order 8 on elliptic curves.

Showed that doubling such points yields new points of order 4.

Identified points of order 4 different from classical references.

Abstract

A formula expressing a point of order 8 on an elliptic curve, in terms of the roots of the associated cubic polynomial, is given. Doubling such a point yields a point of order 4 distinct from the well-known points of order 4 given in standard references such as "A course of Modern Analysis" by Whittaker and Watson.