Points of Finite Order

An element P of any group is said to have order m if $$\displaystyle{mP =\mathop{\underbrace{ P + P + \cdots + P}}\limits _{\mbox{ $m$ summands}} = \mathcal{O},}$$ but \(m'P\neq \mathcal{O}\) for all integers 1 ≤ m′ < m. If such an m exists, then P has finite order, otherwise it has infinite order. We begin our study of points of finite order on cubic curves by looking at points of order two and order three. As usual, we will assume that our non-singular cubic curve is given by a Weierstrass equation $$\displaystyle{y^{2} = f(x) = x^{3} + ax^{2} + bx + c,}$$ and that the point at infinity \(\mathcal{O}\) is taken to be the zero element for the group law.