Nagell-Lutz theorem

The following theorem, proved independently by E. Lutz and T.Nagell, gives a very efficient method to compute the torsionsubgroup of an elliptic curve defined over $\mathbb{Q}$.

Theorem 1 (Nagell-Lutz Theorem).

Let $E\mathrm{/}\mathrm{Q}$ be an elliptic curve with Weierstrass equation:

$$y^2=x^3+Ax+B,A,B\in \mathbb{Z}$$

Then for all non-zero torsion points $P$ we have:

1. The coordinates of $P$ are in $\mathbb{Z}$, i.e.

   $$x(P),y(P)\in \mathbb{Z}$$

2. If $P$ is of order greater than $2$, then

   $$y{(P)}^2\mathit{\hspace{1em}}divides\mathit{\hspace{1em}}4A^3+27B^2$$

3. If $P$ is of order $2$ then

   $$y(P)=0\mathit{\hspace{1em}}and\mathit{\hspace{1em}}x{(P)}^3+Ax(P)+B=0$$