Let $E/\mathbb{Q}$ and let $\hat{h}$ be the canonical height on $E$. Then:

1. 1.

   The height $\hat{h}$ satisfies the parallelogram law:

   $$\hat{h}(P+Q)+\hat{h}(P-Q)=2\hat{h}(P)+2\hat{h}(Q)$$

   for all $P,Q\in E(\overline{\mathbb{Q}})$.

2. 2.

   For all $m\in\mathbb{Z}$ and all $P\in E(\overline{\mathbb{Q}})$:

   $$\hat{h}([m]P)=m^{2}\hat{h}(P).$$

3. 3.

   The height $\hat{h}$ is even and the pairing:

   $$\langle\cdot,\cdot\rangle:E(\overline{\mathbb{Q}})\times E(\overline{\mathbb{Q}})\to\mathbb{R},\quad\langle P,Q\rangle=\hat{h}(P+Q)-\hat{h}(P)-\hat{h}(Q)$$

   is bilinear (usually called the Néron-Tate pairing on $E/\mathbb{Q}$).

4. 4.

   For all $P\in E(\overline{\mathbb{Q}})$ one has $\hat{h}(P)\geq 0$ and $\hat{h}(P)=0$ if and only if $P$ is a torsion point.