 Canonical height of an elliptic curve

This is the elliptic-curve specialization of the Néron–Tate height. For an associated logarithmic projective height $h$, one normalization is $\hat{h}(P)={\lim }_{no\infty }{4}^{-n}h([2^n]P)$.

The canonical height isIt is a quadratic form, vanishes on torsion points, and satisfies $\hat{h}(\varphi P)=\mathrm{deg}(\varphi )\hat{h}(P)$ for every isogeny $\varphi$.