Heights of points on elliptic curves over $\mathbb{Q}$

In this note we obtain effective lower bounds for the canonical heights of non-torsion points on $E(\mathbb{Q})$ by making use of suitable elliptic curve ideal class pairings $${\mathrm{\Psi }}_{E,-D}:E(\mathbb{Q})\times E_{-D}(\mathbb{Q})\mapsto \mathrm{C}\mathrm{L}(-D)\mathrm{.}$$ In terms of the class number $H(-D)$ and $T_E(-D)$, a logarithmic function in $D$, we prove $$\widehat{h}(P)>\frac{\mathrm{\mid }{E}_{\mathrm{t}\mathrm{o}\mathrm{r}}(\mathbb{Q}){\mathrm{\mid }}^2}{{(H(-D)+\mathrm{\mid }{E}_{\mathrm{t}\mathrm{o}\mathrm{r}}(\mathbb{Q})\mathrm{\mid })}^2}\cdot T_E(-D)\mathrm{.}$$