  For positive rank $r$ elliptic curves $E(\mathbb{Q})$, we employ ideal class
pairings
  $$ E(\mathbb{Q})\times E_{-D}(\mathbb{Q}) \rightarrow \mathrm{CL}(-D), $$ for
quadratic twists $E_{-D}(\mathbb{Q})$ with a suitable ``small $y$-height''
rational point, to obtain effective class number lower bounds. For the curves
$E^{(a)}: \ y^2=x^3-a,$ with rank $r(a),$ this gives $$
  h(-D) \geq \frac{1}{10}\cdot
\frac{|E_{\mathrm{tor}}(\mathbb{Q})|}{\sqrt{R_{\mathbb{Q}}(E)}}\cdot
\frac{\Gamma\left (\frac{r(a)}{2}+1\right)}{(4\pi)^{\frac{r(a)}{2}}} \cdot
\frac{\log(D)^{\frac{r(a)}{2}}}{\log \log D}, $$ representing an improvement to
the classical lower bound of Goldfeld, Gross and Zagier when $r(a)\geq 3$. We
prove that the number of twists $E_{-D}^{(a)}(\mathbb{Q})$ with such a point
(resp. with such a point and rank $\geq 2$ under the Parity Conjecture) is
$\gg_{a,\varepsilon} X^{\frac{1}{2}-\varepsilon}.$
  We give infinitely many cases where $r(a)\geq 6$. These results can be viewed
as an analogue of the classical estimate of Gouv\^ea and Mazur for the number
of rank $\geq 2$ quadratic twists, where in addition we obtain ``log-power''
improvements to the Goldfeld-Gross-Zagier class number lower bound.
