Let $\hat{h}_d(\cdot)$ be the Neron-Tate canonicial height function for the curve $E_d:y^2 = x^3+d$ ($d$ is 6th power free). I also assume that $E_d$ has rank one so that, by the quadratic behavior of $\hat{h}$, that for a generator $Q_d$ of $E_d(\mathbb{Q})$ the minimal non-zero value of $|\hat{h}_d(\cdot)|$ is $|\hat{h}_d(Q_d)|$. (Note: I also would like a way to verify that this statement is true, as the quadratic behavior is only up to $O(1)$.)
What I have so far regarding the question is rudimentary data gathered from computations on Sage, from which I found that this minimal value seems to increase on average as the generator grows in Weil height. This is according to my expectation, although the growth seems quite erratic.
My question is if there is any idea of how the value $|\hat{h}_d(Q_d)|$ evolves as $d$ ranges over the 6th-power free integers, where values are "plotted" against $d$ of increasing magnitude: 0, $\pm 1$, $\pm 2$, and so on. I understand this is a question whose answer is not really concretely known, but I would like to get any results I can about it, such as asymptotic behavior or average value. As I am also researching this distribution, any links or titles to works the community knows that address this topic or anything similar would be much appreciated!