According to Helfgott-Venkatesh, Let $E(D)$ denote the elliptic curve $y^2 = x^3 + D$, then $h_3(Q(\sqrt D))$, which is the 3-part of the class number of the Quadratic Field with discriminant $D$, or the class number of binary cubic forms with discriminant $D$, can be bounded from above by the number of integrals points on elliptic curves of the form $E_{D, \delta^2}$ where $1 \le \delta \ll D^{1/4}$

I'm curious about the proof of this statement, and whether or not we can make the upper bound completely explicit, i.e, $\delta \le C D^{1/4}$ for some absolute constant $C$.

For some more context, I know that using a result of Pierce et al. that we can bound $h_3(D)$ by counting the number of squares of the form $4x^3 - dz^2$ and applying a square sieve. I also know that the number of integral points on an elliptic curve on E is atmost $10 h_3(D)$, but I'm still curious about the bounds/proof for $\delta$.

Thank you!