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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!

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    $\begingroup$ The proof is given in the first three paragraphs of the proof of Theorem 4.2 of the linked paper. It is possible to make everything explicit. One just needs to insert into their formula the explicit form of Minkowski's bound given in en.wikipedia.org/wiki/Minkowski%27s_bound $\endgroup$
    – Will Sawin
    Commented Apr 5 at 14:02

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