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Let $E: y^2 = x^3 + Ax + B$ be a quasi-minimal elliptic curve over $\mathbb{Q}$, i.e. $\gcd(a^3, b^2)$ is $12$th power free. Furthermore, let $\operatorname{rank}(E) = 1$ and $j(E)=\frac{1728 \times 4A^3}{4A^3 + 27B^2} \in \mathbb{Z}$.

Given these conditions, is the subset of these elliptic curves conjectured or known to have a positive proportion among all elliptic curves over $\mathbb{Q}$? Initial heuristic approaches suggest no, but definitive insights or references would be highly appreciated.

Thanks!

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    $\begingroup$ $j(E)\in\mathbb{Z}$ is a very strong assumption. $\endgroup$
    – Chris Wuthrich
    Commented Jun 6 at 16:01
  • $\begingroup$ It seems like this comes down to showing that the proportion of rationals $A,B$ of a bounded height satisfying $4A^3 + 27B^2 \mid 1728\cdot 4A^3$ goes to 0. $\endgroup$
    – Kevin Casto
    Commented Jun 6 at 16:20
  • $\begingroup$ @ChrisWuthrich I agree. I'm interested in this because of a theorem by Silverman which gives an explicit upper bound for the number of integral solutions for these curves. $\endgroup$
    – Navvye
    Commented Jun 6 at 16:23

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