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Is there a criterion for the (presumably infinite) set of $D \in \mathbb{Z}\setminus \{0\}$ such that

$$Dy^2 = x^3-1728$$

has an integral point over $\mathbb{Q}$ with $y \neq 0$? I'd also be interested in results about the density of such $D$ (natural or otherwise).

A quick Sage computation suggests that this set isn't especially sparse, but it's hard to go very far because computing integral points is difficult. The sequence of such $D$ between $-30$ and $30$ that we do get doesn't turn up on OEIS, for what it's worth :P

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    $\begingroup$ These are quadratic twist families and there is a lot known (at least conditionally) for distributions of ranks. Searching for these words will turn up a lot of material. $\endgroup$
    – Asvin
    Commented Sep 29, 2020 at 5:26
  • $\begingroup$ In general it is very hard to detect whether a given elliptic curve has an integral point, since the answer depends on the specific model. Precisely, any elliptic curve with a rational point different from $\infty$ will have a model over $\mathbb{Q}$ with an integral point, simply by eliminating denominators of a rational point via a linear change of variables. Since the answer depends on the model there's likely no good answer. For an elliptic curve given by a specific Weierstrass model say, one can use Baker's method to bound the height of potential integral points. $\endgroup$
    – Stanley Yao Xiao
    Commented Sep 29, 2020 at 19:49

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The simplest answer is the following: for any integer $a\neq 12$ take $D=a^3-1728$. Then, the point $P=(a,1)$ is a point with integer coordinates on your curve. Moreover, the point $P$ is of infinite order.

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