I have already tried a somewhat exhaustive search of the literature, but couldn't find anything close to the problem that I am working on.

My question is: When does Mordell's Equation

$$y^2 = x^3 + k$$

have only FINITELY many solutions over the field of rational numbers, if we allow $k$ itself to be a rational number?

I've seen a criterion related to the class number of the (real/imaginary) quadratic field $Q(\sqrt(k))$, but it is limited only to $k$ being either 1 or 2 modulo 4.

The actual criterion (as stated in the Japanese[?] paper that I allude to) is:

Mordell's equation $y^2 = x^3 + k$ has finitely many solutions in $Q$ if and only if

(1) $-k$ is not of the form $3t^2 + 1$ or $3t^2 - 1$; AND

(2) $k \equiv 1 (mod 4)$ or $k \equiv 2 (mod 4)$; AND

(3) $3$ does not divide the class number of the (real/imaginary) quadratic field $Q(\sqrt(k))$.

Thanks to Kevin for pointing out the error in the third condition. I was considering the case $k > 0$ (i.e. for real quadratic fields).