This is about the frequency of integral solutions to $$ b^2 - 4 a^3 = \Delta, $$ when $\Delta < 0$ is a discriminant of positive binary quadratic forms such that the class number is divisible by 3. My observation is that integral solutions are pretty frequent when $|\Delta|$ is small. For instance, there are 25 discriminants with class number exactly 3. Of these, in 19 cases we can write the non-principal, order three form as $$ \langle a, \pm b, a^2 \rangle, $$ the failures seeming to be $$ \Delta = -124, \; -172, \; -307, \; -547, \; -652, \; 907,$$ with the first two proved impossible in integers, maybe all six.

Note that there may be multiple solutions for a given $\Delta,$ which usually means the forms reduce to the same thing or the principal form, as in
$$ \langle 2, 1, 4 \rangle = \langle 4, 15, 16 \rangle $$ but
$$ \langle 10, 63, 100 \rangle = \langle 1, 1, 8 \rangle. $$
On the other hand, in the relatively rare event of 3-rank larger than one, we may have genuinely distinct cube roots of 1, and for $\Delta = -3299$ we find

$$ \langle 11,45, 121 \rangle = \langle 11, 1, 75 \rangle, $$
$$ \langle 15, 101, 225 \rangle = \langle 15, 11, 57 \rangle, $$

$$ \langle 23, 213, 529 \rangle = \langle 23, -17, 39 \rangle. $$

So, the question is, how likely is it that a positive form of order three in the class group can be written as $$ f(x,y) = a x^2 + b x y + a^2 y^2 ? $$

EDIT: of course this becomes Mordell's equation when $\Delta$ is even, which is how I know there are proofs for $-124, -172$ but a rational solution $a = 1177/36, \; b = 40355/108$ for $-172.$ Hope I got the factor of 2 in the right place.

EDIT TOOOOO: forgot. by Dirichlet's "united forms" description of composition, $$ \langle a,b,a^2 \rangle^2 = \langle a^2, b,a \rangle, $$ so such a form is either the principal form or of order three, as long as $$ \gcd(a,b) = 1. $$

EDIT THREE, Saturday: The importance of this for my other recent questions appeared when I found that $ \langle 4,1,9 \rangle$ integrally represents 77, but $ \langle 4,1,9 \rangle + \langle 4,1,9 \rangle,$ or $g(x,y,z,w) = 4 x^2 + x y + 9 y^2 + 4 z^2 + z w + 9 w^2,$ does not represent $77^2 = 5929.$ Now, $ \langle 4,1,9 \rangle$ has order five, so the set of values it represents is not required to be multiplicative. Similarly, if $f$ is of order 2 "ambiguous," it seems always true that $f$ represents some $n$ (often prime) while $f(x,y) + f(z,w)$ fails to represent $n^2.$ Note that the original form in question 88905 is of order three, indeed $$ \langle 4,2, 3 \rangle = \langle 3,-2,4 \rangle = \langle 3,-8,9 \rangle. $$ So three seems to be the order of the day.