The class number of $\mathbb{Q}(\sqrt{-23})$ is $3$, and the form $$2x^2 + xy + 3y^2 = z^3$$ is one of the two reduced non-principal forms with discriminant $-23$.
There are the obvious non-primitive solutions $(a(2a^2+ab+3b^2),b(2a^2+ab+3b^2), (2a^2+ab+3b^2))$.
I'm pretty sure there aren't any primitive solutions, but can't seem to find an easy argument. Are there?
In general, is it possible for a non-principal form to represent an $h$-th power primitively (where $h$ is the class number of the associated quadratic field)?
[EDIT]
I think I've solved the first question and have posted an answer below.
Since the proof is very technical I don't see how it can generalize to the greater question above ($h$-th powers).