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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).

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    please don't rely on the title for content; the body of the message should be self-contained, as a service to your readers.2011-01-25
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    You may want to post that as an answer, rather than an edit to the question. It is perfectly acceptable to post answers to your own questions if you manage to figure them out. Then you can add just the closing paragraph of your addition to the question, referencing your work below.2011-01-25

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