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Can I efficiently verify if given $h$ is the class number of a quadratic field?

Can I efficiently verify if given $h$ is the class number of a quadratic field?

Computing the class number is not tractable.

I tried pari's Qfb and did some experiments with Lidia, but I must be missing something.

What I tried is for random $a$ compute $a^h$ but I don't get the identity and in Lidia I can't check if it is principal.