The number field $E=\mathbb Q(\sqrt{\sqrt{2}-3})$ is a CM-field since it is a totally imaginary extension of the totally real field $\mathbb Q(\sqrt 2)$.
Is there a way to construct an abelian surface defined over $E$ with CM by $E$?
I suppose the problem is intractable in general if $E$ is an arbitrary CM-field, but I would like to know if at least we can do something in dimension two (maybe we could try computing the Jacobian of some conveniently chosen genus 2 curve over $E$?). The field $E$ is an example, but if it's easier I would be happy to see an example of an abelian surface with CM by any other quartic field (preferably non-Galois).