Data from here, and graph from here lead to the following conjecture.
Diagonal Governing Groups Conjecture:
For all \(k \in \mathbb{N}^*\), there exists a fields \(M_{0k}\) such that \(M_{0k}\) is a governing field for \(a_{0k}\), and \(G_{0k} = \text{Gal}(M_{0k}/\mathbb{Q})\) is dihedral.
For all \(k \in \mathbb{N}^*\), there exists a fields \(M_{k0}\) such that \(M_{k0}\) is a governing field for \(a_{k0}\), and \(G_{k0}\) is dihedral.
Moreover \(M_{k0} \neq M_{0k}\) in general, but \(G_{k0} = G_{0k}\).
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