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While the moments of a conjectural distribution in this case had previously been described, no method to calculate the distribution given the moments was known. We resolve this issue by introducing new invariants associated to the class group, $\\psi_{L/K}$ and $\\omega_{L/K},$ and study the distribution of $(G_{L/K}, \\psi_{L/K}, \\omega_{L/K})$ using a linear random matrix model. 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