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By a conjecture of Malle, we expect that $N(n, A_n, X) \\sim C_n X^{1/2} (\\log X)^{b_n}$, for constants $b_n$ and $C_n$. For $5 < n < 84394$, the best known upper bound is $N(n, A_n, X) \\ll X^{\\frac{n + 2}{4}}$; this bound follows from Schmidt's Theorem, which implies there are $\\ll X^{\\frac{n + 2}{4}}$ number fields of degree $n$. (For $n > 84393$, there are better bounds due to Ellenberg and Venkatesh.) 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