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We show that the Hilbert modular group $PSL_2(\\mathfrak o_{k_{49}})\\subset PSL_2(\\mathbb R)^3$, with $k_{49}$ the totally real cubic field of discriminant $49$ has the minimal covolume with respect to $\\chi$ among all irreducible lattices in $PSL_2(\\mathbb R)^n$ for $n\\geq 2$ and is unique such lattice up to conjugation. 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More precisely, let $\\mu$ be the Euler-Poincar\\'e measure on $PSL_2(\\mathbb R)^n$ and $\\chi=\\mu/2^n$. We show that the Hilbert modular group $PSL_2(\\mathfrak o_{k_{49}})\\subset PSL_2(\\mathbb R)^3$, with $k_{49}$ the totally real cubic field of discriminant $49$ has the minimal covolume with respect to $\\chi$ among all irreducible lattices in $PSL_2(\\mathbb R)^n$ for $n\\geq 2$ and is unique such lattice up to conjugation. 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