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In this paper we establish lower and upper bounds for the cardinality of theprofinite genus of the fundamental group $\pi_{1}(M_A)\cong (\mathbb{Z} \times\mathbb{Z})\rtimes_{A}\mathbb{Z}$ of a torus bundle $M_{A}$ in terms of thenumber of ideal classes of the order $\mathbb{Z}[\lambda]$, where $\lambda$ isan eigenvalue of the matrix $A$ in $\mathrm{GL}_{2}(\mathbb{Z})$.

Profinite genus of fundamental groups of torus bundles