We present two class number formulas associated to orders in totally definite quaternion algebras in the spirit of the Eichler class number formula. More precisely, let $F$ be a totally real number field, $D$ be a totally definite quaternion $F$-algebra, and $mathcal{O}$ be an $O_F$-order in $D$. Assume that $mathcal{O}$ has nonzero Eichler invariants at all finite places of $F$ (e.g. $mathcal{O}$ is an Eichler order of arbitrary level). We derive explicit formulas for the following two class numbers associated to $mathcal{O}$: (1) the class number of the reduced norm one group with respect to $mathcal{O}$, namely, the cardinality of the double coset space $D^1backslashwidehat{D}^1/widehat{mathcal{O}}^1$; (2) the number of locally principal right $mathcal{O}$-ideal classes within the spinor class of the principal right $mathcal{O}$-ideals, that is, the cardinality of $D^timesbackslashbig(D^timeswidehat{D}^1widehat{mathcal{O}}^timesbig)/widehat{mathcal{O}}^times$. Both class numbers depend only on the spinor genus of $mathcal{O}$, hence the title of the present paper. The proofs are made possible by optimal spinor selectivity for quaternion orders.
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