Type number of a quaternion algebra

Let $B$ be a quaternion algebra over a number field $K$ , with $R=\mathcal O_K$ . Its type number is the number of conjugacy classes of maximal $R$ -orders in $B$ , where

\mathcal O\sim\mathcal O' \quad\Longleftrightarrow\quad \mathcal O'=a\mathcal Oa^{-1}\text{ for some }a\in B^\times.

This number is finite. In this setting, conjugacy is equivalent to isomorphism as $R$ -algebras.

Compared with class number

The class number of a fixed maximal order counts ideal classes. Type number counts maximal orders. Sending a right ideal class to the conjugacy class of its left order gives a surjection from the former set onto the latter; it need not be injective.

The rational Hamilton algebra has type number one: every maximal order is conjugate to the Hurwitz order. This does not say that all maximal orders are literally the same subset of the algebra.

Convention

One can also count types within the genus of a nonmaximal order. This knowl uses the maximal-order type number of the ambient algebra.

References

John Voight, Quaternion Algebras, author's text, §17.4, especially Lemma 17.4.13, and Main Theorem 17.7.1.