Hurwitz quaternion order

The Hurwitz quaternion order is a specific order in a quaternion algebra over a suitable number field. The order is of particular importance in Riemann surface theory, in connection with surfaces with maximal symmetry, namely the Hurwitz surfaces. The Hurwitz quaternion order was studied in 1967 by Goro Shimura, but first explicitly described by Noam Elkies in 1998. For an alternative use of the term, see Hurwitz quaternion (both usages are current in the literature).

Let ${\displaystyle K}$ be the maximal real subfield of ${\displaystyle \mathbb {Q} }$${\displaystyle (\rho )}$ where ${\displaystyle \rho }$ is a 7th-primitive root of unity. The ring of integers of ${\displaystyle K}$ is ${\displaystyle \mathbb {Z} [\eta ]}$, where the element ${\displaystyle \eta =\rho +{\bar {\rho }}}$ can be identified with the positive real ${\displaystyle 2\cos({\tfrac {2\pi }{7}})}$. Let ${\displaystyle D}$ be the quaternion algebra, or symbol algebra

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