First Hurwitz triplet

In the mathematical theory of Riemann surfaces, the first Hurwitz triplet is a triple of distinct Hurwitz surfaces with the identical automorphism group of the lowest possible genus, namely 14 (genera 3 and 7 each admit a unique Hurwitz surface, respectively the Klein quartic and the Macbeath surface). The explanation for this phenomenon is arithmetic. Namely, in the ring of integers of the appropriate number field, the rational prime 13 splits as a product of three distinct prime ideals. The principal congruence subgroups defined by the triplet of primes produce Fuchsian groups corresponding to the triplet of Riemann surfaces.

Let ${\displaystyle K}$ be the real subfield of ${\displaystyle \mathbb {Q} [\rho ]}$ where ${\displaystyle \rho }$ is a 7th-primitive root of unity. The ring of integers of K is ${\displaystyle \mathbb {Z} [\eta ]}$, where ${\displaystyle \eta =2\cos({\tfrac {2\pi }{7}})}$. Let ${\displaystyle D}$ be the quaternion algebra, or symbol algebra ${\displaystyle (\eta ,\eta )_{K}}$. Also Let ${\displaystyle \tau =1+\eta +\eta ^{2}}$ and ${\displaystyle j'={\tfrac {1}{2}}(1+\eta i+\tau j)}$. Let ${\displaystyle {\mathcal {Q}}_{\mathrm {Hur} }=\mathbb {Z} [\eta ][i,j,j']}$. Then ${\displaystyle {\mathcal {Q}}_{\mathrm {Hur} }}$ is a maximal order of ${\displaystyle D}$ (see Hurwitz quaternion order), described explicitly by Noam Elkies [1].

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