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].

In order to construct the first Hurwitz triplet, consider the prime decomposition of 13 in ${\displaystyle \mathbb {Z} [\eta ]}$ , namely

${\displaystyle 13=\eta (\eta +2)(2\eta -1)(3-2\eta )(\eta +3),}$ 

where ${\displaystyle \eta (\eta +2)}$  is invertible. Also consider the prime ideals generated by the non-invertible factors. The principal congruence subgroup defined by such a prime ideal I is by definition the group

${\displaystyle {\mathcal {Q}}_{\mathrm {Hur} }^{1}(I)=\{x\in {\mathcal {Q}}_{\mathrm {Hur} }^{1}:x\equiv 1{\pmod {I{\mathcal {Q}}_{\mathrm {Hur} }}}\},}$ 

namely, the group of elements of reduced norm 1 in ${\displaystyle {\mathcal {Q}}_{\mathrm {Hur} }}$  equivalent to 1 modulo the ideal ${\displaystyle I{\mathcal {Q}}_{\mathrm {H} ur}}$ . The corresponding Fuchsian group is obtained as the image of the principal congruence subgroup under a representation to PSL(2,R).

Each of the three Riemann surfaces in the first Hurwitz triplet can be formed as a Fuchsian model, the quotient of the hyperbolic plane by one of these three Fuchsian groups.

The Gauss–Bonnet theorem states that

${\displaystyle \chi (\Sigma )={\frac {1}{2\pi }}\int _{\Sigma }K(u)\,dA,}$ 

where ${\displaystyle \chi (\Sigma )}$  is the Euler characteristic of the surface and ${\displaystyle K(u)}$  is the Gaussian curvature . In the case ${\displaystyle g=14}$  we have

${\displaystyle \chi (\Sigma )=-26}$  and ${\displaystyle K(u)=-1,}$ 

thus we obtain that the area of these surfaces is

${\displaystyle 52\pi }$ .

The lower bound on the systole as specified in [2], namely

${\displaystyle {\frac {4}{3}}\log(g(\Sigma )),}$ 

is 3.5187.

Some specific details about each of the surfaces are presented in the following tables (the number of systolic loops is taken from [3]). The term Systolic Trace refers to the least reduced trace of an element in the corresponding subgroup ${\displaystyle {\mathcal {Q}}_{Hur}^{1}(I)}$ . The systolic ratio is the ratio of the square of the systole to the area.

• (2,3,7) triangle group

• Elkies, N. (1999). The Klein quartic in number theory. The eightfold way. Math. Sci. Res. Inst. Publ. Vol. 35. Cambridge: Cambridge Univ. Press. pp. 51–101.
• Katz, M.; Schaps, M.; Vishne, U. (2007). "Logarithmic growth of systole of arithmetic Riemann surfaces along congruence subgroups". J. Differential Geom. 76 (3): 399–422. arXiv:math.DG/0505007. doi:10.4310/jdg/1180135693. S2CID 18152345.
• Vogeler, R. (2003). On the geometry of Hurwitz surfaces (Thesis). Florida State University.