---
tag: KM59PWHJHH
kind: example
name: Hurwitz quaternion order
msc: [11R52, 16H10, 30F10]
instance_of: [E5DH1CSW1N]
uses: [ZWXW0T3EAC, 1ZBX4TKVZ9, MWBDFS2KJ5]
sources:
  - url: https://en.wikipedia.org/wiki/Hurwitz_quaternion_order
same_as:
  wikidata: Q16002489
  wikipedia: Hurwitz quaternion order
---

## Example

Let $\eta = 2\cos(2\pi/7)$ and let $K = \Q(\eta)$, the real cubic subfield of the seventh cyclotomic field, an [[algebraic number field|MWBDFS2KJ5]] with ring of integers $\Z[\eta]$.
The number $\eta$ is a root of $t^3 + t^2 - 2t - 1$.
Let $D$ be the [[quaternion algebra|ZWXW0T3EAC]] $(\eta, \eta)_K$, with $K$-basis $1, i, j, ij$ and relations $i^2 = j^2 = \eta$ and $ij = -ji$.
Put $\tau = 1 + \eta + \eta^2$ and $j' = \tfrac{1}{2}(1 + \eta i + \tau j)$.
The *Hurwitz quaternion order* is the $\Z[\eta]$-[[order|1ZBX4TKVZ9]]
$$\mathcal{Q}_{\mathrm{Hur}} = \Z[\eta][i, j, j']$$
generated by $i$, $j$ and $j'$.
It is a maximal order of $D$.

A check that $j'$ is integral: an element $a + bi + cj + d\,ij$ of $D$ has reduced trace $2a$ and reduced norm $a^2 - \eta b^2 - \eta c^2 + \eta^2 d^2$.
So $j'$ has reduced trace $1$ and reduced norm $\tfrac{1}{4}(1 - \eta^3 - \eta\tau^2)$.
Using $\eta^3 = -\eta^2 + 2\eta + 1$ one finds $\eta \tau^2 = \eta^2 + 10\eta + 4$, so the reduced norm is $\tfrac{1}{4}(-4 - 12\eta) = -1 - 3\eta \in \Z[\eta]$.
Hence $j'$ satisfies the monic equation $x^2 - x - 1 - 3\eta = 0$ over $\Z[\eta]$.

## Remarks

The group of elements of reduced norm $1$ in $\mathcal{Q}_{\mathrm{Hur}}$, modulo $\set{\pm 1}$, is isomorphic to the $(2,3,7)$ triangle group.
Quotients of the hyperbolic plane by its principal congruence subgroups give Hurwitz surfaces, the compact Riemann surfaces with the largest possible automorphism group for their genus.

## Explanations

### L1 human

This is a specific ring of quaternion-like numbers with coefficients from the field generated by $2\cos(2\pi/7)$.
It plays the role that the integers play inside the rationals, and it cannot be enlarged to a bigger such ring.
Its unit group acts on the hyperbolic plane like a very symmetric tiling by triangles with angles $\pi/2$, $\pi/3$ and $\pi/7$.
This is why it is used to build the most symmetric Riemann surfaces, such as the Klein quartic.