# How do you find maximal orders in quaternion algebras?

Let A be the 4-dimensional algebra over Q with basis 1,i,j,k, and multiplication table $$i^2 = -1 \quad j^2 = -11 \quad k^2 = -11 \quad ij = k \quad jk = 11i \quad ki = j$$

So, A is the unique "definite quaternion algebra of discriminant 11" over the rational numbers. I'm led to believe that there are two conjugacy classes of maximal orders in A, but I had a hard time finding them explicitly.

A has elements of multiplicative order 4 (e.g. $i$) and $6$, (e.g. $1/2 + i/4 + j/4$). I presume then that $(1,i)$ can be extended to a basis for a maximal order, and that $(1,1/2 + i/4 + j/4)$ can be extended to a basis for a non-conjugate maximal order. What are these bases?

I'd appreciate an answer even just for discriminant 11. Of course any information about how to answer this kind of question systematically would be a nice bonus for me.

PS. Thanks Aeryk and Aurel. One of the two orders, say $O_1$, has basis $$1, \quad i,\quad 1/2+j/2, \quad i/2 + k/2$$ Can anyone tell me the other one? Are the Ivanyos/Ronyai/Voight algorithms good for finding all maximal orders, or would they output something conjugate to $O_1$ again?

• Magma also has algorithms for computing the set of conjugacy classes of maximal orders. In your case, it tells me there are two conjugacy classes, and it computes a non conjugate order with basis [1/2 + 1/2*j + k, 1/4*i + 1/2*j + 3/4*k, j, 2*k]. – Aurel Oct 16 '13 at 16:00

For semisimple algebras over $$\mathbb{Q}$$, there is a general algorithm due to Gabor Ivanyos and Lajos Ronyai, described in Finding maximal orders in semisimple algebras over $$\mathbb{Q}$$ and implemented in the computer algebra system Pari/gp. For quaternion algebras, there is a dedicated algorithm due to John Voight, described in Identifying the matrix ring: algorithms for quaternion algebras and quadratic forms and implemented in the computer algebra system Magma.
Let $D =\left(\frac{−1,−11}{\mathbb{Q}}\right)$ be the quaternion algebra with discrimimant 11. A discriminant computation shows that a maximal order $O$ is given by $\mathbb{Z}[z] + i\mathbb{Z}[z]$ where $z = \frac{1+j}{2}$.