Discriminant of a quaternion order (awaiting review)

Let $B$ be a quaternion algebra over $\Q$, and let $O\subset B$ be an order. Let $x_1,x_2,x_3,x_4 \in O$ be a $\Z$-basis of $O$. The discriminant of $O$ is the integer \[ \operatorname{disc}(O) \colonequals \left|\det(\operatorname{trd}(x_ix_j))_{i,j}\right|. \] where $\operatorname{trd} : B \to \Q$ is the reduced trace.

The reduced discriminant of $O$ is the positive integer $\operatorname{discrd}(O)$ such that \[ \operatorname{discrd}(O)^2 = \operatorname{disc}(O). \]