I'm referring On the imbeddings of imaginary quadratic orders in definite quaternion orders by Brzezinski and Eichler here.

Let $B$ be a definite quaternion algebra over $\mathbb{Q}$. Given an order $\mathcal{O}$ in $B$, its embedding number $e(\mathcal{O})$ counts the number of maximal orders in $B$ containing it. By the local-to-global principle $e(\mathcal{O})$ is the product of $e_p(\mathcal{O})$, where $p$ runs over primes at which $B$ is not ramified and $e_p(\mathcal{O})$ is the local embedding number; the number of local maximal orders in $B_p$ containing $\mathcal{O}_p$. There is a formulation for the local embedding number in case $\mathcal{O}$ is Bass; if $p$ divides the discriminant of $\mathcal{O}$ and $B$ is not ramified in $p$, then

$$e_p(\mathcal{O})=\begin{cases}1 & \text{if} \ (\mathcal{O}/p)=-1 \ \text{or} \ (\mathcal{O}/p)=0 \ B_p \ \text{is a skew field},\\ 2 & \text{if} \ (\mathcal{O}/p)=0 \ \text{and}\ B_p \ \text{is a matrix algebra}, \\ v_p(D)+1 & \text{if} \ (\mathcal(O)/p)=1,\end{cases}$$ where $(\mathcal{O}/p)$ is the Eichler symbol of $\mathcal{O}$.

Let $q, d>0$ be square-free integers. Suppose $\mathbb{Q}(\sqrt{-q})$ and $\mathbb{Q}(\sqrt{-d})$ simultaneously embed into $B$ and their maximal orders form an order in $B$ (conditions for this happening is given in Proposition 1). Then there exists $\omega_1, \omega_2$ in $B$ such that \begin{equation}\label{eq} \omega_1^2=-q, \omega_2^2=-d \ \text{and}\ \omega_1\omega_2+\omega_2\omega_1=s \in \mathbb{Z} \end{equation} For a fixed $s$, let $\mathcal{O}(s)$ be the least order containing $\omega_1$ and $\omega_2$. In Proposition 4, $e_p(\mathcal{O}(s))$ was computed using the application of the formulation above.

Now I'm interested in counting the number of types of maximal orders containing $\mathcal{O}(s)$ (for a fixed $s$). In Proposition 6, by counting the number of pairs $\omega_1, \omega_2$ satisfying $\omega_1^2=-q, \omega_2^2=-d \ \text{and}\ \omega_1\omega_2+\omega_2\omega_1=s \in \mathbb{Z}$, they showed for any maximal order $\mathcal{O}$ containing $\mathcal{O}(s)$, the number of maximal orders containing $\mathcal{O}(s)$ and isomorphic to $\mathcal{O}$ is same. Say this number is $m$ so that $t_s m = e(\mathcal{O}(s))$, where $t_s$ is the number of types of maximal orders containing $\mathcal{O}(s)$.

I did some computation on Sage and I was trying to see if $t_s$ was significantly smaller than $e(\mathcal{O}(s))$ when $s=0$. Is there a reference computing $m$?