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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 $m$ or $t_s$?

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  • $\begingroup$ Can't you just compute the maximal orders, and see which ones contain $\mathcal O(s)$? $\endgroup$
    – Kimball
    Commented Feb 1 at 13:49
  • $\begingroup$ @Kimball I'm interested in proving if $t_s$ is fixed or in size $\mathcal{O}(\log p)$ when $B$ is a quaternion algebra ramified at $p$ and $\infty$ (if that is the case) in general. By computing the maximal orders, do you mean computing a representation for each isomorphism class of maximal orders? I suppose even if the representation does not contain $\mathcal{O}(s)$, there might be a conjugate of it containing $\mathcal{O}(s)$. $\endgroup$
    – Andy
    Commented Feb 1 at 16:28
  • $\begingroup$ Yes, but it shouldn't be too hard to check if there's an embedding. $\endgroup$
    – Kimball
    Commented Feb 2 at 0:08
  • $\begingroup$ @Kimball Could you elaborate on that? $\endgroup$
    – Andy
    Commented Feb 2 at 5:40
  • $\begingroup$ Pick a basis of $\mathcal O(s)$. For each basis element $\alpha$, you can enumerate the elements of $\mathcal O$ with the same charpoly. So look at the possible images of the basis and see which give ring monomorphisms. Practically, you should first check to see if this is already implemented in Magma. $\endgroup$
    – Kimball
    Commented Feb 3 at 0:35

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