@Kimball Could you elaborate on that?

@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 $\mathcal{O}(s)$.

@Kimbell I was wondering if there are non-canonical bases, but yeah I'm good with abx's answer.

@abx That's a good example. Can you add more number of generators while being central simple?