# How to Reduce a Quaternion Ideal into Power Smoothness?

(TL;DR) How exactly do we reduce a quaternion ideal into another powersmooth one?

Given a supersingular elliptic curve, it is known that its endomorphism ring is non-commutative. Specifically, there is Deuring correspondence associating supersingular $$j$$-invariant up to a twist and a maximal order of quaternion algebra $$B$$ ramified at $$p,\infty$$.

Suppose $$\mathcal O\leftrightarrow j(E)$$ is associated by the correspondence, i.e. $$\mathsf {End}(E)=\mathcal O$$ and there is a left $$\mathcal O$$-ideal $$I\subseteq \mathcal{O}$$ with norm $$\mathsf {Nrd}(I)=N$$ and generators $$\alpha,N$$. It is said that $$I\sim J$$ are equivalent if $$\exists r\in B^\times$$ that $$Ir=J$$. I was reading this article saying there's an efficient algorithm to find a powersmooth representative (up to $$\sim$$) of ideal $$I$$ with large norm. I couldn't understand how exactly it is performed. The main goal in the article is to solve for a $$\beta,\lambda$$ such that $$\beta=\lambda\alpha\mod{N\mathcal O}$$ in page 8. But how exactly solving $$\beta$$ would help one reduce the ideal norm?

Is it possible to write down the explicit procedure? If not, is there any other way to reduce the ideal norm?