inclusion chains: $${\mathbb{Z}}\langle{(1+i)/2,j}\rangle 
  \subsetneq 
  \left\{
  \begin{array}{l}
  {\displaystyle}{\mathbb{Z}}\langle{\frac{1+i}{2}{\raisebox{0.4ex}{,}}\, j\,{\raisebox{0.4ex}{,}}\frac{ci+k}{q}}\rangle{\raisebox{0.4ex}{,}}\\[2.5mm]
  {\displaystyle}{\mathbb{Z}}\langle{\frac{1+i}{2}{\raisebox{0.4ex}{,}}\, j\,{\raisebox{0.4ex}{,}}\frac{ci-k}{q}}\rangle{\raisebox{0.4ex}{,}}\end{array}
  \right.$$ where $c$ is any root of $x^2 + p \bmod q$. In particular $R + Rj$ is an Eichler order.

Under the generalized Riemann hypothesis, for $p \equiv 1 \bmod 4$, the smallest $q$ satisfying the conditions of the last lemma is $O(\log(p)^2)$ by a result of Ankeny [@Ankeny1952] (or explicitly $q < 2\log(p)^2$ by Bach [@Bach1990]). In the remainder of this paper, we will assume that ${B_{p,\infty}}$, ${\mathcal{O}}$, and $R$ are suitably constructed from these lemmas with ${\mathrm{disc}}(R)$ the minimal discriminant in which $p$ is inert in the sequence $-3$, $-4$, $-7$, $-8$, or $-q$ for $q \equiv 3 \bmod 4$ prime.

Reduced norms and ideal morphisms {#sec:ideals:properties}
---------------------------------

Now suppose that ${\mathcal{O}}$ is any maximal order. We recall that the reduced norm on ${B_{p,\infty}}$ induces a reduced norm on left ideals defined by any of the equivalent conditions $${\mathrm{Nrd}}(I) :=  \sqrt{|{\mathcal{O}}/I|} 
= \gcd\left(\{\, {\mathrm{Nrd}}(\alpha) \;:\; \alpha \in I \,\}\right),$$ or by $I\bar{I} = {\mathrm{Nrd}}(I){\mathcal{O}}$. It follows that the reduced norm on ideals is multiplicative and compatible with the reduced norm on elements ${\mathrm{Nrd}}(\alpha) = 
{\mathrm{Nrd}}(\alpha{\mathcal{O}}) = {\mathrm{Nrd}}({\mathcal{O}}\alpha)$. If $I$ and $J$ are left ${\mathcal{O}}$-ideals, a homomorphism of 