Working in an order $\mathcal{O}$ in an imaginary quadratic field $K = \mathbb{Q}(\sqrt{d})$ and given an invertible ideal $\mathfrak{a}\subseteq \mathcal{O}$, I would like to produce another integral ideal $\mathfrak{b}$ in the same equivalence class of the ideal class group, i.e. $\mathfrak{b}=\alpha\mathfrak{a}$ for some $\alpha\in K$, such that the norm of $\mathfrak{b}$ is coprime to the conductor of $\mathcal{O}$.


We know that this is possible by, for instance, Cox's *Primes of the Form $x^2 +ny^2$* Corollary 7.17. However, the proof of this fact is existential, using the isomorphism to the form class group, and does not give an explicit method of finding such a representative.


I remember reading a way to do this long ago, and maybe it's a consequence of Artin-Whaples Lemma? But I have been unable to get anywhere myself.


Any and all help is appreciated.


Edit:
In this paper, [ON SINGULAR MODULI FOR ARBITRARY DISCRIMINANTS](https://arxiv.org/pdf/1206.6942.pdf), on page 15, the author seems to say that, if we start with a $\mathfrak{p}$-primary ideal $\mathfrak{a}$ that is locally principally generated by some integral $\alpha$ whose norm is supported only at $\mathfrak{p}$ over the conductor (which I am not sure how one would find such an element), then $\mathfrak{a}\sim \tilde{\mathfrak{a}}:= \mathcal{O}\_p\cap\bigcap \alpha\mathcal{O}\_\mathfrak{q}$.


Why must such an $\alpha$ exist? Further, what principal ideal makes these in the same ideal class? It feels like it should be $\alpha^{-1}$; am I being silly?



Representative in ideal class group coprime to the conductor
There is probably better algorithmically but 


* $C = \{ b\in O\_K,bO\_K\subset O\}$ is a $O$ and $O\_K$ ideal.


Choose some $O\_K$-representatives of $O\_K/C$ and let $f(a+C) = a O\_K \cap O$ whose class doesn't depend on the representative $a$.


$f(O\_K/C)$ contains the kernel of $\phi : Cl(O) \to Cl(O\_K), \phi(I) =IO\_K$.


* The invertible classes of $O$-ideals are in the image of $f(O\_K/C^\times)$ (I don't know how to prove this, it is p.12 of [kconrad](http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/conductor.pdf))
* If a $O\_K$ ideal $J$ is coprime with $C$ then $J = (J \cap O) O\_K$.


For an invertible ideal $I \subset O$ there is $b/d\in K$ such that $J=\frac{b}{d} IO\_K$ has norm coprime with $N(C)$, 


then $I \in (J\cap O)\ker(\phi)$ which means for some $a \in O\_K/C^\times$, $I \sim (aO\_K \cap O) (J\cap O)$


