Relationship between the conductor of an order and the conductor of a number field extension

Question

What is the relationship between the conductor $\mathfrak{f}_{\mathcal{o}}$ of an order $\mathcal{o}\subset \mathcal{O}_K$ and the conductor $\mathfrak{f}_{L/K}$ of a field extension in the classical formulation of Artin reciprocity from class field theory? Considering they have the same name and are frequently denoted by the same symbols in the same area of math, it seems obvious they should be the same concept (or at least related) somehow, but I can't really see how - one encodes at which primes a specific order fails to be a Dedekind domain, and the other encodes the ramification data of an extension. Indeed, it seems impossible; the conductor from CFT is relative while the conductor of an order doesn't mention a field extension, and the conductor from CFT has "data at infinity" (it's a modulus) while the conductor of an order is genuinely just an ideal.

The only inkling of a connection I have is that the conductor of an order is the "difference of the differents" between the order and the full number ring (e.g. see Neukirch chapter III theorem 2.5), and differents also encode ramification data. But obviously differents (primarily) encode the degree of ramification, whereas the conductor modulus encodes the wildness of the ramification, so they're still qualitatively different - and I don't know of any explicit link between the different and the conductor modulus either.

Answer 1

Let $D$ be a squarefree integer, and let $\mathcal{O}$ be the ring of integers in the quadratic field $\mathbb{Q}(\sqrt{D})$. For positive integer $f$ define the order of conductor $f$, $\mathcal{O}_f$ by $$ \mathcal{O}_f = \mathbb{Z}[f\omega] = \{ a + bf\omega \mid a,b \in \mathbb{Z} \}, $$

where

$$\omega = \begin{cases} \sqrt{D} & \text{if } D \equiv 2 \text{ or } 3 \pmod 4 \\ \frac{1+\sqrt{D}}{2} & \text{if } D \equiv 1 \pmod 4. \end{cases}$$

In this case $\mathcal{O}_f$ is a subring of $\mathcal{O}$.