Norm forms, slicing, and ideal classes

Let $$K$$ be a number field, which we may suppose satisfies $$n = [K : \mathbb{Q}] \geq 3$$. Let $$\mathcal{O}_K$$ be the ring of integers of $$K$$, and let $$\{\omega_1, \cdots, \omega_{n}\}$$ be a basis of $$\mathcal{O}_K$$. Define the norm form of $$K$$ to be the homogeneous polynomial defined by

$$\displaystyle N_{K/\mathbb{Q}}(\mathbf{x}) = \prod_{\sigma : K \hookrightarrow \mathbb{C}} \left(\sigma(\omega_1) x_1 + \cdots + \sigma(\omega_n) x_n\right),$$

where the product runs over embeddings $$\sigma: K \hookrightarrow \mathbb{C}$$.

We can extend this definition of norm form to any order $$\mathcal{O}$$ contained in $$\mathcal{O}_K$$, by replacing the basis $$\{\omega_1, \cdots, \omega_n\}$$ with a basis for $$\mathcal{O}$$.

Now suppose that $$\mathcal{O}$$ is an order, with basis $$\{\nu_1, \cdots, \nu_n\}$$, say, and let $$N_{\mathcal{O}}(\mathbf{x})$$ be the corresponding norm form. Let us take an element $$A \in \text{GL}_n(\mathbb{Z})$$, and consider $$G : = N_{\mathcal{O}}(A \mathbf{x})$$. Now restrict $$N_{\mathcal{O}}, G$$ to the 2-dimensional space defined by $$x_3 = \cdots = x_n = 0$$, to obtain binary $$n$$-ic forms $$f(x_1, x_2),g(x_1, x_2)$$ say.

What can we say about $$f,g$$ in general? In particular, are their discriminants connected in any way? What about the rings/ideal classes they represent?