nt.number theory – 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

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

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?