The article "Nonsingular Plane Cubic Curves over Finite Fields" by Schoof has the following step:
It seems to imply that the discriminant of a quadratic order $\mathbb{Z}\left[ \alpha \right]$ equals the discriminant of the minimal polynomial of $\alpha$, that is, $x^2 - (\alpha + \hat{\alpha})x + \alpha \hat{\alpha} = x^2 - T(\alpha)x + N(\alpha)$.
The way the discriminant of an order was previously defined was as the square of the conductor times the discriminant of the quadratic field. How can i show those definitions are equivalent?