Let $K$ be a quadratic imaginary field where the rational prime $N$ splits: $\mathfrak{n}\cdot\bar{\mathfrak{n}}=(N)$ and denote with $H_K$ the Hilbert class field of $K$.
At the beginnig of the proof of proposition 14.1 of his article "Heegner points on $X_0(N)$", Gross writes that if $\mathfrak{a}$ is a fractional ideal of $K$, then $\Delta(\mathfrak{a}) / \Delta(\mathfrak{n}\mathfrak{a})$ is an integer of $H_K$ which generates the ideal $\mathfrak{n}^{12}$ of $K$.
Why this is true? I tried viewing the discriminant cusp form $\Delta$ as function on lattices satisfying $\Delta(\alpha\cdot\mathfrak{a})=\alpha^{-12}\Delta(\mathfrak{a})$, for every $\alpha\in \mathbb{C}^{*}$ and using the fact that every ideal of $\mathcal{O}_K$ becomes principal in $H_K$. But, if we write $\mathfrak{n}\mathcal{O}_{H_K}=(\beta)$, in general I don't think it's true that $\mathfrak{n}\mathfrak{a}=\beta\cdot\mathfrak{a}$.