I need to give an example of $\mathbb Z_p$(p-adic integers) lattice $N$ in $\mathbb Q_p\oplus\mathbb Q_p$ ($\mathbb Q_p$ - p-adic numbers), such that discriminant: $$\mathfrak{d}_{\mathbb Z_p}(N)\neq \mathfrak{d}_{\mathbb Z_p} (\pi_1(N))\mathfrak{d}_{\mathbb Z_p} (\pi_2(N)),$$ where $\pi_1$ and $\pi_2$ - projection of $N$ on the first and second components of $\mathbb Q_p\oplus\mathbb Q_p$.
My attempt: Discriminant by definition is determinant of the Gram matrix of basis $N$. So, the Gram matrix must be not diagonalizable. Then we need to find such example of p-adic integers free module $N$ in $\mathbb Q_p\oplus\mathbb Q_p$ such that $Tr(e_ie_j)$ is not diagonalizable, where $e_i$ - basis of $N$ in $\mathbb Q_p\oplus\mathbb Q_p$.
Please help.
$$N = (a,b)\Bbb{Z}_p\oplus (c,d)\Bbb{Z}_p, \qquad a,b,c,d\in \Bbb{Q}_p,ad-bc\ne 0$$
$Disc(N)$ is the image of $ \det(\pmatrix{a&b\\c&d})^2= (ad-bc)^2$ in $\Bbb{Q}_p^*/ (\Bbb{Z}_p^\times)^2$, note that it doesn't depend on the chosen basis of $N$ (which can be changed with a multiplication by a $GL_2(\Bbb{Z}_p)$ matrix, whose determinant is in $\Bbb{Z}_p^\times$).
$ad-bc=p^{v_p(ad-bc)} u$ with $u\in \Bbb{Z}_p^\times$ so $$Disc(N)=p^{2 v_p(ad-bc)}$$ Next $$\pi_1(N) = p^{\min (v_p(a),v_p(c))}\Bbb{Z}_p,\qquad Disc(\pi_1(N))= p^{2\min (v_p(a),v_p(c))}$$ and similarly $Disc(\pi_2(N)) = p^{2\min (v_p(b),v_p(d))}$.
Whence you want $$v_p(ad-bc)\ne \min (v_p(a),v_p(c))+\min (v_p(b),v_p(d))$$
For this take $$\pmatrix{a&b\\c&d} = \pmatrix{1&1+p\\1&1}$$