Easy way to calculate discriminant

Question

I am trying to calculate the discriminant of of ring of integers with integral basis $$1,\sqrt{p}, \frac{1+\sqrt{pq}}{2}, \frac{\sqrt{p}+\sqrt{q}}{2}$$ The standard definition is to look at the matrix with entries $\sigma_i(\alpha_j)$ and then take the square of the determinant. However that would seem to involve taking the determinant of a $4$x$4$ matrix which is very ugly. Is there a quicker way of doing this?