  We determine the density of curves having squarefree discriminant in some
families of curves that arise from Vinberg representations, showing that the
global density is the product of the local densities. We do so using the
framework of Thorne and Laga's PhD theses and Bhargava's orbit-counting
techniques. This paper generalises a previous result by Bhargava, Shankar and
Wang. As an application, we give an asymptotic for the number of reducible
orbits of our representations.
