In this short note we prove two theorems; the first one is a sharpening of a result of Lange and Sernesi (Int J Math 13(3):227–244, 2002): the discriminant curve W of a general Abelian surface A endowed with an irreducible polarization D of type \((1,3)\) is an irreducible curve of degree 18 whose singularities are exactly 36 nodes and 72 cusps. Moreover, we analyze the degeneration of the discriminant curve W and its singularities as A tends to the product of two equal elliptic curves. The second theorem, using the first one in order to prove a transversality assertion, shows that the general element of a family of surfaces constructed by Alessandro and Catanese (On the components of the Main Stream of the moduli space of surfaces of general type with \(p_g=q=2\) , preprint Dec 2022) is a smooth surface, thereby proving the existence of a new family of minimal surfaces of general type with \(p_g=q=2, K^2=6\) and Albanese map of degree 3.