Discriminant criterion for nonsingularity

The point $(x_0,y_0)$ is singular iff $F=y^2-x^3-ax-b=0$, $\partial F/\partial y = 2y_0 = 0$, and $\partial F/\partial x = -3x_0^2-a=0$ all hold. The second equation forces $y_0=0$, so $x_0$ must be a root of $x^3+ax+b$; the third equation forces $3x_0^2+a=0$, i.e. $x_0$ is also a root of the derivative $3x^2+a$. A polynomial and its derivative share a common root exactly at a repeated root of the polynomial, so $E$ is singular iff $x^3+ax+b$ has a repeated root. The classical discriminant of the depressed cubic $x^3+ax+b$ is $-4a^3-27b^2$, which vanishes exactly when the cubic has a repeated root; multiplying by the normalization constant $-16$ gives $\Delta = -16(4a^3+27b^2)$, so $E$ is nonsingular iff $\Delta \neq 0$.