Binary quartic ($f(X,Y)={\sum }_{j=0}^4{a}_j{X}^{4-j}{Y}^j$)

A binary quartic is a homogeneous degree-four polynomial in two variables. Over the complex numbers it determines four points of the projective line, counted with multiplicity. Its discriminant is nonzero exactly when the four points are distinct; this is also its geometric invariant theory stability condition under $S{L}_2$.