Classes of rings C[x,y]/(x²+cy²+ey+f)

Question

I have a question. I would like to describe the classes of rings that appear in $\mathbb{C}[x,y]/I$ up to isomorphism, where $I=(Q)$, $Q=x²+cy²+ey+f$, $c,e,f\in\mathbb{C}$. $Q$ comes from $Q'=ax²+bxy+cy²+dx+ey+f\in\mathbb{C}[x,y], \ (a,b,c)\neq (0,0,0)$ after a suitable change of coordinates. Until now I thought that the classes are going to be the well-known conic sections but when I was playing around yesterday (after a tip here) I came to the conclusion that the hyperbola and the circle/ellipse are actually isomorphic so now I have to reconsider the the approach. Any help or tips?