I'm trying to work with polynomials on Elliptic curves. Thus polynomials are elements of the ring \begin{equation*} \frac{K[X,Y]}{Y^2-X^3-aX-b} \end{equation*} (Field characteristic is supposed to be different from 2 or 3)
How can I define the degree of a polynomial in this ring? My problem is that if I put x and y to be degree 1 the relation $Y^2=X^3-aX-b$ does not hold anymore in terms of the degree. Does anyone have an idea how to overcome this problem?