# generating the ring for schoof's division polynomials

hello,

i want to generate the ring $F_p[x,y]/(y^2-x^3-ax-b)$, which is necessary to compute the division polynomials in schoof's algorithm to compute the amount of elements of a elliptic curve of the form $y^2=x^3+ax+b$ over $F_p$.

R.<x,y>=PolynomialRing(Zmod(p))
F=R.quo(y^2-x^3-ax-b)
x,y=F.gens()


When i computed some division polynomials in the generated F and matched them with division polynomials which sage computed by this:

F=Zmod(p)
E=EllipticCurve(F,[a,b])
E.division_polynomial()


i saw that my computed division polynomials must be wrong. Does anyone have an idea for generating the Ring $F_p[x,y]/(y^2-x^3-ax-b)$ or generating a "pseudo" elliptic curve in sage, because this code

F=Zmod(p)
E=EllipticCurve(F,[a,b])
E.division_polynomial()


doesn't work if $p$ isn't prime of course.

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