There's a bit of mathematical confusion in your question, since you cannot quotient a field by an ideal; you mean to quotient certain coordinate rings by an ideal.
So you have: F = ground field (finite in your case) f(x,y) in F[x,y], the defining polynomial of the curve E, possibly of the form y^2-(x^3+a*x+b) R=F[x,y]/(f), the coordinate ring of E (so R is an integral domain whose quotient field is the function field of E) g_n(x,y) in F[x,y] (or its image in R)) is the n'th division poly of E The quotients you want to work in are, I think, R/(g_n) and R/(g_n,g_m). Are you assuming that m and n are coprime to the characteristic of F? I don't think it is relevant for this construction that they do not divide |E|. Without thinking about this for two long I would guess that you get a quotient isomorphic to R/(g_d) where d=gcd(m,n) (and in particular if m,n are coprime you get the trivial ring. In a nutshell it's all rings and not fields. I hope this helps, even though I have said nothing Sage-specific. John On 27/09/2007, Dror Speiser <[EMAIL PROTECTED]> wrote: > > The main thing I'm looking for, and couldn't achieve on my own is the > following: > > I have an elliptic curve E over a finite field of size q. > At first, I want to work with elements of the function field of E over > the finite field extended by an irreducible polynomial (specifically, > an nth division polynomial, with n not dividing |E|). > Then, finally, I want to divide the last function field (its > multiplicative group) by the ideal generated by another irreducible > polynomial (specifically, an mth division polynomial, with (m,n,| > E|)=1). > > In the end, I think I should get something isomorphic to a > multiplicative group of a finite field of size p^( (n^2/2)*(m^2/2) ). > > Any ideas on how to get this? > > > > > -- John Cremona --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to [email protected] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/sage-forum URLs: http://sage.math.washington.edu/sage/ and http://sage.scipy.org/sage/ -~----------~----~----~----~------~----~------~--~---
