I got it in another version! Thank you very much!
On Thursday, April 17, 2014 6:18:56 PM UTC+2, John Cremona wrote: > > Your code works ok with Sage 6.1.1 (apart from a small slip in the > code you posted: the line > FFps=PolynomialRing(Fps) > should say > FFps<X>=PolynomialRing(Fps) > > and the result is > > x^6 + (973912*b + 2535329)*x^5 + (416282*b + 3608920)*x^4 + (686636*b > + 908282)*x^3 + (2100014*b + 2063451)*x^2 + (2563113*b + 751714)*x + > 2687623*b + 1658379 > > > On 17 April 2014 09:05, Irene <irene....@gmail.com <javascript:>> wrote: > > p=3700001 > > Fp=GF(p) > > E=EllipticCurve([Fp(3),Fp(5)]) > > j=E.j_invariant() > > l=13#Atkin prime > > n=((l-1)/2).round() > > r=2# Phi_13 factorize in factors of degree 2 > > s=12#Psi_13 factorize in factors of degree 12 > > > > #repsq(a,n) computes a^n > > def repsq(a,n): > > B = Integer(n).binary() > > C=list(B) > > k=len(B)-1 > > bk=a > > i=1 > > while i <= k: > > if C[i]=="1": > > bk=(bk^2)*a > > else: > > bk=bk^2 > > i=i+1 > > return bk > > > > d=E.division_polynomial(13) > > Fps=GF(repsq(p,s),'a') > > a=Fps.gen() > > Fpr=GF(repsq(p,r),'b') > > b=Fpr.gen() > > FFps=PolynomialRing(Fps) > > Fl=GF(l) > > c=GF(2) > > rts=d.roots(Fps,multiplicities=False) > > Px=rts[0] > > Py2=Px^3+3*Px+5 > > c=Fl.multiplicative_generator() > > > > def produx(n,Qx): > > if is_odd(n): > > > > > pro=Qx-(E.division_polynomial(n-1,(Qx,1),two_torsion_multiplicity=1)*E.division_polynomial(n+1,(Qx,1),two_torsion_multiplicity=1)* > > > > > (Qx^3+3*Qx+5))/((E.division_polynomial(n,(Qx,1),two_torsion_multiplicity=1)^2)) > > > > else: > > > > > pro=Qx-(E.division_polynomial(n-1,(Qx,1),two_torsion_multiplicity=1)*E.division_polynomial(n+1,(Qx,1),two_torsion_multiplicity=1))/((Qx^3+3*Qx+5)*E.division_polynomial(n,(Qx,1),two_torsion_multiplicity=1)^2) > > > > return pro > > > > > Ep=(X-produx(2,Px))*(X-produx(4,Px))*(X-produx(8,Px))*(X-produx(3,Px))*(X-produx(6,Px))*(X-produx(12,Px)) > > > > > > bb=b.minpoly().roots(Fps)[0][0] > > i=Fpr.hom([bb],Fps) > > j=i.section() > > PolynomialRing(Fpr,'x')([j(c) for c in Ep.coeffs()]) > > > > That was the code, and the version is 5.11. Maybe the problem is because > the > > version is too old, but I was using another version and I got many > problems > > and right now I don't have another option. > > > > On Thursday, April 17, 2014 5:47:32 PM UTC+2, John Cremona wrote: > >> > >> On 17 April 2014 08:43, Irene <irene....@gmail.com> wrote: > >> > I think that this is exactly what I need. Nevertheless I cannot use > >> > neither > >> > i.section() nor i.inverse_image(). The second one because of the same > >> > reason > >> > as you, and the first one when I try it is says "TypeError: > 'NoneType' > >> > object is not callable". > >> > >> You'll have to post your actual code (and say which Sage version) for > >> us to help debug that! > >> > >> John > >> > >> > > >> > > >> > On Thursday, April 17, 2014 12:07:18 PM UTC+2, John Cremona wrote: > >> >> > >> >> OK, that makes sense now. It boils down to this: given an element > of > >> >> F12=GF(p^12) which happens to lie in F2 = GF(p^2), how to express it > >> >> in terms of a generator of F2. This is not quite as easy as it > should > >> >> be but this works (assuming that you have defined F12 with generator > a > >> >> and F2 with generator b): > >> >> > >> >> sage: bb = b.minpoly().roots(F12)[0][0] > >> >> sage: i = F2.hom([bb],F12) > >> >> sage: j = i.section() > >> >> > >> >> Here we have defined an embedding i of F2 into F12 by find a place > to > >> >> map b (called bb) and set j to be an inverse to i. (I think we > should > >> >> be use i.inverse_image() but that gave me a NotImplementedError, > which > >> >> is a pity since I have used sort of construction easily in > extensions > >> >> of number fields). > >> >> > >> >> Now if f is your polynomial in F12[x] whose coefficients lie in F2 > you > >> >> can > >> >> say > >> >> > >> >> sage: PolynomialRing(F2,'X')([j(c) for c in f.coeffs()]) > >> >> > >> >> to get what you want, I hope! > >> >> > >> >> John > >> >> > >> >> On 17 April 2014 02:52, Irene <irene....@gmail.com> wrote: > >> >> > Sorry, I didn't write it correctly. I meant GF(p^12,'a') instead > of > >> >> > GF(p^13,'a'). As 2 divides 12, GF(p^12,'a') is an extension of > >> >> > GF(p^2,'b'). > >> >> > My question is the same now with the correct data. > >> >> > > >> >> > On Thursday, April 17, 2014 11:04:40 AM UTC+2, John Cremona wrote: > >> >> >> > >> >> >> On 17 April 2014 01:55, Irene <irene....@gmail.com> wrote: > >> >> >> > Hello! > >> >> >> > > >> >> >> > I want to define a polynomial that I know lies in > GF(p^2,'b')[x], > >> >> >> > p=3700001. > >> >> >> > The problem is that I have to define it as a product > >> >> >> > E=(X-a_1)*(X-a_2)*(X-a_3)*(X-a_4)*(X-a_5)*(X-a_6), where every > a_j > >> >> >> > is > >> >> >> > in > >> >> >> > GF(p^13,'a')[X]. > >> >> >> > I tried to do GF(p^2,'b')[x](E), but then Sage just changes the > >> >> >> > generator > >> >> >> > 'a' and writes the same expression with the generator 'b'. > >> >> >> > Any idea about how to do this? > >> >> >> > Thank you!! > >> >> >> > >> >> >> Did you write that correctly? GF(p^13) is not an extension of > >> >> >> GF(p^2). If a1 is in GF(p^13) then a1.minpoly() will give its > min > >> >> >> poly, in GF(p)[x]. > >> >> >> > >> >> >> John Cremona > >> >> >> > >> >> >> > > >> >> >> > -- > >> >> >> > You received this message because you are subscribed to the > Google > >> >> >> > Groups > >> >> >> > "sage-support" group. > >> >> >> > To unsubscribe from this group and stop receiving emails from > it, > >> >> >> > send > >> >> >> > an > >> >> >> > email to sage-support...@googlegroups.com. > >> >> >> > To post to this group, send email to sage-s...@googlegroups.com. > > >> >> >> > Visit this group at http://groups.google.com/group/sage-support. > > >> >> >> > For more options, visit https://groups.google.com/d/optout. > >> >> > > >> >> > -- > >> >> > You received this message because you are subscribed to the Google > >> >> > Groups > >> >> > "sage-support" group. > >> >> > To unsubscribe from this group and stop receiving emails from it, > >> >> > send > >> >> > an > >> >> > email to sage-support...@googlegroups.com. > >> >> > To post to this group, send email to sage-s...@googlegroups.com. > >> >> > Visit this group at http://groups.google.com/group/sage-support. > >> >> > For more options, visit https://groups.google.com/d/optout. > >> > > >> > -- > >> > You received this message because you are subscribed to the Google > >> > Groups > >> > "sage-support" group. > >> > To unsubscribe from this group and stop receiving emails from it, > send > >> > an > >> > email to sage-support...@googlegroups.com. > >> > To post to this group, send email to sage-s...@googlegroups.com. > >> > Visit this group at http://groups.google.com/group/sage-support. > >> > For more options, visit https://groups.google.com/d/optout. > > > > -- > > You received this message because you are subscribed to the Google > Groups > > "sage-support" group. > > To unsubscribe from this group and stop receiving emails from it, send > an > > email to sage-support...@googlegroups.com <javascript:>. > > To post to this group, send email to > > sage-s...@googlegroups.com<javascript:>. > > > Visit this group at http://groups.google.com/group/sage-support. > > For more options, visit https://groups.google.com/d/optout. > -- You received this message because you are subscribed to the Google Groups "sage-support" group. 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