Extension of a field extension

[Irene]

Hello,

I have the following defined:

p=3700001

Fp=GF(p)

E=EllipticCurve([Fp(3),Fp(5)])

j_inv=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

Fps=GF(repsq(p,s),'a')

a=Fps.gen()

Fpr=GF(repsq(p,r),'b')

b=Fpr.gen()

FFps.<X>=PolynomialRing(Fps)

FFpr.<x>=PolynomialRing(Fpr)

EP=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

A1.<theta>=Fpr.extension(EP)

and now I want to "add" to A1 the square root of theta^3+3*theta+5.

The problem is that when I consider the following:

gamma2=theta^3+3*theta+5

AA1.<xbar>=PolynomialRing(A1)

AA.<gamma>=A1.extension(xbar^2-gamma2)

(xbar^2-gamma2).roots(AA,multiplicities=False)

it gives me a NotImplementedError. Any idea? Thank you in advance.

Irene

[Irene]

I forgot to write what is repsq():

#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

[John Cremona]

On 21 April 2014 13:58, Irene <irene....@gmail.com> wrote:
> I forgot to write what is repsq():

You could use the builtin function power_mod:

sage: power_mod?
Type: function
String Form:<function power_mod at 0x1f78668>
File: /usr/local/sage/sage-6.1.1/local/lib/python2.7/site-packages/sage/rings/arith.py
Definition: power_mod(a, n, m)
Docstring:
The n-th power of a modulo the integer m.

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[Irene]

Ok! Thank you! Do you have some idea about the first question?

I want to "add" to A1 the square root of theta^3+3*theta+5.

The problem is that when I consider the following:

gamma2=theta^3+3*theta+5

AA1.<xbar>=PolynomialRing(A1)

AA.<gamma>=A1.extension(xbar^2-gamma2)

(xbar^2-gamma2).roots(AA,multiplicities=False)

it gives me a NotImplementedError. Any idea? Thank you in advance.

Irene