ASKSAGE: Sage Q&A Forum - Latest question feedhttp://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Thu, 16 Nov 2017 04:43:57 -0600How to defining a twist on an elliptic curvehttp://ask.sagemath.org/question/39597/how-to-defining-a-twist-on-an-elliptic-curve/Hey,
I would like to do map points of the ellitptic curve $E(\mathbb F_{p^{k}})$ to its twist. I am able to define the twist on a mathematical way, but it returns always errors. I will give you a M(n)WE:
Aim: Get a generator of a r-torsion subgroup of E, lift that to E16 and twist it down to E4, to get a r-torsion subgroup-generator of E4.
#MWE
## Mathematical definition:
$E(\mathbb F_{p^{16}}): y^2=x^3+x$ and its quartic twist $E'(\mathbb F_{p^4}): y^2=x^3+2^{-1/4}x$. The point mapping is defined as $\psi: E\to E', (x,y,z)\mapsto (2^{-1/2}x, 2^{-3/4}y, z)$.
## Define the fields and curves
p= 13
r=5
R=GF(p)
_.<x> = PolynomialRing(R)
R4.<x> = R.extension(x^4 - 2, 'x')
_.<y> = PolynomialRing(R)
R16.<y> = R.extension(y^16 -2, 'y')
_.<z> = PolynomialRing(R4)
R16_over_R4.<z> = R4.extension(z^4-x, 'z')
E = EllipticCurve(R, [1,0]) # y^2 = x^3+x
E4 = EllipticCurve(R4, [x,0])
E16 = EllipticCurve(R16, [1,0])
## Defining the point and pre-computations
k= ZZ(E.order()/r) # since E.order()*P = (0:1:0), we can trick with that
b=R16(2^(-1)); b= sqrt(sqrt(2)) #the twisting parameter, that is a square in R4 and R16
P= E.gens()[0]
Q=k*P # check if Q != E((0,1,0)), if yes its a r-torsion point.
Q16 = E16(Q) #raise Q
## The not working twist
#twist
E4( (Q16[0]*b^2, Q16[1]*b^3))
Sage is able to compute the quartic twist of its own, but I do not recieve the right twist, that I had computed by hand ( $E'$ ). Using
E4.quartic_twist(v^-1)ShalecThu, 16 Nov 2017 04:43:57 -0600http://ask.sagemath.org/question/39597/