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2014-08-25 17:55:29 +0200 asked a question division_polynomial with integer coefficients?!

Hi, I would like to get a Division-polynomial for an elliptic curve. The curve is


I used the commands

E3 = E.change_ring(QQbar)
p = E3.division_polynomial(3, two_torsion_multiplicity=0)

and I obtained

3*x^4 - 105/2*x^2 - 147*x - 1225/16

Is there a way to get a division-polynomial which has integer coefficients and which is normalized?

2014-08-25 17:47:15 +0200 commented question elliptic curve complex numbers

Hi Frederic, I'm really happy with your answer. It helped me a lot! Thank you. I just klicked the green button on the left; is that what you mean by "accept" ?

2014-08-25 17:44:24 +0200 received badge  Scholar (source)
2014-08-22 09:46:35 +0200 asked a question elliptic curve complex numbers

Hi, I want to look at the curve E=EllipticCurve(CC,[-35/4,-49/4]) over the complex numbers. I want to find the 3-Torsion Points on the curve, so I tried to use the function E.division_polynomial(3, two_torsion_multiplicity=0) which gave me the 3-Division-Polynomial g=3x^4 - 105/2x^2 - 147x - 1225/16 which is an univariate Polynomial. The zeros of this Polynomial should be the x-coordinates of the 3-Torsion-Points. One of the zeros is a=5.26556730825188 Then I tried to compute the y-coordinates via the curve-equation y^2 = x^3 + (-8.75000000000000)x + (-12.2500000000000) The point I got was P=(5.26556730825188 , 9.36325015678742) which is clearly lying on the curve, because it fulfills the equation of the curve E, what I have tested.

So I wanted to use the function P = E(5.26556730825188 , 9.36325015678742)

Here I got an error, telling me "TypeError: Coordinates [5.26556730825188, 9.36325015678742, 1.00000000000000] do not define a point on Elliptic Curve defined by y^2 = x^3 + (-8.75000000000000)*x + (-12.2500000000000) over Complex Field with 53 bits of precision" Why does that happen?

Next problem is the following: If I use the function Q = E(0); Q.division_points(3)

this should give me the 3-torsion-points, but the x-coordinates of the points I get by this metod are different from the method with the 3-divison-polynomial! actually the function does not find any 3-torsion points! How can that happen? Sorry, I'm a sage-beginner from germany and my english is terrible! But this is really really important for me, so I would be very very thankful for any help!!!

greetings pittersen!!