# 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=3*x^4 - 105/2*x^2 - 147*x - 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!!

could you please accept my answer if you think it is good ?

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" ?

yes. Thanks