ASKSAGE: Sage Q&A Forum - Individual question feedhttp://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Mon, 25 Aug 2014 11:52:13 -0500elliptic curve complex numbershttp://ask.sagemath.org/question/23895/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!!
Fri, 22 Aug 2014 02:46:35 -0500http://ask.sagemath.org/question/23895/elliptic-curve-complex-numbers/Comment by FrédéricC for <p>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<em>x^4 - 105/2</em>x^2 - 147<em>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)</em>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.</p>
<p>So I wanted to use the function
P = E(5.26556730825188 , 9.36325015678742)</p>
<p>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?</p>
<p>Next problem is the following: If I use the function
Q = E(0);
Q.division_points(3)</p>
<p>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!!!</p>
<p>greetings pittersen!! </p>
http://ask.sagemath.org/question/23895/elliptic-curve-complex-numbers/?comment=23929#post-id-23929yes. ThanksMon, 25 Aug 2014 11:52:13 -0500http://ask.sagemath.org/question/23895/elliptic-curve-complex-numbers/?comment=23929#post-id-23929Comment by FrédéricC for <p>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<em>x^4 - 105/2</em>x^2 - 147<em>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)</em>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.</p>
<p>So I wanted to use the function
P = E(5.26556730825188 , 9.36325015678742)</p>
<p>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?</p>
<p>Next problem is the following: If I use the function
Q = E(0);
Q.division_points(3)</p>
<p>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!!!</p>
<p>greetings pittersen!! </p>
http://ask.sagemath.org/question/23895/elliptic-curve-complex-numbers/?comment=23907#post-id-23907could you please accept my answer if you think it is good ?Sat, 23 Aug 2014 01:24:47 -0500http://ask.sagemath.org/question/23895/elliptic-curve-complex-numbers/?comment=23907#post-id-23907Comment by pittersen for <p>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<em>x^4 - 105/2</em>x^2 - 147<em>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)</em>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.</p>
<p>So I wanted to use the function
P = E(5.26556730825188 , 9.36325015678742)</p>
<p>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?</p>
<p>Next problem is the following: If I use the function
Q = E(0);
Q.division_points(3)</p>
<p>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!!!</p>
<p>greetings pittersen!! </p>
http://ask.sagemath.org/question/23895/elliptic-curve-complex-numbers/?comment=23927#post-id-23927Hi 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" ?Mon, 25 Aug 2014 10:47:15 -0500http://ask.sagemath.org/question/23895/elliptic-curve-complex-numbers/?comment=23927#post-id-23927Answer by FrédéricC for <p>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<em>x^4 - 105/2</em>x^2 - 147<em>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)</em>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.</p>
<p>So I wanted to use the function
P = E(5.26556730825188 , 9.36325015678742)</p>
<p>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?</p>
<p>Next problem is the following: If I use the function
Q = E(0);
Q.division_points(3)</p>
<p>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!!!</p>
<p>greetings pittersen!! </p>
http://ask.sagemath.org/question/23895/elliptic-curve-complex-numbers/?answer=23896#post-id-23896Maybe something like that:
sage: E = EllipticCurve(QQ,[-35/4,-49/4])
sage: E2 = E.change_ring(CC)
sage: p = E.torsion_polynomial(3)
sage: p.complex_roots()
[-0.682991613296036,
5.26556730825188,
-2.29128784747792 - 1.35880032042306*I,
-2.29128784747792 + 1.35880032042306*I]
sage: x = p.complex_roots()[1]
sage: E2.lift_x(x)
(5.26556730825188 : 9.36325015678740 : 1.00000000000000)
Sometimes it is better (but maybe slower) to work over `QQbar` for exact results.
sage: E3 = E.change_ring(QQbar)
sage: p = E3.torsion_polynomial(3)
sage: p.roots()
[(-0.6829916132960358?, 1),
(5.265567308251876?, 1),
(-2.291287847477920? - 1.358800320423061?*I, 1),
(-2.291287847477920? + 1.358800320423061?*I, 1)]
sage: x = p.roots()[1][0]
sage: g = E3.lift_x(x); g
(5.265567308251876? : 9.363250156787399? : 1)
sage: g+g
(5.265567308251876? : -9.363250156787399? : 1)
sage: g+g+g
(0 : 1 : 0)
You could also work over the splitting field of the division polynomial.Fri, 22 Aug 2014 03:21:55 -0500http://ask.sagemath.org/question/23895/elliptic-curve-complex-numbers/?answer=23896#post-id-23896