Modular parametrization of elliptic curves over \(\QQ\)¶
By the work of Taylor–Wiles et al. it is known that there is a surjective morphism
from the modular curve \(X_0(N)\), where \(N\) is the conductor of \(E\). The map sends the cusp \(\infty\) to the origin of \(E\).
EXAMPLES:
sage: phi = EllipticCurve('11a1').modular_parametrization()
sage: phi
Modular parameterization from the upper half plane to Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field
sage: phi(0.5+CDF(I))
(285684.320516... + 7.0...e-11*I : 1.526964169...e8 + 5.6...e-8*I : 1.00000000000000)
sage: phi.power_series(prec = 7)
(q^-2 + 2*q^-1 + 4 + 5*q + 8*q^2 + q^3 + 7*q^4 + O(q^5), -q^-3 - 3*q^-2 - 7*q^-1 - 13 - 17*q - 26*q^2 - 19*q^3 + O(q^4))
AUTHORS:
- Chris Wuthrich (02/10) - moved from ell_rational_field.py.
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class
sage.schemes.elliptic_curves.modular_parametrization.ModularParameterization(E)¶ This class represents the modular parametrization of an elliptic curve
\[\phi_E: X_0(N) \rightarrow E.\]Evaluation is done by passing through the lattice representation of \(E\).
EXAMPLES:
sage: phi = EllipticCurve('11a1').modular_parametrization() sage: phi Modular parameterization from the upper half plane to Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field
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curve()¶ Return the curve associated to this modular parametrization.
EXAMPLES:
sage: E = EllipticCurve('15a') sage: phi = E.modular_parametrization() sage: phi.curve() is E True
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map_to_complex_numbers(z, prec=None)¶ Evaluate
selfat a point \(z \in X_0(N)\) where \(z\) is given by a representative in the upper half plane, returning a point in the complex numbers.All computations are done with
precbits of precision. Ifprecis not given, use the precision of \(z\). Use self(z) to compute the image of z on the Weierstrass equation of the curve.EXAMPLES:
sage: E = EllipticCurve('37a'); phi = E.modular_parametrization() sage: tau = (sqrt(7)*I - 17)/74 sage: z = phi.map_to_complex_numbers(tau); z 0.929592715285395 - 1.22569469099340*I sage: E.elliptic_exponential(z) (...e-16 - ...e-16*I : ...e-16 + ...e-16*I : 1.00000000000000) sage: phi(tau) (...e-16 - ...e-16*I : ...e-16 + ...e-16*I : 1.00000000000000)
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power_series(prec=20)¶ Return the power series of this modular parametrization.
The curve must be a a minimal model. The prec parameter determines the number of significant terms. This means that X will be given up to O(q^(prec-2)) and Y will be given up to O(q^(prec-3)).
OUTPUT: A list of two Laurent series
[X(x),Y(x)]of degrees -2, -3 respectively, which satisfy the equation of the elliptic curve. There are modular functions on \(\Gamma_0(N)\) where \(N\) is the conductor.The series should satisfy the differential equation
\[\frac{\mathrm{d}X}{2Y + a_1 X + a_3} = \frac{f(q)\, \mathrm{d}q}{q}\]where \(f\) is
self.curve().q_expansion().EXAMPLES:
sage: E = EllipticCurve('389a1') sage: phi = E.modular_parametrization() sage: X,Y = phi.power_series(prec=10) sage: X q^-2 + 2*q^-1 + 4 + 7*q + 13*q^2 + 18*q^3 + 31*q^4 + 49*q^5 + 74*q^6 + 111*q^7 + O(q^8) sage: Y -q^-3 - 3*q^-2 - 8*q^-1 - 17 - 33*q - 61*q^2 - 110*q^3 - 186*q^4 - 320*q^5 - 528*q^6 + O(q^7) sage: X,Y = phi.power_series() sage: X q^-2 + 2*q^-1 + 4 + 7*q + 13*q^2 + 18*q^3 + 31*q^4 + 49*q^5 + 74*q^6 + 111*q^7 + 173*q^8 + 251*q^9 + 379*q^10 + 560*q^11 + 824*q^12 + 1199*q^13 + 1773*q^14 + 2548*q^15 + 3722*q^16 + 5374*q^17 + O(q^18) sage: Y -q^-3 - 3*q^-2 - 8*q^-1 - 17 - 33*q - 61*q^2 - 110*q^3 - 186*q^4 - 320*q^5 - 528*q^6 - 861*q^7 - 1383*q^8 - 2218*q^9 - 3472*q^10 - 5451*q^11 - 8447*q^12 - 13020*q^13 - 19923*q^14 - 30403*q^15 - 46003*q^16 + O(q^17)
The following should give 0, but only approximately:
sage: q = X.parent().gen() sage: E.defining_polynomial()(X,Y,1) + O(q^11) == 0 True
Note that below we have to change variable from \(x\) to \(q\):
sage: a1,_,a3,_,_ = E.a_invariants() sage: f = E.q_expansion(17) sage: q = f.parent().gen() sage: f/q == (X.derivative()/(2*Y+a1*X+a3)) True
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