Modular parametrization of elliptic curves over \(\QQ\)¶

By the work of Taylor–Wiles et al. it is known that there is a surjective morphism

\[\phi_E: X_0(N) \rightarrow E.\]

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

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))

Python

>>> from sage.all import *
>>> phi = EllipticCurve('11a1').modular_parametrization()
>>> 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
>>> phi(RealNumber('0.5')+CDF(I))
(285684.320516... + 7.0...e-11*I : 1.526964169...e8 + 5.6...e-8*I : 1.00000000000000)
>>> phi.power_series(prec = Integer(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))

Sage Live

phi = EllipticCurve('11a1').modular_parametrization()
phi
phi(0.5+CDF(I))
phi.power_series(prec = 7)

AUTHORS:

Chris Wuthrich (02/10): moved from ell_rational_field.py.

class sage.schemes.elliptic_curves.modular_parametrization.ModularParameterization(E)[source]¶

Bases: object

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

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

Python

>>> from sage.all import *
>>> phi = EllipticCurve('11a1').modular_parametrization()
>>> 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 Live

phi = EllipticCurve('11a1').modular_parametrization()
phi

curve()[source]¶

Return the curve associated to this modular parametrization.

EXAMPLES:

Sage

sage: E = EllipticCurve('15a')
sage: phi = E.modular_parametrization()
sage: phi.curve() is E
True

Python

>>> from sage.all import *
>>> E = EllipticCurve('15a')
>>> phi = E.modular_parametrization()
>>> phi.curve() is E
True

Sage Live

E = EllipticCurve('15a')
phi = E.modular_parametrization()
phi.curve() is E

map_to_complex_numbers(z, prec=None)[source]¶

Evaluate self at 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 prec bits of precision. If prec is 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

sage: # needs sage.symbolic
sage: E = EllipticCurve('37a'); phi = E.modular_parametrization()
sage: x = polygen(ZZ, 'x')
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)

Python

>>> from sage.all import *
>>> # needs sage.symbolic
>>> E = EllipticCurve('37a'); phi = E.modular_parametrization()
>>> x = polygen(ZZ, 'x')
>>> tau = (sqrt(Integer(7))*I - Integer(17))/Integer(74)
>>> z = phi.map_to_complex_numbers(tau); z
0.929592715285395 - 1.22569469099340*I
>>> E.elliptic_exponential(z)
(...e-16 - ...e-16*I : ...e-16 + ...e-16*I : 1.00000000000000)
>>> phi(tau)
(...e-16 - ...e-16*I : ...e-16 + ...e-16*I : 1.00000000000000)

Sage Live

# needs sage.symbolic
E = EllipticCurve('37a'); phi = E.modular_parametrization()
x = polygen(ZZ, 'x')
tau = (sqrt(7)*I - 17)/74
z = phi.map_to_complex_numbers(tau); z
E.elliptic_exponential(z)
phi(tau)

power_series(prec=20)[source]¶

Return the power series of this modular parametrization.

The curve must be 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

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)

Python

>>> from sage.all import *
>>> E = EllipticCurve('389a1')
>>> phi = E.modular_parametrization()
>>> X, Y = phi.power_series(prec=Integer(10))
>>> 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)
>>> 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)
>>> X,Y = phi.power_series()
>>> 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)
>>> 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)

Sage Live

E = EllipticCurve('389a1')
phi = E.modular_parametrization()
X, Y = phi.power_series(prec=10)
X
Y
X,Y = phi.power_series()
X
Y

The following should give 0, but only approximately:

Sage

sage: q = X.parent().gen()
sage: E.defining_polynomial()(X,Y,1) + O(q^11) == 0
True

Python

>>> from sage.all import *
>>> q = X.parent().gen()
>>> E.defining_polynomial()(X,Y,Integer(1)) + O(q**Integer(11)) == Integer(0)
True

Sage Live

q = X.parent().gen()
E.defining_polynomial()(X,Y,1) + O(q^11) == 0

Note that below we have to change variable from \(x\) to \(q\):

Sage

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

Python

>>> from sage.all import *
>>> a1,_,a3,_,_ = E.a_invariants()
>>> f = E.q_expansion(Integer(17))
>>> q = f.parent().gen()
>>> f/q == (X.derivative()/(Integer(2)*Y+a1*X+a3))
True

Sage Live

a1,_,a3,_,_ = E.a_invariants()
f = E.q_expansion(17)
q = f.parent().gen()
f/q == (X.derivative()/(2*Y+a1*X+a3))