Modular parametrization of elliptic curves over $\mathbf{Q}$

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

$${\varphi }_E:X_0(N)\to E.$$

from the modular curve $X_0(N)$, where $N$ is the conductor of $E$. The map sends the cusp $\mathrm{\infty }$ 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)

Bases: object

This class represents the modular parametrization of an elliptic curve

$${\varphi }_E:X_0(N)\to E.$$

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()

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)

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)

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 ${\mathrm{\Gamma }}_0(N)$ 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))

Make live