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# 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 \(E\).

EXMAPLES:

    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.

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

  `curve`()

    Returns the curve associated to this modular parametrization.

    EXAMPLES:

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

  `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 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: 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)

  `power_series`(prec=20)

    Computes and returns 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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