Coverage for local/lib/python2.7/site-packages/sage/modular/modform/l_series_gross_zagier.py : 97%

from __future__ import absolute_import 

from sage.rings.integer import Integer 

from sage.structure.sage_object import SageObject 

from sage.lfunctions.dokchitser import Dokchitser 

from .l_series_gross_zagier_coeffs import gross_zagier_L_series 

from sage.modular.dirichlet import kronecker_character 

class GrossZagierLseries(SageObject): 

def __init__(self, E, A, prec=53): 

r""" 

Class for the Gross-Zagier L-series. 

This is attached to a pair `(E,A)` where `E` is an elliptic curve over 

`\QQ` and `A` is an ideal class in an imaginary quadratic number field. 

For the exact definition, in the more general setting of modular forms 

instead of elliptic curves, see section IV of [GrossZagier]_. 

INPUT: 

- ``E`` -- an elliptic curve over `\QQ` 

- ``A`` -- an ideal class in an imaginary quadratic number field 

- ``prec`` -- an integer (default 53) giving the required precision 

EXAMPLES:: 

sage: e = EllipticCurve('37a') 

sage: K.<a> = QuadraticField(-40) 

sage: A = K.class_group().gen(0) 

sage: from sage.modular.modform.l_series_gross_zagier import GrossZagierLseries 

sage: G = GrossZagierLseries(e, A) 

TESTS:: 

sage: K.<b> = QuadraticField(131) 

sage: A = K.class_group().one() 

sage: G = GrossZagierLseries(e, A) 

Traceback (most recent call last): 

... 

ValueError: A is not an ideal class in an imaginary quadratic field 

""" 

self._E = E 

self._N = N = E.conductor() 

self._A = A 

ideal = A.ideal() 

K = A.gens()[0].parent() 

D = K.disc() 

if not(K.degree() == 2 and D < 0): 

raise ValueError("A is not an ideal class in an" 

" imaginary quadratic field") 

Q = ideal.quadratic_form().reduced_form() 

epsilon = - kronecker_character(D)(N) 

self._dokchister = Dokchitser(N ** 2 * D ** 2, 

[0, 0, 1, 1], 

weight=2, eps=epsilon, prec=prec) 

self._nterms = nterms = Integer(self._dokchister.num_coeffs()) 

if nterms > 1e6: 

# just takes way to long 

raise ValueError("Too many terms: {}".format(nterms)) 

zeta_ord = ideal.number_field().zeta_order() 

an_list = gross_zagier_L_series(E.anlist(nterms + 1), Q, N, zeta_ord) 

self._dokchister.gp().set('a', an_list[1:]) 

self._dokchister.init_coeffs('a[k]', 1) 

def __call__(self, s, der=0): 

r""" 

Return the value at `s`. 

INPUT: 

- `s` -- complex number 

- ``der`` -- ? (default 0) 

EXAMPLES:: 

sage: e = EllipticCurve('37a') 

sage: K.<a> = QuadraticField(-40) 

sage: A = K.class_group().gen(0) 

sage: from sage.modular.modform.l_series_gross_zagier import GrossZagierLseries 

sage: G = GrossZagierLseries(e, A) 

sage: G(3) 

-0.272946890617590 

""" 

return self._dokchister(s, der) 

def taylor_series(self, s=1, series_prec=6, var='z'): 

r""" 

Return the Taylor series at `s`. 

INPUT: 

- `s` -- complex number (default 1) 

- ``series_prec`` -- number of terms (default 6) in the Taylor series 

- ``var`` -- variable (default 'z') 

EXAMPLES:: 

sage: e = EllipticCurve('37a') 

sage: K.<a> = QuadraticField(-40) 

sage: A = K.class_group().gen(0) 

sage: from sage.modular.modform.l_series_gross_zagier import GrossZagierLseries 

sage: G = GrossZagierLseries(e, A) 

sage: G.taylor_series(2,3) 

-0.613002046122894 + 0.490374999263514*z - 0.122903033710382*z^2 + O(z^3) 

""" 

return self._dokchister.taylor_series(s, series_prec, var) 

def _repr_(self): 

""" 

Return the string representation. 

EXAMPLES:: 

sage: e = EllipticCurve('37a') 

sage: K.<a> = QuadraticField(-40) 

sage: A = K.class_group().gen(0) 

sage: from sage.modular.modform.l_series_gross_zagier import GrossZagierLseries 

sage: GrossZagierLseries(e, A) 

Gross Zagier L-series attached to Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field with ideal class Fractional ideal class (2, 1/2*a) 

""" 

msg = "Gross Zagier L-series attached to {} with ideal class {}" 

return msg.format(self._E, self._A)