def __init__(self,
conductor,
gammaV,
weight,
eps,
poles=None,
residues='automatic',
prec=53,
init=None):
"""
Initialization of Dokchitser calculator EXAMPLES::
sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1')
sage: L.num_coeffs()
4
"""
self.conductor = conductor
self.gammaV = gammaV
self.weight = weight
self.eps = eps
self.poles = poles if poles is not None else []
self.residues = residues
self.prec = prec
self.__CC = ComplexField(self.prec)
self.__RR = self.__CC._real_field()
self.__initialized = False
if init is not None:
self.init_coeffs(init)
def zeta_symmetric(s):
r"""
Completed function `\xi(s)` that satisfies
`\xi(s) = \xi(1-s)` and has zeros at the same points as the
Riemann zeta function.
INPUT:
- ``s`` - real or complex number
If s is a real number the computation is done using the MPFR
library. When the input is not real, the computation is done using
the PARI C library.
More precisely,
.. math::
xi(s) = \gamma(s/2 + 1) * (s-1) * \pi^{-s/2} * \zeta(s).
EXAMPLES::
sage: zeta_symmetric(0.7)
0.497580414651127
sage: zeta_symmetric(1-0.7)
0.497580414651127
sage: RR = RealField(200)
sage: zeta_symmetric(RR(0.7))
0.49758041465112690357779107525638385212657443284080589766062
sage: C.<i> = ComplexField()
sage: zeta_symmetric(0.5 + i*14.0)
0.000201294444235258 + 1.49077798716757e-19*I
sage: zeta_symmetric(0.5 + i*14.1)
0.0000489893483255687 + 4.40457132572236e-20*I
sage: zeta_symmetric(0.5 + i*14.2)
-0.0000868931282620101 + 7.11507675693612e-20*I
REFERENCE:
- I copied the definition of xi from
http://web.viu.ca/pughg/RiemannZeta/RiemannZetaLong.html
"""
if not (is_ComplexNumber(s) or is_RealNumber(s)):
s = ComplexField()(s)
R = s.parent()
if s == 1: # deal with poles, hopefully
return R(0.5)
return (s / 2 + 1).gamma() * (s - 1) * (R.pi() ** (-s / 2)) * s.zeta()
def zeta_symmetric(s):
r"""
Completed function `\xi(s)` that satisfies
`\xi(s) = \xi(1-s)` and has zeros at the same points as the
Riemann zeta function.
INPUT:
- ``s`` - real or complex number
If s is a real number the computation is done using the MPFR
library. When the input is not real, the computation is done using
the PARI C library.
More precisely,
.. MATH::
xi(s) = \gamma(s/2 + 1) * (s-1) * \pi^{-s/2} * \zeta(s).
EXAMPLES::
sage: zeta_symmetric(0.7)
0.497580414651127
sage: zeta_symmetric(1-0.7)
0.497580414651127
sage: RR = RealField(200)
sage: zeta_symmetric(RR(0.7))
0.49758041465112690357779107525638385212657443284080589766062
sage: C.<i> = ComplexField()
sage: zeta_symmetric(0.5 + i*14.0)
0.000201294444235258 + 1.49077798716757e-19*I
sage: zeta_symmetric(0.5 + i*14.1)
0.0000489893483255687 + 4.40457132572236e-20*I
sage: zeta_symmetric(0.5 + i*14.2)
-0.0000868931282620101 + 7.11507675693612e-20*I
REFERENCE:
- I copied the definition of xi from
http://web.viu.ca/pughg/RiemannZeta/RiemannZetaLong.html
"""
if not (is_ComplexNumber(s) or is_RealNumber(s)):
s = ComplexField()(s)
R = s.parent()
if s == 1: # deal with poles, hopefully
return R(0.5)
return (s / 2 + 1).gamma() * (s - 1) * (R.pi()**(-s / 2)) * s.zeta()
def _evalf_(self, x, y, parent=None, algorithm='mpmath'):
"""
EXAMPLES::
sage: gamma_inc_lower(3,2.)
0.646647167633873
sage: gamma_inc_lower(3,2).n(200)
0.646647167633873081060005050275155...
sage: gamma_inc_lower(0,2.)
+infinity
"""
R = parent or s_parent(x)
# C is the complex version of R
# prec is the precision of R
if R is float:
prec = 53
C = complex
else:
try:
prec = R.precision()
except AttributeError:
prec = 53
try:
C = R.complex_field()
except AttributeError:
C = R
if algorithm == 'pari':
try:
v = ComplexField(prec)(x).gamma() - ComplexField(prec)(
x).gamma_inc(y)
except AttributeError:
if not (is_ComplexNumber(x)):
if is_ComplexNumber(y):
C = y.parent()
else:
C = ComplexField()
x = C(x)
v = ComplexField(prec)(x).gamma() - ComplexField(prec)(
x).gamma_inc(y)
else:
import mpmath
v = ComplexField(prec)(mpmath_utils.call(mpmath.gammainc,
x,
0,
y,
parent=R))
if v.is_real():
return R(v)
else:
return C(v)
def _do_sqrt(x, prec=None, extend=True, all=False):
r"""
Used internally to compute the square root of x.
INPUT:
- ``x`` - a number
- ``prec`` - None (default) or a positive integer
(bits of precision) If not None, then compute the square root
numerically to prec bits of precision.
- ``extend`` - bool (default: True); this is a place
holder, and is always ignored since in the symbolic ring everything
has a square root.
- ``extend`` - bool (default: True); whether to extend
the base ring to find roots. The extend parameter is ignored if
prec is a positive integer.
- ``all`` - bool (default: False); whether to return
a list of all the square roots of x.
EXAMPLES::
sage: from sage.functions.other import _do_sqrt
sage: _do_sqrt(3)
sqrt(3)
sage: _do_sqrt(3,prec=10)
1.7
sage: _do_sqrt(3,prec=100)
1.7320508075688772935274463415
sage: _do_sqrt(3,all=True)
[sqrt(3), -sqrt(3)]
Note that the extend parameter is ignored in the symbolic ring::
sage: _do_sqrt(3,extend=False)
sqrt(3)
"""
from sage.rings.all import RealField, ComplexField
if prec:
if x >= 0:
return RealField(prec)(x).sqrt(all=all)
else:
return ComplexField(prec)(x).sqrt(all=all)
if x == -1:
from sage.symbolic.pynac import I
z = I
else:
z = SR(x)**one_half
if all:
if z:
return [z, -z]
else:
return [z]
return z
def map_to_complex_numbers(self, 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: 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)
"""
if prec is None:
try:
prec = z.parent().prec()
except AttributeError:
prec = 53
CC = ComplexField(prec)
if z in QQ:
raise NotImplementedError
z = CC(z)
if z.imag() <= 0:
raise ValueError("Point must be in the upper half plane")
# TODO: for very small imaginary part, maybe try to transform under
# \Gamma_0(N) to a better representative?
q = (2 * CC.gen() * CC.pi() * z).exp()
# nterms'th term is less than 2**-(prec+10) (c.f. eclib code)
nterms = (-(prec + 10) / q.abs().log2()).ceil()
# Use Horner's rule to sum the integral of the form
enumerated_an = list(enumerate(self._E.anlist(nterms)))[1:]
lattice_point = 0
for n, an in reversed(enumerated_an):
lattice_point += an / n
lattice_point *= q
return lattice_point
def _evalf_(self, x, parent_d=None):
"""
EXAMPLES::
sage: arg(0.0)
0.000000000000000
sage: arg(3.0)
0.000000000000000
sage: arg(3.00000000000000000000000000)
0.00000000000000000000000000
sage: arg(3.00000000000000000000000000).prec()
90
sage: arg(ComplexIntervalField(90)(3)).prec()
90
sage: arg(ComplexIntervalField(90)(3)).parent()
Real Interval Field with 90 bits of precision
sage: arg(3.0r)
0.000000000000000
sage: arg(RDF(3))
0.0
sage: arg(RDF(3)).parent()
Real Double Field
sage: arg(-2.5)
3.14159265358979
sage: arg(2.0+3*i)
0.982793723247329
"""
try:
return x.arg()
except AttributeError:
pass
# try to find a parent that support .arg()
if parent_d is None:
parent_d = parent(x)
try:
parent_d = parent_d.complex_field()
except AttributeError:
from sage.rings.complex_field import ComplexField
try:
parent_d = ComplexField(x.prec())
except AttributeError:
parent_d = ComplexField()
return parent_d(x).arg()
def _evalf_(self, x, y, parent=None, algorithm='mpmath'):
"""
EXAMPLES::
sage: gamma_inc_lower(3,2.)
0.646647167633873
sage: gamma_inc_lower(3,2).n(200)
0.646647167633873081060005050275155...
sage: gamma_inc_lower(0,2.)
+infinity
"""
R = parent or s_parent(x)
# C is the complex version of R
# prec is the precision of R
if R is float:
prec = 53
C = complex
else:
try:
prec = R.precision()
except AttributeError:
prec = 53
try:
C = R.complex_field()
except AttributeError:
C = R
if algorithm == 'pari':
try:
v = ComplexField(prec)(x).gamma() - ComplexField(prec)(x).gamma_inc(y)
except AttributeError:
if not (is_ComplexNumber(x)):
if is_ComplexNumber(y):
C = y.parent()
else:
C = ComplexField()
x = C(x)
v = ComplexField(prec)(x).gamma() - ComplexField(prec)(x).gamma_inc(y)
else:
import mpmath
v = ComplexField(prec)(mpmath_utils.call(mpmath.gammainc, x, 0, y, parent=R))
if v.is_real():
return R(v)
else:
return C(v)
def __init__(self, lfun, prec=None):
"""
Initialization of the L-function from a PARI L-function.
INPUT:
- lfun -- a PARI :pari:`lfun` object or an instance of :class:`lfun_generic`
- prec -- integer (default: 53) number of *bits* of precision
EXAMPLES::
sage: from sage.lfunctions.pari import lfun_generic, LFunction
sage: lf = lfun_generic(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], v=pari('k->vector(k,n,1)'))
sage: L = LFunction(lf)
sage: L.num_coeffs()
4
"""
if isinstance(lfun, lfun_generic):
# preparation using motivic data
self._L = lfun.__pari__()
if prec is None:
prec = lfun.prec
elif isinstance(lfun, Gen):
# already some PARI lfun
self._L = lfun
else:
# create a PARI lfunction from other input data
self._L = pari.lfuncreate(lfun)
self._conductor = ZZ(self._L[4]) # needs check
self._weight = ZZ(self._L[3]) # needs check
if prec is None:
self.prec = 53
else:
self.prec = PyNumber_Index(prec)
self._RR = RealField(self.prec)
self._CC = ComplexField(self.prec)
# Complex field used for inputs, which ensures exact 1-to-1
# conversion to/from PARI. Since PARI objects have a precision
# in machine words (not bits), this is typically higher. For
# example, the default of 53 bits of precision would become 64.
self._CCin = ComplexField(pari.bitprecision(self._RR(1)))
def _evalf_(self, x, y, parent=None):
"""
EXAMPLES::
sage: gamma_inc(0,2)
-Ei(-2)
sage: gamma_inc(0,2.)
0.0489005107080611
sage: gamma_inc(3,2).n()
1.35335283236613
"""
try:
return x.gamma_inc(y)
except AttributeError:
if not (is_ComplexNumber(x)):
if is_ComplexNumber(y):
C = y.parent()
else:
C = ComplexField()
x = C(x)
return x.gamma_inc(y)
def __init__(self, conductor, gammaV, weight, eps, \
poles=[], residues='automatic', prec=53,
init=None):
"""
Initialization of Dokchitser calculator EXAMPLES::
sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1')
sage: L.num_coeffs()
4
"""
self.conductor = conductor
self.gammaV = gammaV
self.weight = weight
self.eps = eps
self.poles = poles
self.residues = residues
self.prec = prec
self.__CC = ComplexField(self.prec)
self.__RR = self.__CC._real_field()
self.__initialized = False
if init is not None:
self.init_coeffs(init)
def _evalf_(self, x, y, parent=None, algorithm='pari'):
"""
EXAMPLES::
sage: gamma_inc(0,2)
-Ei(-2)
sage: gamma_inc(0,2.)
0.0489005107080611
sage: gamma_inc(0,2).n(algorithm='pari')
0.0489005107080611
sage: gamma_inc(0,2).n(200)
0.048900510708061119567239835228...
sage: gamma_inc(3,2).n()
1.35335283236613
TESTS:
Check that :trac:`7099` is fixed::
sage: R = RealField(1024)
sage: gamma(R(9), R(10^-3)) # rel tol 1e-308
40319.99999999999999999999999999988898884344822911869926361916294165058203634104838326009191542490601781777105678829520585311300510347676330951251563007679436243294653538925717144381702105700908686088851362675381239820118402497959018315224423868693918493033078310647199219674433536605771315869983788442389633
sage: numerical_approx(gamma(9, 10^(-3)) - gamma(9), digits=40) # abs tol 1e-36
-1.110111598370794007949063502542063148294e-28
Check that :trac:`17328` is fixed::
sage: gamma_inc(float(-1), float(-1))
(-0.8231640121031085+3.141592653589793j)
sage: gamma_inc(RR(-1), RR(-1))
-0.823164012103109 + 3.14159265358979*I
sage: gamma_inc(-1, float(-log(3))) - gamma_inc(-1, float(-log(2))) # abs tol 1e-15
(1.2730972164471142+0j)
Check that :trac:`17130` is fixed::
sage: r = gamma_inc(float(0), float(1)); r
0.21938393439552029
sage: type(r)
<... 'float'>
"""
R = parent or s_parent(x)
# C is the complex version of R
# prec is the precision of R
if R is float:
prec = 53
C = complex
else:
try:
prec = R.precision()
except AttributeError:
prec = 53
try:
C = R.complex_field()
except AttributeError:
C = R
if algorithm == 'pari':
v = ComplexField(prec)(x).gamma_inc(y)
else:
import mpmath
v = ComplexField(prec)(mpmath_utils.call(mpmath.gammainc, x, y, parent=R))
if v.is_real():
return R(v)
else:
return C(v)
def _evalf_(self, x, y, parent=None, algorithm='pari'):
"""
EXAMPLES::
sage: gamma_inc(0,2)
-Ei(-2)
sage: gamma_inc(0,2.)
0.0489005107080611
sage: gamma_inc(0,2).n(algorithm='pari')
0.0489005107080611
sage: gamma_inc(0,2).n(200)
0.048900510708061119567239835228...
sage: gamma_inc(3,2).n()
1.35335283236613
TESTS:
Check that :trac:`7099` is fixed::
sage: R = RealField(1024)
sage: gamma(R(9), R(10^-3)) # rel tol 1e-308
40319.99999999999999999999999999988898884344822911869926361916294165058203634104838326009191542490601781777105678829520585311300510347676330951251563007679436243294653538925717144381702105700908686088851362675381239820118402497959018315224423868693918493033078310647199219674433536605771315869983788442389633
sage: numerical_approx(gamma(9, 10^(-3)) - gamma(9), digits=40) # abs tol 1e-36
-1.110111598370794007949063502542063148294e-28
Check that :trac:`17328` is fixed::
sage: gamma_inc(float(-1), float(-1))
(-0.8231640121031085+3.141592653589793j)
sage: gamma_inc(RR(-1), RR(-1))
-0.823164012103109 + 3.14159265358979*I
sage: gamma_inc(-1, float(-log(3))) - gamma_inc(-1, float(-log(2))) # abs tol 1e-15
(1.2730972164471142+0j)
Check that :trac:`17130` is fixed::
sage: r = gamma_inc(float(0), float(1)); r
0.21938393439552029
sage: type(r)
<... 'float'>
"""
R = parent or s_parent(x)
# C is the complex version of R
# prec is the precision of R
if R is float:
prec = 53
C = complex
else:
try:
prec = R.precision()
except AttributeError:
prec = 53
try:
C = R.complex_field()
except AttributeError:
C = R
if algorithm == 'pari':
v = ComplexField(prec)(x).gamma_inc(y)
else:
import mpmath
v = ComplexField(prec)(mpmath_utils.call(mpmath.gammainc,
x,
y,
parent=R))
if v.is_real():
return R(v)
else:
return C(v)
def _coerce_map_from_(self, P):
r"""
Return whether ``P`` coerces into this symbolic subring.
INPUT:
- ``P`` -- a parent.
OUTPUT:
A boolean or ``None``.
TESTS::
sage: from sage.symbolic.subring import GenericSymbolicSubring
sage: GenericSymbolicSubring(vars=tuple()).has_coerce_map_from(SR) # indirect doctest # not tested see #19231
False
::
sage: from sage.symbolic.subring import SymbolicSubring
sage: C = SymbolicSubring(no_variables=True)
sage: C.has_coerce_map_from(ZZ) # indirect doctest
True
sage: C.has_coerce_map_from(QQ) # indirect doctest
True
sage: C.has_coerce_map_from(RR) # indirect doctest
True
sage: C.has_coerce_map_from(RIF) # indirect doctest
True
sage: C.has_coerce_map_from(CC) # indirect doctest
True
sage: C.has_coerce_map_from(CIF) # indirect doctest
True
sage: C.has_coerce_map_from(AA) # indirect doctest
True
sage: C.has_coerce_map_from(QQbar) # indirect doctest
True
sage: C.has_coerce_map_from(SR) # indirect doctest
False
"""
if P == SR:
# Workaround; can be deleted once #19231 is fixed
return False
from sage.rings.real_mpfr import mpfr_prec_min
from sage.rings.all import (ComplexField, RLF, CLF, AA, QQbar,
InfinityRing)
from sage.rings.real_mpfi import is_RealIntervalField
from sage.rings.complex_interval_field import is_ComplexIntervalField
if isinstance(P, type):
return SR._coerce_map_from_(P)
elif RLF.has_coerce_map_from(P) or \
CLF.has_coerce_map_from(P) or \
AA.has_coerce_map_from(P) or \
QQbar.has_coerce_map_from(P):
return True
elif (P is InfinityRing or is_RealIntervalField(P)
or is_ComplexIntervalField(P)):
return True
elif ComplexField(mpfr_prec_min()).has_coerce_map_from(P):
return P not in (RLF, CLF, AA, QQbar)
def hilbert_class_polynomial(D, algorithm=None):
r"""
Return the Hilbert class polynomial for discriminant `D`.
INPUT:
- ``D`` (int) -- a negative integer congruent to 0 or 1 modulo 4.
- ``algorithm`` (string, default None).
OUTPUT:
(integer polynomial) The Hilbert class polynomial for the
discriminant `D`.
ALGORITHM:
- If ``algorithm`` = "arb" (default): Use Arb's implementation which uses complex interval arithmetic.
- If ``algorithm`` = "sage": Use complex approximations to the roots.
- If ``algorithm`` = "magma": Call the appropriate Magma function (if available).
AUTHORS:
- Sage implementation originally by Eduardo Ocampo Alvarez and
AndreyTimofeev
- Sage implementation corrected by John Cremona (using corrected precision bounds from Andreas Enge)
- Magma implementation by David Kohel
EXAMPLES::
sage: hilbert_class_polynomial(-4)
x - 1728
sage: hilbert_class_polynomial(-7)
x + 3375
sage: hilbert_class_polynomial(-23)
x^3 + 3491750*x^2 - 5151296875*x + 12771880859375
sage: hilbert_class_polynomial(-37*4)
x^2 - 39660183801072000*x - 7898242515936467904000000
sage: hilbert_class_polynomial(-37*4, algorithm="magma") # optional - magma
x^2 - 39660183801072000*x - 7898242515936467904000000
sage: hilbert_class_polynomial(-163)
x + 262537412640768000
sage: hilbert_class_polynomial(-163, algorithm="sage")
x + 262537412640768000
sage: hilbert_class_polynomial(-163, algorithm="magma") # optional - magma
x + 262537412640768000
TESTS::
sage: all([hilbert_class_polynomial(d, algorithm="arb") == \
....: hilbert_class_polynomial(d, algorithm="sage") \
....: for d in range(-1,-100,-1) if d%4 in [0,1]])
True
"""
if algorithm is None:
algorithm = "arb"
D = Integer(D)
if D >= 0:
raise ValueError("D (=%s) must be negative" % D)
if not (D % 4 in [0, 1]):
raise ValueError("D (=%s) must be a discriminant" % D)
if algorithm == "arb":
import sage.libs.arb.arith
return sage.libs.arb.arith.hilbert_class_polynomial(D)
if algorithm == "magma":
magma.eval("R<x> := PolynomialRing(IntegerRing())")
f = str(magma.eval("HilbertClassPolynomial(%s)" % D))
return IntegerRing()['x'](f)
if algorithm != "sage":
raise ValueError("%s is not a valid algorithm" % algorithm)
from sage.quadratic_forms.binary_qf import BinaryQF_reduced_representatives
from sage.rings.all import RR, ComplexField
from sage.functions.all import elliptic_j
# get all primitive reduced quadratic forms, (necessary to exclude
# imprimitive forms when D is not a fundamental discriminant):
rqf = BinaryQF_reduced_representatives(D, primitive_only=True)
# compute needed precision
#
# NB: [https://arxiv.org/abs/0802.0979v1], quoting Enge (2006), is
# incorrect. Enge writes (2009-04-20 email to John Cremona) "The
# source is my paper on class polynomials
# [https://hal.inria.fr/inria-00001040] It was pointed out to me by
# the referee after ANTS that the constant given there was
# wrong. The final version contains a corrected constant on p.7
# which is consistent with your example. It says:
# "The logarithm of the absolute value of the coefficient in front
# of X^j is bounded above by
#
# log (2*k_2) * h + pi * sqrt(|D|) * sum (1/A_i)
#
# independently of j", where k_2 \approx 10.163.
h = len(rqf) # class number
c1 = 3.05682737291380 # log(2*10.63)
c2 = sum([1 / RR(qf[0]) for qf in rqf], RR(0))
prec = c2 * RR(3.142) * RR(D).abs().sqrt() + h * c1 # bound on log
prec = prec * 1.45 # bound on log_2 (1/log(2) = 1.44..)
prec = 10 + prec.ceil() # allow for rounding error
# set appropriate precision for further computing
Dsqrt = D.sqrt(prec=prec)
R = ComplexField(prec)['t']
t = R.gen()
pol = R(1)
for qf in rqf:
a, b, c = list(qf)
tau = (b + Dsqrt) / (a << 1)
pol *= (t - elliptic_j(tau))
coeffs = [cof.real().round() for cof in pol.coefficients(sparse=False)]
return IntegerRing()['x'](coeffs)
def hilbert_class_polynomial(D, algorithm=None):
r"""
Returns the Hilbert class polynomial for discriminant `D`.
INPUT:
- ``D`` (int) -- a negative integer congruent to 0 or 1 modulo 4.
- ``algorithm`` (string, default None).
OUTPUT:
(integer polynomial) The Hilbert class polynomial for the
discriminant `D`.
ALGORITHM:
- If ``algorithm`` = "arb" (default): Use Arb's implementation which uses complex interval arithmetic.
- If ``algorithm`` = "sage": Use complex approximations to the roots.
- If ``algorithm`` = "magma": Call the appropriate Magma function (if available).
AUTHORS:
- Sage implementation originally by Eduardo Ocampo Alvarez and
AndreyTimofeev
- Sage implementation corrected by John Cremona (using corrected precision bounds from Andreas Enge)
- Magma implementation by David Kohel
EXAMPLES::
sage: hilbert_class_polynomial(-4)
x - 1728
sage: hilbert_class_polynomial(-7)
x + 3375
sage: hilbert_class_polynomial(-23)
x^3 + 3491750*x^2 - 5151296875*x + 12771880859375
sage: hilbert_class_polynomial(-37*4)
x^2 - 39660183801072000*x - 7898242515936467904000000
sage: hilbert_class_polynomial(-37*4, algorithm="magma") # optional - magma
x^2 - 39660183801072000*x - 7898242515936467904000000
sage: hilbert_class_polynomial(-163)
x + 262537412640768000
sage: hilbert_class_polynomial(-163, algorithm="sage")
x + 262537412640768000
sage: hilbert_class_polynomial(-163, algorithm="magma") # optional - magma
x + 262537412640768000
TESTS::
sage: all([hilbert_class_polynomial(d, algorithm="arb") == \
....: hilbert_class_polynomial(d, algorithm="sage") \
....: for d in range(-1,-100,-1) if d%4 in [0,1]])
True
"""
if algorithm is None:
algorithm = "arb"
D = Integer(D)
if D >= 0:
raise ValueError("D (=%s) must be negative"%D)
if not (D%4 in [0,1]):
raise ValueError("D (=%s) must be a discriminant"%D)
if algorithm == "arb":
import sage.libs.arb.arith
return sage.libs.arb.arith.hilbert_class_polynomial(D)
if algorithm == "magma":
magma.eval("R<x> := PolynomialRing(IntegerRing())")
f = str(magma.eval("HilbertClassPolynomial(%s)"%D))
return IntegerRing()['x'](f)
if algorithm != "sage":
raise ValueError("%s is not a valid algorithm"%algorithm)
from sage.quadratic_forms.binary_qf import BinaryQF_reduced_representatives
from sage.rings.all import RR, ZZ, ComplexField
from sage.functions.all import elliptic_j
# get all primitive reduced quadratic forms, (necessary to exclude
# imprimitive forms when D is not a fundamental discriminant):
rqf = BinaryQF_reduced_representatives(D, primitive_only=True)
# compute needed precision
#
# NB: [http://arxiv.org/abs/0802.0979v1], quoting Enge (2006), is
# incorrect. Enge writes (2009-04-20 email to John Cremona) "The
# source is my paper on class polynomials
# [http://hal.inria.fr/inria-00001040] It was pointed out to me by
# the referee after ANTS that the constant given there was
# wrong. The final version contains a corrected constant on p.7
# which is consistent with your example. It says:
# "The logarithm of the absolute value of the coefficient in front
# of X^j is bounded above by
#
# log (2*k_2) * h + pi * sqrt(|D|) * sum (1/A_i)
#
# independently of j", where k_2 \approx 10.163.
h = len(rqf) # class number
c1 = 3.05682737291380 # log(2*10.63)
c2 = sum([1/RR(qf[0]) for qf in rqf], RR(0))
prec = c2*RR(3.142)*RR(D).abs().sqrt() + h*c1 # bound on log
prec = prec * 1.45 # bound on log_2 (1/log(2) = 1.44..)
prec = 10 + prec.ceil() # allow for rounding error
# set appropriate precision for further computing
Dsqrt = D.sqrt(prec=prec)
R = ComplexField(prec)['t']
t = R.gen()
pol = R(1)
for qf in rqf:
a, b, c = list(qf)
tau = (b+Dsqrt)/(a<<1)
pol *= (t - elliptic_j(tau))
coeffs = [cof.real().round() for cof in pol.coefficients(sparse=False)]
return IntegerRing()['x'](coeffs)
class Dokchitser(SageObject):
r"""
Dokchitser's `L`-functions Calculator
Create a Dokchitser `L`-series with
Dokchitser(conductor, gammaV, weight, eps, poles, residues, init,
prec)
where
- ``conductor`` - integer, the conductor
- ``gammaV`` - list of Gamma-factor parameters, e.g. [0] for
Riemann zeta, [0,1] for ell.curves, (see examples).
- ``weight`` - positive real number, usually an integer e.g. 1 for
Riemann zeta, 2 for `H^1` of curves/`\QQ`
- ``eps`` - complex number; sign in functional equation
- ``poles`` - (default: []) list of points where `L^*(s)` has
(simple) poles; only poles with `Re(s)>weight/2` should be
included
- ``residues`` - vector of residues of `L^*(s)` in those poles or
set residues='automatic' (default value)
- ``prec`` - integer (default: 53) number of *bits* of precision
RIEMANN ZETA FUNCTION:
We compute with the Riemann Zeta function.
::
sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1')
sage: L
Dokchitser L-series of conductor 1 and weight 1
sage: L(1)
Traceback (most recent call last):
...
ArithmeticError
sage: L(2)
1.64493406684823
sage: L(2, 1.1)
1.64493406684823
sage: L.derivative(2)
-0.937548254315844
sage: h = RR('0.0000000000001')
sage: (zeta(2+h) - zeta(2.))/h
-0.937028232783632
sage: L.taylor_series(2, k=5)
1.64493406684823 - 0.937548254315844*z + 0.994640117149451*z^2 - 1.00002430047384*z^3 + 1.00006193307...*z^4 + O(z^5)
RANK 1 ELLIPTIC CURVE:
We compute with the `L`-series of a rank `1`
curve.
::
sage: E = EllipticCurve('37a')
sage: L = E.lseries().dokchitser(); L
Dokchitser L-function associated to Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
sage: L(1)
0.000000000000000
sage: L.derivative(1)
0.305999773834052
sage: L.derivative(1,2)
0.373095594536324
sage: L.num_coeffs()
48
sage: L.taylor_series(1,4)
0.000000000000000 + 0.305999773834052*z + 0.186547797268162*z^2 - 0.136791463097188*z^3 + O(z^4)
sage: L.check_functional_equation()
6.11218974700000e-18 # 32-bit
6.04442711160669e-18 # 64-bit
RANK 2 ELLIPTIC CURVE:
We compute the leading coefficient and Taylor expansion of the
`L`-series of a rank `2` curve.
::
sage: E = EllipticCurve('389a')
sage: L = E.lseries().dokchitser()
sage: L.num_coeffs ()
156
sage: L.derivative(1,E.rank())
1.51863300057685
sage: L.taylor_series(1,4)
-1.27685190980159e-23 + (7.23588070754027e-24)*z + 0.759316500288427*z^2 - 0.430302337583362*z^3 + O(z^4) # 32-bit
-2.72911738151096e-23 + (1.54658247036311e-23)*z + 0.759316500288427*z^2 - 0.430302337583362*z^3 + O(z^4) # 64-bit
RAMANUJAN DELTA L-FUNCTION:
The coefficients are given by Ramanujan's tau function::
sage: L = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1)
sage: pari_precode = 'tau(n)=(5*sigma(n,3)+7*sigma(n,5))*n/12 - 35*sum(k=1,n-1,(6*k-4*(n-k))*sigma(k,3)*sigma(n-k,5))'
sage: L.init_coeffs('tau(k)', pari_precode=pari_precode)
We redefine the default bound on the coefficients: Deligne's
estimate on tau(n) is better than the default
coefgrow(n)=`(4n)^{11/2}` (by a factor 1024), so
re-defining coefgrow() improves efficiency (slightly faster).
::
sage: L.num_coeffs()
12
sage: L.set_coeff_growth('2*n^(11/2)')
sage: L.num_coeffs()
11
Now we're ready to evaluate, etc.
::
sage: L(1)
0.0374412812685155
sage: L(1, 1.1)
0.0374412812685155
sage: L.taylor_series(1,3)
0.0374412812685155 + 0.0709221123619322*z + 0.0380744761270520*z^2 + O(z^3)
"""
def __init__(self, conductor, gammaV, weight, eps, \
poles=[], residues='automatic', prec=53,
init=None):
"""
Initialization of Dokchitser calculator EXAMPLES::
sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1')
sage: L.num_coeffs()
4
"""
self.conductor = conductor
self.gammaV = gammaV
self.weight = weight
self.eps = eps
self.poles = poles
self.residues = residues
self.prec = prec
self.__CC = ComplexField(self.prec)
self.__RR = self.__CC._real_field()
if not init is None:
self.init_coeffs(init)
self.__init = init
else:
self.__init = False
def __reduce__(self):
D = copy.copy(self.__dict__)
if '_Dokchitser__gp' in D:
del D['_Dokchitser__gp']
return reduce_load_dokchitser, (D, )
def _repr_(self):
z = "Dokchitser L-series of conductor %s and weight %s"%(
self.conductor, self.weight)
return z
def __del__(self):
self.gp().quit()
def gp(self):
"""
Return the gp interpreter that is used to implement this Dokchitser
L-function.
EXAMPLES::
sage: E = EllipticCurve('11a')
sage: L = E.lseries().dokchitser()
sage: L(2)
0.546048036215014
sage: L.gp()
PARI/GP interpreter
"""
try:
return self.__gp
except AttributeError:
logfile = None
# For debugging
import os
from sage.env import DOT_SAGE
logfile = os.path.join(DOT_SAGE, 'dokchitser.log')
g = sage.interfaces.gp.Gp(script_subdirectory='dokchitser', logfile=logfile)
g.read('computel.gp')
self.__gp = g
self._gp_eval('default(realprecision, %s)'%(self.prec//3 + 2))
self._gp_eval('conductor = %s'%self.conductor)
self._gp_eval('gammaV = %s'%self.gammaV)
self._gp_eval('weight = %s'%self.weight)
self._gp_eval('sgn = %s'%self.eps)
self._gp_eval('Lpoles = %s'%self.poles)
self._gp_eval('Lresidues = %s'%self.residues)
g._dokchitser = True
return g
def _gp_eval(self, s):
try:
t = self.gp().eval(s)
except (RuntimeError, TypeError):
raise RuntimeError("Unable to create L-series, due to precision or other limits in PARI.")
if '***' in t:
raise RuntimeError("Unable to create L-series, due to precision or other limits in PARI.")
return t
def __check_init(self):
if not self.__init:
raise ValueError("you must call init_coeffs on the L-function first")
def num_coeffs(self, T=1):
"""
Return number of coefficients `a_n` that are needed in
order to perform most relevant `L`-function computations to
the desired precision.
EXAMPLES::
sage: E = EllipticCurve('11a')
sage: L = E.lseries().dokchitser()
sage: L.num_coeffs()
26
sage: E = EllipticCurve('5077a')
sage: L = E.lseries().dokchitser()
sage: L.num_coeffs()
568
sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1')
sage: L.num_coeffs()
4
"""
return Integer(self.gp().eval('cflength(%s)'%T))
def init_coeffs(self, v, cutoff=1,
w=None,
pari_precode='',
max_imaginary_part=0,
max_asymp_coeffs=40):
"""
Set the coefficients `a_n` of the `L`-series. If
`L(s)` is not equal to its dual, pass the coefficients of
the dual as the second optional argument.
INPUT:
- ``v`` - list of complex numbers or string (pari
function of k)
- ``cutoff`` - real number = 1 (default: 1)
- ``w`` - list of complex numbers or string (pari
function of k)
- ``pari_precode`` - some code to execute in pari
before calling initLdata
- ``max_imaginary_part`` - (default: 0): redefine if
you want to compute L(s) for s having large imaginary part,
- ``max_asymp_coeffs`` - (default: 40): at most this
many terms are generated in asymptotic series for phi(t) and
G(s,t).
EXAMPLES::
sage: L = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1)
sage: pari_precode = 'tau(n)=(5*sigma(n,3)+7*sigma(n,5))*n/12 - 35*sum(k=1,n-1,(6*k-4*(n-k))*sigma(k,3)*sigma(n-k,5))'
sage: L.init_coeffs('tau(k)', pari_precode=pari_precode)
Evaluate the resulting L-function at a point, and compare with the answer that
one gets "by definition" (of L-function attached to a modular form)::
sage: L(14)
0.998583063162746
sage: a = delta_qexp(1000)
sage: sum(a[n]/float(n)^14 for n in range(1,1000))
0.9985830631627459
Illustrate that one can give a list of complex numbers for v
(see :trac:`10937`)::
sage: L2 = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1)
sage: L2.init_coeffs(list(delta_qexp(1000))[1:])
sage: L2(14)
0.998583063162746
TESTS:
Verify that setting the `w` parameter does not raise an error
(see :trac:`10937`). Note that the meaning of `w` does not seem to
be documented anywhere in Dokchitser's package yet, so there is
no claim that the example below is meaningful! ::
sage: L2 = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1)
sage: L2.init_coeffs(list(delta_qexp(1000))[1:], w=[1..1000])
"""
if isinstance(v, tuple) and w is None:
v, cutoff, w, pari_precode, max_imaginary_part, max_asymp_coeffs = v
self.__init = (v, cutoff, w, pari_precode, max_imaginary_part, max_asymp_coeffs)
gp = self.gp()
if pari_precode != '':
self._gp_eval(pari_precode)
RR = self.__CC._real_field()
cutoff = RR(cutoff)
if isinstance(v, str):
if w is None:
self._gp_eval('initLdata("%s", %s)'%(v, cutoff))
return
self._gp_eval('initLdata("%s",%s,"%s")'%(v,cutoff,w))
return
if not isinstance(v, (list, tuple)):
raise TypeError("v (=%s) must be a list, tuple, or string"%v)
CC = self.__CC
v = ','.join([CC(a)._pari_init_() for a in v])
self._gp_eval('Avec = [%s]'%v)
if w is None:
self._gp_eval('initLdata("Avec[k]", %s)'%cutoff)
return
w = ','.join([CC(a)._pari_init_() for a in w])
self._gp_eval('Bvec = [%s]'%w)
self._gp_eval('initLdata("Avec[k]",%s,"Bvec[k]")'%cutoff)
def __to_CC(self, s):
s = s.replace('.E','.0E').replace(' ','')
return self.__CC(sage_eval(s, locals={'I':self.__CC.gen(0)}))
def _clear_value_cache(self):
del self.__values
def __call__(self, s, c=None):
r"""
INPUT:
- ``s`` - complex number
.. note::
Evaluation of the function takes a long time, so each
evaluation is cached. Call ``self._clear_value_cache()`` to
clear the evaluation cache.
EXAMPLES::
sage: E = EllipticCurve('5077a')
sage: L = E.lseries().dokchitser(100)
sage: L(1)
0.00000000000000000000000000000
sage: L(1+I)
-1.3085436607849493358323930438 + 0.81298000036784359634835412129*I
"""
self.__check_init()
s = self.__CC(s)
try:
return self.__values[s]
except AttributeError:
self.__values = {}
except KeyError:
pass
z = self.gp().eval('L(%s)'%s)
if 'pole' in z:
print(z)
raise ArithmeticError
elif '***' in z:
print(z)
raise RuntimeError
elif 'Warning' in z:
i = z.rfind('\n')
msg = z[:i].replace('digits','decimal digits')
verbose(msg, level=-1)
ans = self.__to_CC(z[i+1:])
self.__values[s] = ans
return ans
ans = self.__to_CC(z)
self.__values[s] = ans
return ans
def derivative(self, s, k=1):
r"""
Return the `k`-th derivative of the `L`-series at
`s`.
.. warning::
If `k` is greater than the order of vanishing of
`L` at `s` you may get nonsense.
EXAMPLES::
sage: E = EllipticCurve('389a')
sage: L = E.lseries().dokchitser()
sage: L.derivative(1,E.rank())
1.51863300057685
"""
self.__check_init()
s = self.__CC(s)
k = Integer(k)
z = self.gp().eval('L(%s,,%s)'%(s,k))
if 'pole' in z:
raise ArithmeticError(z)
elif 'Warning' in z:
i = z.rfind('\n')
msg = z[:i].replace('digits','decimal digits')
verbose(msg, level=-1)
return self.__CC(z[i:])
return self.__CC(z)
def taylor_series(self, a=0, k=6, var='z'):
r"""
Return the first `k` terms of the Taylor series expansion
of the `L`-series about `a`.
This is returned as a series in ``var``, where you
should view ``var`` as equal to `s-a`. Thus
this function returns the formal power series whose coefficients
are `L^{(n)}(a)/n!`.
INPUT:
- ``a`` - complex number (default: 0); point about
which to expand
- ``k`` - integer (default: 6), series is
`O(``var``^k)`
- ``var`` - string (default: 'z'), variable of power
series
EXAMPLES::
sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1')
sage: L.taylor_series(2, 3)
1.64493406684823 - 0.937548254315844*z + 0.994640117149451*z^2 + O(z^3)
sage: E = EllipticCurve('37a')
sage: L = E.lseries().dokchitser()
sage: L.taylor_series(1)
0.000000000000000 + 0.305999773834052*z + 0.186547797268162*z^2 - 0.136791463097188*z^3 + 0.0161066468496401*z^4 + 0.0185955175398802*z^5 + O(z^6)
We compute a Taylor series where each coefficient is to high
precision.
::
sage: E = EllipticCurve('389a')
sage: L = E.lseries().dokchitser(200)
sage: L.taylor_series(1,3)
-9.094...e-82 + (5.1538...e-82)*z + 0.75931650028842677023019260789472201907809751649492435158581*z^2 + O(z^3)
"""
self.__check_init()
a = self.__CC(a)
k = Integer(k)
try:
z = self.gp()('Vec(Lseries(%s,,%s))'%(a,k-1))
except TypeError as msg:
raise RuntimeError("%s\nUnable to compute Taylor expansion (try lowering the number of terms)"%msg)
r = repr(z)
if 'pole' in r:
raise ArithmeticError(r)
elif 'Warning' in r:
i = r.rfind('\n')
msg = r[:i].replace('digits','decimal digits')
verbose(msg, level=-1)
v = list(z)
K = self.__CC
v = [K(repr(x)) for x in v]
R = self.__CC[[var]]
return R(v,len(v))
def check_functional_equation(self, T=1.2):
r"""
Verifies how well numerically the functional equation is satisfied,
and also determines the residues if ``self.poles !=
[]`` and residues='automatic'.
More specifically: for `T>1` (default 1.2),
``self.check_functional_equation(T)`` should ideally
return 0 (to the current precision).
- if what this function returns does not look like 0 at all,
probably the functional equation is wrong (i.e. some of the
parameters gammaV, conductor etc., or the coefficients are wrong),
- if checkfeq(T) is to be used, more coefficients have to be
generated (approximately T times more), e.g. call cflength(1.3),
initLdata("a(k)",1.3), checkfeq(1.3)
- T=1 always (!) returns 0, so T has to be away from 1
- default value `T=1.2` seems to give a reasonable
balance
- if you don't have to verify the functional equation or the
L-values, call num_coeffs(1) and initLdata("a(k)",1), you need
slightly less coefficients.
EXAMPLES::
sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1')
sage: L.check_functional_equation()
-1.35525271600000e-20 # 32-bit
-2.71050543121376e-20 # 64-bit
If we choose the sign in functional equation for the
`\zeta` function incorrectly, the functional equation
doesn't check out.
::
sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=-11, poles=[1], residues=[-1], init='1')
sage: L.check_functional_equation()
-9.73967861488124
"""
self.__check_init()
z = self.gp().eval('checkfeq(%s)'%T).replace(' ','')
return self.__CC(z)
def set_coeff_growth(self, coefgrow):
r"""
You might have to redefine the coefficient growth function if the
`a_n` of the `L`-series are not given by the
following PARI function::
coefgrow(n) = if(length(Lpoles),
1.5*n^(vecmax(real(Lpoles))-1),
sqrt(4*n)^(weight-1));
INPUT:
- ``coefgrow`` - string that evaluates to a PARI
function of n that defines a coefgrow function.
EXAMPLE::
sage: L = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1)
sage: pari_precode = 'tau(n)=(5*sigma(n,3)+7*sigma(n,5))*n/12 - 35*sum(k=1,n-1,(6*k-4*(n-k))*sigma(k,3)*sigma(n-k,5))'
sage: L.init_coeffs('tau(k)', pari_precode=pari_precode)
sage: L.set_coeff_growth('2*n^(11/2)')
sage: L(1)
0.0374412812685155
"""
if not isinstance(coefgrow, str):
raise TypeError("coefgrow must be a string")
g = self.gp()
g.eval('coefgrow(n) = %s'%(coefgrow.replace('\n',' ')))
class Dokchitser(SageObject):
r"""
Dokchitser's `L`-functions Calculator
Create a Dokchitser `L`-series with
Dokchitser(conductor, gammaV, weight, eps, poles, residues, init,
prec)
where
- ``conductor`` - integer, the conductor
- ``gammaV`` - list of Gamma-factor parameters, e.g. [0] for
Riemann zeta, [0,1] for ell.curves, (see examples).
- ``weight`` - positive real number, usually an integer e.g. 1 for
Riemann zeta, 2 for `H^1` of curves/`\QQ`
- ``eps`` - complex number; sign in functional equation
- ``poles`` - (default: []) list of points where `L^*(s)` has
(simple) poles; only poles with `Re(s)>weight/2` should be
included
- ``residues`` - vector of residues of `L^*(s)` in those poles or
set residues='automatic' (default value)
- ``prec`` - integer (default: 53) number of *bits* of precision
RIEMANN ZETA FUNCTION:
We compute with the Riemann Zeta function.
::
sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1')
sage: L
Dokchitser L-series of conductor 1 and weight 1
sage: L(1)
Traceback (most recent call last):
...
ArithmeticError
sage: L(2)
1.64493406684823
sage: L(2, 1.1)
1.64493406684823
sage: L.derivative(2)
-0.937548254315844
sage: h = RR('0.0000000000001')
sage: (zeta(2+h) - zeta(2.))/h
-0.937028232783632
sage: L.taylor_series(2, k=5)
1.64493406684823 - 0.937548254315844*z + 0.994640117149451*z^2 - 1.00002430047384*z^3 + 1.00006193307...*z^4 + O(z^5)
RANK 1 ELLIPTIC CURVE:
We compute with the `L`-series of a rank `1`
curve.
::
sage: E = EllipticCurve('37a')
sage: L = E.lseries().dokchitser(); L
Dokchitser L-function associated to Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
sage: L(1)
0.000000000000000
sage: L.derivative(1)
0.305999773834052
sage: L.derivative(1,2)
0.373095594536324
sage: L.num_coeffs()
48
sage: L.taylor_series(1,4)
0.000000000000000 + 0.305999773834052*z + 0.186547797268162*z^2 - 0.136791463097188*z^3 + O(z^4)
sage: L.check_functional_equation()
6.11218974700000e-18 # 32-bit
6.04442711160669e-18 # 64-bit
RANK 2 ELLIPTIC CURVE:
We compute the leading coefficient and Taylor expansion of the
`L`-series of a rank `2` curve.
::
sage: E = EllipticCurve('389a')
sage: L = E.lseries().dokchitser()
sage: L.num_coeffs ()
156
sage: L.derivative(1,E.rank())
1.51863300057685
sage: L.taylor_series(1,4)
-1.27685190980159e-23 + (7.23588070754027e-24)*z + 0.759316500288427*z^2 - 0.430302337583362*z^3 + O(z^4) # 32-bit
-2.72911738151096e-23 + (1.54658247036311e-23)*z + 0.759316500288427*z^2 - 0.430302337583362*z^3 + O(z^4) # 64-bit
RAMANUJAN DELTA L-FUNCTION:
The coefficients are given by Ramanujan's tau function::
sage: L = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1)
sage: pari_precode = 'tau(n)=(5*sigma(n,3)+7*sigma(n,5))*n/12 - 35*sum(k=1,n-1,(6*k-4*(n-k))*sigma(k,3)*sigma(n-k,5))'
sage: L.init_coeffs('tau(k)', pari_precode=pari_precode)
We redefine the default bound on the coefficients: Deligne's
estimate on tau(n) is better than the default
coefgrow(n)=`(4n)^{11/2}` (by a factor 1024), so
re-defining coefgrow() improves efficiency (slightly faster).
::
sage: L.num_coeffs()
12
sage: L.set_coeff_growth('2*n^(11/2)')
sage: L.num_coeffs()
11
Now we're ready to evaluate, etc.
::
sage: L(1)
0.0374412812685155
sage: L(1, 1.1)
0.0374412812685155
sage: L.taylor_series(1,3)
0.0374412812685155 + 0.0709221123619322*z + 0.0380744761270520*z^2 + O(z^3)
"""
def __init__(self, conductor, gammaV, weight, eps, \
poles=[], residues='automatic', prec=53,
init=None):
"""
Initialization of Dokchitser calculator EXAMPLES::
sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1')
sage: L.num_coeffs()
4
"""
self.conductor = conductor
self.gammaV = gammaV
self.weight = weight
self.eps = eps
self.poles = poles
self.residues = residues
self.prec = prec
self.__CC = ComplexField(self.prec)
self.__RR = self.__CC._real_field()
if not init is None:
self.init_coeffs(init)
self.__init = init
else:
self.__init = False
def __reduce__(self):
D = copy.copy(self.__dict__)
if '_Dokchitser__gp' in D:
del D['_Dokchitser__gp']
return reduce_load_dokchitser, (D, )
def _repr_(self):
z = "Dokchitser L-series of conductor %s and weight %s"%(
self.conductor, self.weight)
return z
def __del__(self):
self.gp().quit()
def gp(self):
"""
Return the gp interpreter that is used to implement this Dokchitser
L-function.
EXAMPLES::
sage: E = EllipticCurve('11a')
sage: L = E.lseries().dokchitser()
sage: L(2)
0.546048036215014
sage: L.gp()
PARI/GP interpreter
"""
try:
return self.__gp
except AttributeError:
logfile = None
# For debugging
import os
from sage.env import DOT_SAGE
logfile = os.path.join(DOT_SAGE, 'dokchitser.log')
g = sage.interfaces.gp.Gp(script_subdirectory='dokchitser', logfile=logfile)
g.read('computel.gp')
self.__gp = g
self._gp_eval('default(realprecision, %s)'%(self.prec//3 + 2))
self._gp_eval('conductor = %s'%self.conductor)
self._gp_eval('gammaV = %s'%self.gammaV)
self._gp_eval('weight = %s'%self.weight)
self._gp_eval('sgn = %s'%self.eps)
self._gp_eval('Lpoles = %s'%self.poles)
self._gp_eval('Lresidues = %s'%self.residues)
g._dokchitser = True
return g
def _gp_eval(self, s):
try:
t = self.gp().eval(s)
except (RuntimeError, TypeError):
raise RuntimeError("Unable to create L-series, due to precision or other limits in PARI.")
if '***' in t:
raise RuntimeError("Unable to create L-series, due to precision or other limits in PARI.")
return t
def __check_init(self):
if not self.__init:
raise ValueError("you must call init_coeffs on the L-function first")
def num_coeffs(self, T=1):
"""
Return number of coefficients `a_n` that are needed in
order to perform most relevant `L`-function computations to
the desired precision.
EXAMPLES::
sage: E = EllipticCurve('11a')
sage: L = E.lseries().dokchitser()
sage: L.num_coeffs()
26
sage: E = EllipticCurve('5077a')
sage: L = E.lseries().dokchitser()
sage: L.num_coeffs()
568
sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1')
sage: L.num_coeffs()
4
"""
return Integer(self.gp().eval('cflength(%s)'%T))
def init_coeffs(self, v, cutoff=1,
w=None,
pari_precode='',
max_imaginary_part=0,
max_asymp_coeffs=40):
"""
Set the coefficients `a_n` of the `L`-series. If
`L(s)` is not equal to its dual, pass the coefficients of
the dual as the second optional argument.
INPUT:
- ``v`` - list of complex numbers or string (pari
function of k)
- ``cutoff`` - real number = 1 (default: 1)
- ``w`` - list of complex numbers or string (pari
function of k)
- ``pari_precode`` - some code to execute in pari
before calling initLdata
- ``max_imaginary_part`` - (default: 0): redefine if
you want to compute L(s) for s having large imaginary part,
- ``max_asymp_coeffs`` - (default: 40): at most this
many terms are generated in asymptotic series for phi(t) and
G(s,t).
EXAMPLES::
sage: L = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1)
sage: pari_precode = 'tau(n)=(5*sigma(n,3)+7*sigma(n,5))*n/12 - 35*sum(k=1,n-1,(6*k-4*(n-k))*sigma(k,3)*sigma(n-k,5))'
sage: L.init_coeffs('tau(k)', pari_precode=pari_precode)
Evaluate the resulting L-function at a point, and compare with the answer that
one gets "by definition" (of L-function attached to a modular form)::
sage: L(14)
0.998583063162746
sage: a = delta_qexp(1000)
sage: sum(a[n]/float(n)^14 for n in range(1,1000))
0.9985830631627459
Illustrate that one can give a list of complex numbers for v
(see :trac:`10937`)::
sage: L2 = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1)
sage: L2.init_coeffs(list(delta_qexp(1000))[1:])
sage: L2(14)
0.998583063162746
TESTS:
Verify that setting the `w` parameter does not raise an error
(see :trac:`10937`). Note that the meaning of `w` does not seem to
be documented anywhere in Dokchitser's package yet, so there is
no claim that the example below is meaningful! ::
sage: L2 = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1)
sage: L2.init_coeffs(list(delta_qexp(1000))[1:], w=[1..1000])
"""
if isinstance(v, tuple) and w is None:
v, cutoff, w, pari_precode, max_imaginary_part, max_asymp_coeffs = v
self.__init = (v, cutoff, w, pari_precode, max_imaginary_part, max_asymp_coeffs)
gp = self.gp()
if pari_precode != '':
self._gp_eval(pari_precode)
RR = self.__CC._real_field()
cutoff = RR(cutoff)
if isinstance(v, str):
if w is None:
self._gp_eval('initLdata("%s", %s)'%(v, cutoff))
return
self._gp_eval('initLdata("%s",%s,"%s")'%(v,cutoff,w))
return
if not isinstance(v, (list, tuple)):
raise TypeError("v (=%s) must be a list, tuple, or string"%v)
CC = self.__CC
v = ','.join([CC(a)._pari_init_() for a in v])
self._gp_eval('Avec = [%s]'%v)
if w is None:
self._gp_eval('initLdata("Avec[k]", %s)'%cutoff)
return
w = ','.join([CC(a)._pari_init_() for a in w])
self._gp_eval('Bvec = [%s]'%w)
self._gp_eval('initLdata("Avec[k]",%s,"Bvec[k]")'%cutoff)
def __to_CC(self, s):
s = s.replace('.E','.0E').replace(' ','')
return self.__CC(sage_eval(s, locals={'I':self.__CC.gen(0)}))
def _clear_value_cache(self):
del self.__values
def __call__(self, s, c=None):
r"""
INPUT:
- ``s`` - complex number
.. note::
Evaluation of the function takes a long time, so each
evaluation is cached. Call ``self._clear_value_cache()`` to
clear the evaluation cache.
EXAMPLES::
sage: E = EllipticCurve('5077a')
sage: L = E.lseries().dokchitser(100)
sage: L(1)
0.00000000000000000000000000000
sage: L(1+I)
-1.3085436607849493358323930438 + 0.81298000036784359634835412129*I
"""
self.__check_init()
s = self.__CC(s)
try:
return self.__values[s]
except AttributeError:
self.__values = {}
except KeyError:
pass
z = self.gp().eval('L(%s)'%s)
if 'pole' in z:
print(z)
raise ArithmeticError
elif '***' in z:
print(z)
raise RuntimeError
elif 'Warning' in z:
i = z.rfind('\n')
msg = z[:i].replace('digits','decimal digits')
verbose(msg, level=-1)
ans = self.__to_CC(z[i+1:])
self.__values[s] = ans
return ans
ans = self.__to_CC(z)
self.__values[s] = ans
return ans
def derivative(self, s, k=1):
r"""
Return the `k`-th derivative of the `L`-series at
`s`.
.. warning::
If `k` is greater than the order of vanishing of
`L` at `s` you may get nonsense.
EXAMPLES::
sage: E = EllipticCurve('389a')
sage: L = E.lseries().dokchitser()
sage: L.derivative(1,E.rank())
1.51863300057685
"""
self.__check_init()
s = self.__CC(s)
k = Integer(k)
z = self.gp().eval('L(%s,,%s)'%(s,k))
if 'pole' in z:
raise ArithmeticError(z)
elif 'Warning' in z:
i = z.rfind('\n')
msg = z[:i].replace('digits','decimal digits')
verbose(msg, level=-1)
return self.__CC(z[i:])
return self.__CC(z)
def taylor_series(self, a=0, k=6, var='z'):
r"""
Return the first `k` terms of the Taylor series expansion
of the `L`-series about `a`.
This is returned as a series in ``var``, where you
should view ``var`` as equal to `s-a`. Thus
this function returns the formal power series whose coefficients
are `L^{(n)}(a)/n!`.
INPUT:
- ``a`` - complex number (default: 0); point about
which to expand
- ``k`` - integer (default: 6), series is
`O(``var``^k)`
- ``var`` - string (default: 'z'), variable of power
series
EXAMPLES::
sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1')
sage: L.taylor_series(2, 3)
1.64493406684823 - 0.937548254315844*z + 0.994640117149451*z^2 + O(z^3)
sage: E = EllipticCurve('37a')
sage: L = E.lseries().dokchitser()
sage: L.taylor_series(1)
0.000000000000000 + 0.305999773834052*z + 0.186547797268162*z^2 - 0.136791463097188*z^3 + 0.0161066468496401*z^4 + 0.0185955175398802*z^5 + O(z^6)
We compute a Taylor series where each coefficient is to high
precision.
::
sage: E = EllipticCurve('389a')
sage: L = E.lseries().dokchitser(200)
sage: L.taylor_series(1,3)
-9.094...e-82 + (5.1538...e-82)*z + 0.75931650028842677023019260789472201907809751649492435158581*z^2 + O(z^3)
"""
self.__check_init()
a = self.__CC(a)
k = Integer(k)
try:
z = self.gp()('Vec(Lseries(%s,,%s))'%(a,k-1))
except TypeError as msg:
raise RuntimeError("%s\nUnable to compute Taylor expansion (try lowering the number of terms)"%msg)
r = repr(z)
if 'pole' in r:
raise ArithmeticError(r)
elif 'Warning' in r:
i = r.rfind('\n')
msg = r[:i].replace('digits','decimal digits')
verbose(msg, level=-1)
v = list(z)
K = self.__CC
v = [K(repr(x)) for x in v]
R = self.__CC[[var]]
return R(v,len(v))
def check_functional_equation(self, T=1.2):
r"""
Verifies how well numerically the functional equation is satisfied,
and also determines the residues if ``self.poles !=
[]`` and residues='automatic'.
More specifically: for `T>1` (default 1.2),
``self.check_functional_equation(T)`` should ideally
return 0 (to the current precision).
- if what this function returns does not look like 0 at all,
probably the functional equation is wrong (i.e. some of the
parameters gammaV, conductor etc., or the coefficients are wrong),
- if checkfeq(T) is to be used, more coefficients have to be
generated (approximately T times more), e.g. call cflength(1.3),
initLdata("a(k)",1.3), checkfeq(1.3)
- T=1 always (!) returns 0, so T has to be away from 1
- default value `T=1.2` seems to give a reasonable
balance
- if you don't have to verify the functional equation or the
L-values, call num_coeffs(1) and initLdata("a(k)",1), you need
slightly less coefficients.
EXAMPLES::
sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1')
sage: L.check_functional_equation()
-1.35525271600000e-20 # 32-bit
-2.71050543121376e-20 # 64-bit
If we choose the sign in functional equation for the
`\zeta` function incorrectly, the functional equation
doesn't check out.
::
sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=-11, poles=[1], residues=[-1], init='1')
sage: L.check_functional_equation()
-9.73967861488124
"""
self.__check_init()
z = self.gp().eval('checkfeq(%s)'%T).replace(' ','')
return self.__CC(z)
def set_coeff_growth(self, coefgrow):
r"""
You might have to redefine the coefficient growth function if the
`a_n` of the `L`-series are not given by the
following PARI function::
coefgrow(n) = if(length(Lpoles),
1.5*n^(vecmax(real(Lpoles))-1),
sqrt(4*n)^(weight-1));
INPUT:
- ``coefgrow`` - string that evaluates to a PARI
function of n that defines a coefgrow function.
EXAMPLES::
sage: L = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1)
sage: pari_precode = 'tau(n)=(5*sigma(n,3)+7*sigma(n,5))*n/12 - 35*sum(k=1,n-1,(6*k-4*(n-k))*sigma(k,3)*sigma(n-k,5))'
sage: L.init_coeffs('tau(k)', pari_precode=pari_precode)
sage: L.set_coeff_growth('2*n^(11/2)')
sage: L(1)
0.0374412812685155
"""
if not isinstance(coefgrow, str):
raise TypeError("coefgrow must be a string")
g = self.gp()
g.eval('coefgrow(n) = %s'%(coefgrow.replace('\n',' ')))
"""
Complex Elliptic Curve L-series
"""
from sage.structure.sage_object import SageObject
from sage.rings.all import (RealField, RationalField, ComplexField)
from math import sqrt, exp, ceil
import sage.functions.transcendental as transcendental
R = RealField()
Q = RationalField()
C = ComplexField()
import sage.misc.all as misc
class Lseries_ell(SageObject):
"""
An elliptic curve $L$-series.
EXAMPLES:
"""
def __init__(self, E):
"""
Create an elliptic curve $L$-series.
EXAMPLES:
sage: EllipticCurve([1..5]).lseries()
Complex L-series of the Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational Field
"""
self.__E = E
class Dokchitser(SageObject):
r"""
Dokchitser's `L`-functions Calculator
Create a Dokchitser `L`-series with
Dokchitser(conductor, gammaV, weight, eps, poles, residues, init,
prec)
where
- ``conductor`` -- integer, the conductor
- ``gammaV`` -- list of Gamma-factor parameters, e.g. [0] for
Riemann zeta, [0,1] for ell.curves, (see examples).
- ``weight`` -- positive real number, usually an integer e.g. 1 for
Riemann zeta, 2 for `H^1` of curves/`\QQ`
- ``eps`` -- complex number; sign in functional equation
- ``poles`` -- (default: []) list of points where `L^*(s)` has
(simple) poles; only poles with `Re(s)>weight/2` should be
included
- ``residues`` -- vector of residues of `L^*(s)` in those poles or
set residues='automatic' (default value)
- ``prec`` -- integer (default: 53) number of *bits* of precision
RIEMANN ZETA FUNCTION:
We compute with the Riemann Zeta function. ::
sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1')
sage: L
Dokchitser L-series of conductor 1 and weight 1
sage: L(1)
Traceback (most recent call last):
...
ArithmeticError
sage: L(2)
1.64493406684823
sage: L(2, 1.1)
1.64493406684823
sage: L.derivative(2)
-0.937548254315844
sage: h = RR('0.0000000000001')
sage: (zeta(2+h) - zeta(2.))/h
-0.937028232783632
sage: L.taylor_series(2, k=5)
1.64493406684823 - 0.937548254315844*z + 0.994640117149451*z^2 - 1.00002430047384*z^3 + 1.00006193307...*z^4 + O(z^5)
RANK 1 ELLIPTIC CURVE:
We compute with the `L`-series of a rank `1` curve. ::
sage: E = EllipticCurve('37a')
sage: L = E.lseries().dokchitser(algorithm='gp'); L
Dokchitser L-function associated to Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
sage: L(1)
0.000000000000000
sage: L.derivative(1)
0.305999773834052
sage: L.derivative(1,2)
0.373095594536324
sage: L.num_coeffs()
48
sage: L.taylor_series(1,4)
0.000000000000000 + 0.305999773834052*z + 0.186547797268162*z^2 - 0.136791463097188*z^3 + O(z^4)
sage: L.check_functional_equation() # abs tol 1e-19
6.04442711160669e-18
RANK 2 ELLIPTIC CURVE:
We compute the leading coefficient and Taylor expansion of the
`L`-series of a rank `2` elliptic curve. ::
sage: E = EllipticCurve('389a')
sage: L = E.lseries().dokchitser(algorithm='gp')
sage: L.num_coeffs()
156
sage: L.derivative(1,E.rank())
1.51863300057685
sage: L.taylor_series(1,4)
-1.27685190980159e-23 + (7.23588070754027e-24)*z + 0.759316500288427*z^2 - 0.430302337583362*z^3 + O(z^4) # 32-bit
-2.72911738151096e-23 + (1.54658247036311e-23)*z + 0.759316500288427*z^2 - 0.430302337583362*z^3 + O(z^4) # 64-bit
NUMBER FIELD:
We compute with the Dedekind zeta function of a number field. ::
sage: x = var('x')
sage: K = NumberField(x**4 - x**2 - 1,'a')
sage: L = K.zeta_function(algorithm='gp')
sage: L.conductor
400
sage: L.num_coeffs()
264
sage: L(2)
1.10398438736918
sage: L.taylor_series(2,3)
1.10398438736918 - 0.215822638498759*z + 0.279836437522536*z^2 + O(z^3)
RAMANUJAN DELTA L-FUNCTION:
The coefficients are given by Ramanujan's tau function::
sage: L = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1)
sage: pari_precode = 'tau(n)=(5*sigma(n,3)+7*sigma(n,5))*n/12 - 35*sum(k=1,n-1,(6*k-4*(n-k))*sigma(k,3)*sigma(n-k,5))'
sage: L.init_coeffs('tau(k)', pari_precode=pari_precode)
We redefine the default bound on the coefficients: Deligne's
estimate on tau(n) is better than the default
coefgrow(n)=`(4n)^{11/2}` (by a factor 1024), so
re-defining coefgrow() improves efficiency (slightly faster). ::
sage: L.num_coeffs()
12
sage: L.set_coeff_growth('2*n^(11/2)')
sage: L.num_coeffs()
11
Now we're ready to evaluate, etc. ::
sage: L(1)
0.0374412812685155
sage: L(1, 1.1)
0.0374412812685155
sage: L.taylor_series(1,3)
0.0374412812685155 + 0.0709221123619322*z + 0.0380744761270520*z^2 + O(z^3)
"""
__gp = None
__globals = set()
# set of global variables defined in a run of the
# computel.gp script that are replaced by indexed copies
# in the computel.gp.template
__globals_re = None
__instance = 0 # Monotonically increasing unique instance ID
__n_instances = 0 # Number of currently allocated instances
__template_filename = os.path.join(SAGE_EXTCODE, 'pari', 'dokchitser',
'computel.gp.template')
__init = False
def __new__(cls, *args, **kwargs):
inst = super(Dokchitser, cls).__new__(cls, *args, **kwargs)
inst.__instance = cls.__instance
cls.__n_instances += 1
cls.__instance += 1
return inst
def __init__(self,
conductor,
gammaV,
weight,
eps,
poles=None,
residues='automatic',
prec=53,
init=None):
"""
Initialization of Dokchitser calculator EXAMPLES::
sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1')
sage: L.num_coeffs()
4
"""
self.conductor = conductor
self.gammaV = gammaV
self.weight = weight
self.eps = eps
self.poles = poles if poles is not None else []
self.residues = residues
self.prec = prec
self.__CC = ComplexField(self.prec)
self.__RR = self.__CC._real_field()
self.__initialized = False
if init is not None:
self.init_coeffs(init)
def __reduce__(self):
D = copy.copy(self.__dict__)
if '_Dokchitser__gp' in D:
del D['_Dokchitser__gp']
return reduce_load_dokchitser, (D, )
def _repr_(self):
return "Dokchitser L-series of conductor %s and weight %s" % (
self.conductor, self.weight)
def __del__(self):
self._teardown_gp(self.__instance)
def gp(self):
"""
Return the gp interpreter that is used to implement this Dokchitser
L-function.
EXAMPLES::
sage: E = EllipticCurve('11a')
sage: L = E.lseries().dokchitser(algorithm='gp')
sage: L(2)
0.546048036215014
sage: L.gp()
PARI/GP interpreter
"""
if self.__gp is None:
self._instantiate_gp()
elif self.__initialized:
return self.__gp
self.__initialized = True
with open(self.__template_filename) as tf:
template = string.Template(tf.read())
tmp_script = os.path.join(SAGE_TMP, 'computel_%s.gp' % self.__instance)
with open(tmp_script, 'w') as f:
f.write(template.substitute(i=str(self.__instance)))
try:
self.__gp.read(tmp_script)
finally:
os.unlink(tmp_script)
self._gp_eval('default(realprecision, %s)' % (self.prec // 3 + 2))
self._gp_set_inst('conductor', self.conductor)
self._gp_set_inst('gammaV', self.gammaV)
self._gp_set_inst('weight', self.weight)
self._gp_set_inst('sgn', self.eps)
self._gp_set_inst('Lpoles', self.poles)
self._gp_set_inst('Lresidues', self.residues)
return self.__gp
@classmethod
def _instantiate_gp(cls):
from sage.env import DOT_SAGE
logfile = os.path.join(DOT_SAGE, 'dokchitser.log')
cls.__gp = sage.interfaces.gp.Gp(script_subdirectory='dokchitser',
logfile=logfile)
# Read the script template and parse out all indexed global variables
# (easy because they all end in "_$i" and there's nothing else in the
# script that uses $)
global_re = re.compile(r'([a-zA-Z0-9]+)_\$i')
with open(cls.__template_filename) as f:
for line in f:
for m in global_re.finditer(line):
cls.__globals.add(m.group(1))
cls.__globals_re = re.compile(
'([^a-zA-Z0-9_]|^)(%s)([^a-zA-Z0-9_]|$)' % '|'.join(cls.__globals))
return
@classmethod
def _teardown_gp(cls, instance=None):
cls.__n_instances -= 1
if cls.__n_instances == 0:
cls.__gp.quit()
elif instance is not None:
# Clean up all global variables created by this instance
for varname in cls.__globals:
cls.__gp.eval('kill(%s_%s)' % (varname, instance))
def _gp_call_inst(self, func, *args):
"""
Call the specified PARI function in the GP interpreter, with the
instance number of this `Dokchitser` instance automatically appended.
For example, ``self._gp_call_inst('L', 1)`` is equivalent to
``self.gp().eval('L_N(1)')`` where ``N`` is ``self.__instance``.
"""
cmd = '%s_%d(%s)' % (func, self.__instance, ','.join(
str(a) for a in args))
return self._gp_eval(cmd)
def _gp_set_inst(self, varname, value):
"""
Like ``_gp_call_inst`` but sets the variable given by ``varname`` to
the given value, appending ``_N`` to the variable name.
If ``varname`` is a function (e.g. ``'func(n)'``) then this sets
``func_N(n)``).
"""
if '(' in varname:
funcname, args = varname.split('(', 1)
varname = '%s_%s(%s' % (funcname, self.__instance, args)
else:
varname = '%s_%s' % (varname, self.__instance)
cmd = '%s = %s' % (varname, value)
return self._gp_eval(cmd)
def _gp_eval(self, s):
try:
t = self.gp().eval(s)
except (RuntimeError, TypeError):
raise RuntimeError(
"Unable to create L-series, due to precision or other limits in PARI."
)
if not self.__init and '***' in t:
# After init_coeffs is called, future calls to this method should
# return the full output for further parsing
raise RuntimeError(
"Unable to create L-series, due to precision or other limits in PARI."
)
return t
def __check_init(self):
if not self.__init:
raise ValueError(
"you must call init_coeffs on the L-function first")
def num_coeffs(self, T=1):
"""
Return number of coefficients `a_n` that are needed in
order to perform most relevant `L`-function computations to
the desired precision.
EXAMPLES::
sage: E = EllipticCurve('11a')
sage: L = E.lseries().dokchitser(algorithm='gp')
sage: L.num_coeffs()
26
sage: E = EllipticCurve('5077a')
sage: L = E.lseries().dokchitser(algorithm='gp')
sage: L.num_coeffs()
568
sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1')
sage: L.num_coeffs()
4
Verify that ``num_coeffs`` works with non-real spectral
parameters, e.g. for the L-function of the level 10 Maass form
with eigenvalue 2.7341055592527126::
sage: ev = 2.7341055592527126
sage: L = Dokchitser(conductor=10, gammaV=[ev*i, -ev*i],weight=2,eps=1)
sage: L.num_coeffs()
26
"""
return Integer(self._gp_call_inst('cflength', T))
def init_coeffs(self,
v,
cutoff=1,
w=None,
pari_precode='',
max_imaginary_part=0,
max_asymp_coeffs=40):
"""
Set the coefficients `a_n` of the `L`-series.
If `L(s)` is not equal to its dual, pass the coefficients of
the dual as the second optional argument.
INPUT:
- ``v`` -- list of complex numbers or string (pari function of k)
- ``cutoff`` -- real number = 1 (default: 1)
- ``w`` -- list of complex numbers or string (pari function of k)
- ``pari_precode`` -- some code to execute in pari
before calling initLdata
- ``max_imaginary_part`` -- (default: 0): redefine if
you want to compute L(s) for s having large imaginary part,
- ``max_asymp_coeffs`` -- (default: 40): at most this
many terms are generated in asymptotic series for phi(t) and
G(s,t).
EXAMPLES::
sage: L = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1)
sage: pari_precode = 'tau(n)=(5*sigma(n,3)+7*sigma(n,5))*n/12 - 35*sum(k=1,n-1,(6*k-4*(n-k))*sigma(k,3)*sigma(n-k,5))'
sage: L.init_coeffs('tau(k)', pari_precode=pari_precode)
Evaluate the resulting L-function at a point, and compare with
the answer that one gets "by definition" (of L-function
attached to a modular form)::
sage: L(14)
0.998583063162746
sage: a = delta_qexp(1000)
sage: sum(a[n]/float(n)^14 for n in range(1,1000))
0.9985830631627459
Illustrate that one can give a list of complex numbers for v
(see :trac:`10937`)::
sage: L2 = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1)
sage: L2.init_coeffs(list(delta_qexp(1000))[1:])
sage: L2(14)
0.998583063162746
TESTS:
Verify that setting the `w` parameter does not raise an error
(see :trac:`10937`). Note that the meaning of `w` does not seem to
be documented anywhere in Dokchitser's package yet, so there is
no claim that the example below is meaningful! ::
sage: L2 = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1)
sage: L2.init_coeffs(list(delta_qexp(1000))[1:], w=[1..1000])
"""
if isinstance(v, tuple) and w is None:
v, cutoff, w, pari_precode, max_imaginary_part, max_asymp_coeffs = v
if pari_precode:
# Must have __globals_re instantiated
if self.__gp is None:
self._instantiate_gp()
def repl(m):
return '%s%s_%d%s' % (m.group(1), m.group(2), self.__instance,
m.group(3))
# If any of the pre-code contains references to some of the
# templated global variables we must replace those as well
pari_precode = self.__globals_re.sub(repl, pari_precode)
if pari_precode != '':
self._gp_eval(pari_precode)
RR = self.__CC._real_field()
cutoff = RR(cutoff)
if isinstance(v, str):
if w is None:
self._gp_call_inst('initLdata', '"%s"' % v, cutoff)
else:
self._gp_call_inst('initLdata', '"%s"' % v, cutoff, '"%s"' % w)
elif not isinstance(v, (list, tuple)):
raise TypeError("v (=%s) must be a list, tuple, or string" % v)
else:
CC = self.__CC
v = ','.join([CC(a)._pari_init_() for a in v])
self._gp_eval('Avec = [%s]' % v)
if w is None:
self._gp_call_inst('initLdata', '"Avec[k]"', cutoff)
else:
w = ','.join([CC(a)._pari_init_() for a in w])
self._gp_eval('Bvec = [%s]' % w)
self._gp_call_inst('initLdata', '"Avec[k]"', cutoff,
'"Bvec[k]"')
self.__init = (v, cutoff, w, pari_precode, max_imaginary_part,
max_asymp_coeffs)
def __to_CC(self, s):
s = s.replace('.E', '.0E').replace(' ', '')
return self.__CC(sage_eval(s, locals={'I': self.__CC.gen(0)}))
def _clear_value_cache(self):
del self.__values
def __call__(self, s, c=None):
r"""
INPUT:
- ``s`` -- complex number
.. NOTE::
Evaluation of the function takes a long time, so each
evaluation is cached. Call ``self._clear_value_cache()`` to
clear the evaluation cache.
EXAMPLES::
sage: E = EllipticCurve('5077a')
sage: L = E.lseries().dokchitser(100, algorithm='gp')
sage: L(1)
0.00000000000000000000000000000
sage: L(1+I)
-1.3085436607849493358323930438 + 0.81298000036784359634835412129*I
"""
self.__check_init()
s = self.__CC(s)
try:
return self.__values[s]
except AttributeError:
self.__values = {}
except KeyError:
pass
z = self._gp_call_inst('L', s)
if 'pole' in z:
print(z)
raise ArithmeticError
elif '***' in z:
print(z)
raise RuntimeError
elif 'Warning' in z:
i = z.rfind('\n')
msg = z[:i].replace('digits', 'decimal digits')
verbose(msg, level=-1)
ans = self.__to_CC(z[i + 1:])
self.__values[s] = ans
return ans
ans = self.__to_CC(z)
self.__values[s] = ans
return ans
def derivative(self, s, k=1):
r"""
Return the `k`-th derivative of the `L`-series at `s`.
.. WARNING::
If `k` is greater than the order of vanishing of
`L` at `s` you may get nonsense.
EXAMPLES::
sage: E = EllipticCurve('389a')
sage: L = E.lseries().dokchitser(algorithm='gp')
sage: L.derivative(1,E.rank())
1.51863300057685
"""
self.__check_init()
s = self.__CC(s)
k = Integer(k)
z = self._gp_call_inst('L', s, '', k)
if 'pole' in z:
raise ArithmeticError(z)
elif 'Warning' in z:
i = z.rfind('\n')
msg = z[:i].replace('digits', 'decimal digits')
verbose(msg, level=-1)
return self.__CC(z[i:])
return self.__CC(z)
def taylor_series(self, a=0, k=6, var='z'):
r"""
Return the first `k` terms of the Taylor series expansion
of the `L`-series about `a`.
This is returned as a series in ``var``, where you
should view ``var`` as equal to `s-a`. Thus
this function returns the formal power series whose coefficients
are `L^{(n)}(a)/n!`.
INPUT:
- ``a`` -- complex number (default: 0); point about
which to expand
- ``k`` -- integer (default: 6), series is
`O(``var``^k)`
- ``var`` -- string (default: 'z'), variable of power
series
EXAMPLES::
sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1')
sage: L.taylor_series(2, 3)
1.64493406684823 - 0.937548254315844*z + 0.994640117149451*z^2 + O(z^3)
sage: E = EllipticCurve('37a')
sage: L = E.lseries().dokchitser(algorithm='gp')
sage: L.taylor_series(1)
0.000000000000000 + 0.305999773834052*z + 0.186547797268162*z^2 - 0.136791463097188*z^3 + 0.0161066468496401*z^4 + 0.0185955175398802*z^5 + O(z^6)
We compute a Taylor series where each coefficient is to high
precision. ::
sage: E = EllipticCurve('389a')
sage: L = E.lseries().dokchitser(200, algorithm='gp')
sage: L.taylor_series(1,3)
...e-82 + (...e-82)*z + 0.75931650028842677023019260789472201907809751649492435158581*z^2 + O(z^3)
Check that :trac:`25402` is fixed::
sage: L = EllipticCurve("24a1").modular_form().lseries()
sage: L.taylor_series(-1, 3)
0.000000000000000 - 0.702565506265199*z + 0.638929001045535*z^2 + O(z^3)
Check that :trac:`25965` is fixed::
sage: L2 = EllipticCurve("37a1").modular_form().lseries(); L2
L-series associated to the cusp form q - 2*q^2 - 3*q^3 + 2*q^4 - 2*q^5 + O(q^6)
sage: L2.taylor_series(0,4)
0.000000000000000 - 0.357620466127498*z + 0.273373112603865*z^2 + 0.303362857047671*z^3 + O(z^4)
sage: L2.taylor_series(0,1)
O(z^1)
sage: L2(0)
0.000000000000000
"""
self.__check_init()
a = self.__CC(a)
k = Integer(k)
try:
z = self._gp_call_inst('Lseries', a, '', k - 1)
z = self.gp()('Vecrev(Pol(%s))' % z)
except TypeError as msg:
raise RuntimeError(
"%s\nUnable to compute Taylor expansion (try lowering the number of terms)"
% msg)
r = repr(z)
if 'pole' in r:
raise ArithmeticError(r)
elif 'Warning' in r:
i = r.rfind('\n')
msg = r[:i].replace('digits', 'decimal digits')
verbose(msg, level=-1)
v = list(z)
K = self.__CC
v = [K(repr(x)) for x in v]
R = self.__CC[[var]]
return R(v, k)
def check_functional_equation(self, T=1.2):
r"""
Verifies how well numerically the functional equation is satisfied,
and also determines the residues if ``self.poles !=
[]`` and residues='automatic'.
More specifically: for `T>1` (default 1.2),
``self.check_functional_equation(T)`` should ideally
return 0 (to the current precision).
- if what this function returns does not look like 0 at all,
probably the functional equation is wrong (i.e. some of the
parameters gammaV, conductor etc., or the coefficients are wrong),
- if checkfeq(T) is to be used, more coefficients have to be
generated (approximately T times more), e.g. call cflength(1.3),
initLdata("a(k)",1.3), checkfeq(1.3)
- T=1 always (!) returns 0, so T has to be away from 1
- default value `T=1.2` seems to give a reasonable
balance
- if you don't have to verify the functional equation or the
L-values, call num_coeffs(1) and initLdata("a(k)",1), you need
slightly less coefficients.
EXAMPLES::
sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1')
sage: L.check_functional_equation() # abs tol 1e-19
-2.71050543121376e-20
If we choose the sign in functional equation for the
`\zeta` function incorrectly, the functional equation
doesn't check out. ::
sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=-11, poles=[1], residues=[-1], init='1')
sage: L.check_functional_equation()
-9.73967861488124
"""
self.__check_init()
z = self._gp_call_inst('checkfeq', T)
z = z.replace(' ', '')
return self.__CC(z)
def set_coeff_growth(self, coefgrow):
r"""
You might have to redefine the coefficient growth function if the
`a_n` of the `L`-series are not given by the
following PARI function::
coefgrow(n) = if(length(Lpoles),
1.5*n^(vecmax(real(Lpoles))-1),
sqrt(4*n)^(weight-1));
INPUT:
- ``coefgrow`` -- string that evaluates to a PARI
function of n that defines a coefgrow function.
EXAMPLES::
sage: L = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1)
sage: pari_precode = 'tau(n)=(5*sigma(n,3)+7*sigma(n,5))*n/12 - 35*sum(k=1,n-1,(6*k-4*(n-k))*sigma(k,3)*sigma(n-k,5))'
sage: L.init_coeffs('tau(k)', pari_precode=pari_precode)
sage: L.set_coeff_growth('2*n^(11/2)')
sage: L(1)
0.0374412812685155
"""
if not isinstance(coefgrow, str):
raise TypeError("coefgrow must be a string")
self._gp_set_inst('coefgrow(n)', coefgrow.replace('\n', ' '))
