Python ZZ.factor方法代码示例

# 需要导入模块: from sage.all import ZZ [as 别名]
# 或者: from sage.all.ZZ import factor [as 别名]
class GaloisRepresentation( Lfunction):

    def __init__(self, thingy):
        """
        Class representing a L-function coming from a Galois representation.
        Typically, dirichlet characters, artin reps, elliptic curves,...
        can give rise to such a class.

        It can be used for tensor two such together (mul below) and a
        L-function class can be extracted from it.
        """

        # this is an important global variable.
        # it is the maximum of the imag parts of values s at
        # which we will compute L(s,.)
        self.max_imaginary_part = "40"

        if isinstance(thingy, sage.schemes.elliptic_curves.ell_rational_field.EllipticCurve_rational_field):
            self.init_elliptic_curve(thingy)

        elif isinstance(thingy, lmfdb.WebCharacter.WebDirichletCharacter):
            self.init_dir_char(thingy)

        elif isinstance(thingy, lmfdb.artin_representations.math_classes.ArtinRepresentation):
            self.init_artin_rep(thingy)

        elif (isinstance(thingy, list) and
              len(thingy) == 2 and
              isinstance(thingy[0],lmfdb.modular_forms.elliptic_modular_forms.backend.web_newforms.WebNewForm) and
              isinstance(thingy[1],sage.rings.integer.Integer) ):
            self.init_elliptic_modular_form(thingy[0],thingy[1])

        elif (isinstance(thingy, list) and
              len(thingy) == 2 and
              isinstance(thingy[0], GaloisRepresentation) and
              isinstance(thingy[1], GaloisRepresentation) ):
            self.init_tensor_product(thingy[0], thingy[1])

        else:
            raise ValueError("GaloisRepresentations are currently not implemented for that type (%s) of objects"%type(thingy))

        # set a few common variables
        self.level = self.conductor
        self.degree = self.dim
        self.poles = []
        self.residues = []
        self.algebraic = True
        self.weight = self.motivic_weight + 1

## Various ways to construct such a class

    def init_elliptic_curve(self, E):
        """
        Returns the Galois rep of an elliptic curve over Q
        """

        self.original_object = [E]
        self.object_type = "ellipticcurve"
        self.dim = 2
        self.motivic_weight = 1
        self.conductor = E.conductor()
        self.bad_semistable_primes = [ fa[0] for fa in self.conductor.factor() if fa[1]==1 ]
        self.bad_pot_good = [p for p in self.conductor.prime_factors() if E.j_invariant().valuation(p) > 0 ]
        self.sign = E.root_number()
        self.mu_fe = []
        self.nu_fe = [ZZ(1)/ZZ(2)]
        self.gammaV = [0, 1]
        self.langlands = True
        self.selfdual = True
        self.primitive = True
        self.set_dokchitser_Lfunction()
        self.set_number_of_coefficients()
        self.coefficient_type = 2
        self.coefficient_period = 0
        self.besancon_bound = 50000
        self.ld.gp().quit()

        def eu(p):
            """
            Local Euler factor passed as a function
            whose input is a prime and
            whose output is a polynomial
            such that evaluated at p^-s,
            we get the inverse of the local factor
            of the L-function
            """
            R = PolynomialRing(QQ, "T")
            T = R.gens()[0]
            N = self.conductor
            if N % p != 0 : # good reduction
                return 1 - E.ap(p) * T + p * T**2
            elif N % (p**2) != 0: # multiplicative reduction
                return 1 - E.ap(p) * T
            else:
                return R(1)

        self.local_euler_factor = eu

    def init_dir_char(self, chi):
        """
#.........这里部分代码省略.........

# 需要导入模块: from sage.all import ZZ [as 别名]
# 或者: from sage.all.ZZ import factor [as 别名]
def render_curve_webpage_by_label(label):
    C = base.getDBConnection()
    data = C.ellcurves.curves.find_one({'label': label})
    if data is None:
        return "No such curve"    
    info = {}
    ainvs = [int(a) for a in data['ainvs']]
    E = EllipticCurve(ainvs)
    label=data['label']
    N = ZZ(data['conductor'])
    iso_class = data['iso']
    rank = data['rank']
    j_invariant=E.j_invariant()
    #plot=E.plot()
    discriminant=E.discriminant()
    xintpoints_projective=[E.lift_x(x) for x in xintegral_point(data['x-coordinates_of_integral_points'])]
    xintpoints=proj_to_aff(xintpoints_projective)
    G = E.torsion_subgroup().gens()
    
    if 'gens' in data:
        generator=parse_gens(data['gens'])
    if len(G) == 0:
        tor_struct = 'Trivial'
        tor_group='Trivial'
    else:
        tor_group=' \\times '.join(['\mathbb{Z}/{%s}\mathbb{Z}'%a.order() for a in G])
    if 'torsion_structure' in data:
        info['tor_structure']= ' \\times '.join(['\mathbb{Z}/{%s}\mathbb{Z}'% int(a) for a in data['torsion_structure']])
    else:
        info['tor_structure'] = tor_group
        
    info.update(data)
    if rank >=2:
        lder_tex = "L%s(E,1)" % ("^{("+str(rank)+")}")
    elif rank ==1:
        lder_tex = "L%s(E,1)" % ("'"*rank)
    else:
        assert rank == 0
        lder_tex = "L(E,1)"
    info.update({
        'conductor': N,
        'disc_factor': latex(discriminant.factor()),
        'j_invar_factor':latex(j_invariant.factor()),
        'label': label,
        'isogeny':iso_class,
        'equation': web_latex(E),
        #'f': ajax_more(E.q_eigenform, 10, 20, 50, 100, 250),
        'f' : web_latex(E.q_eigenform(10)),
        'generators':','.join(web_latex(g) for g in generator) if 'gens' in data else ' ',
        'lder'  : lder_tex,
        'p_adic_primes': [p for p in sage.all.prime_range(5,100) if E.is_ordinary(p) and not p.divides(N)],
        'ainvs': format_ainvs(data['ainvs']),
        'tamagawa_numbers': r' \cdot '.join(str(sage.all.factor(c)) for c in E.tamagawa_numbers()),
        'cond_factor':latex(N.factor()),
        'xintegral_points':','.join(web_latex(i_p) for i_p in xintpoints),
        'tor_gens':','.join(web_latex(eval(g)) for g in data['torsion_generators']) if 'torsion_generators' in data else list(G)
                        })
    info['downloads_visible'] = True
    info['downloads'] = [('worksheet', url_for("not_yet_implemented"))]
    info['friends'] = [('Isogeny class', "/EllipticCurve/Q/%s" % iso_class),
                       ('Modular Form', url_for("emf.render_elliptic_modular_form_from_label",label="%s" %(iso_class))),
                       ('L-function', "/L/EllipticCurve/Q/%s" % label)]
    info['learnmore'] = [('Elliptic Curves', url_for("not_yet_implemented"))]
    #info['plot'] = image_src(plot)
    info['plot'] = url_for('plot_ec', label=label)
    info['iso_class'] = data['iso']
    info['download_qexp_url'] = url_for('download_qexp', limit=100, ainvs=','.join([str(a) for a in ainvs]))
    properties2 = [('Label', '%s' % label),
                   (None, '<img src="%s" width="200" height="150"/>' % url_for('plot_ec', label=label) ),
                   ('Conductor', '\(%s\)' % N), 
                   ('Discriminant', '\(%s\)' % discriminant),
                   ('j-invariant', '\(%s\)' % j_invariant),
                   ('Rank', '\(%s\)' % rank),
                   ('Torsion Structure', '\(%s\)' % tor_group)
    ]
    #properties.extend([ "prop %s = %s<br/>" % (_,_*1923) for _ in range(12) ])
    credit = 'John Cremona'
    t = "Elliptic Curve %s" % info['label']
    bread = [('Elliptic Curves ', url_for("rational_elliptic_curves")),('Elliptic curves %s' %info['label'],' ')]

    return render_template("elliptic_curve/elliptic_curve.html", 
         info=info, properties2=properties2, credit=credit,bread=bread, title = t)

# 需要导入模块: from sage.all import ZZ [as 别名]
# 或者: from sage.all.ZZ import factor [as 别名]
def render_curve_webpage_by_label(label):
    C = lmfdb.base.getDBConnection()
    data = C.elliptic_curves.curves.find_one({'lmfdb_label': label})
    if data is None:
        return elliptic_curve_jump_error(label, {})
    info = {}
    ainvs = [int(a) for a in data['ainvs']]
    E = EllipticCurve(ainvs)
    cremona_label = data['label']
    lmfdb_label = data['lmfdb_label']
    N = ZZ(data['conductor'])
    cremona_iso_class = data['iso']  # eg '37a'
    lmfdb_iso_class = data['lmfdb_iso']  # eg '37.a'
    rank = data['rank']
    try:
        j_invariant = QQ(str(data['jinv']))
    except KeyError:
        j_invariant = E.j_invariant()
    if j_invariant == 0:
        j_inv_factored = latex(0)
    else:
        j_inv_factored = latex(j_invariant.factor())
    jinv = unicode(str(j_invariant))
    CMD = 0
    CM = "no"
    EndE = "\(\Z\)"
    if E.has_cm():
        CMD = E.cm_discriminant()
        CM = "yes (\(%s\))"%CMD
        if CMD%4==0:
            d4 = ZZ(CMD)//4
            # r = d4.squarefree_part()
            # f = (d4//r).isqrt()
            # f="" if f==1 else str(f)
            # EndE = "\(\Z[%s\sqrt{%s}]\)"%(f,r)
            EndE = "\(\Z[\sqrt{%s}]\)"%(d4)
        else:            
            EndE = "\(\Z[(1+\sqrt{%s})/2]\)"%CMD

    # plot=E.plot()
    discriminant = E.discriminant()
    xintpoints_projective = [E.lift_x(x) for x in xintegral_point(data['x-coordinates_of_integral_points'])]
    xintpoints = proj_to_aff(xintpoints_projective)
    if 'degree' in data:
        modular_degree = data['degree']
    else:
        try:
            modular_degree = E.modular_degree()
        except RuntimeError:
            modular_degree = 0  # invalid, will be displayed nicely

    G = E.torsion_subgroup().gens()
    minq = E.minimal_quadratic_twist()[0]
    if E == minq:
        minq_label = lmfdb_label
    else:
        minq_ainvs = [str(c) for c in minq.ainvs()]
        minq_label = C.elliptic_curves.curves.find_one({'ainvs': minq_ainvs})['lmfdb_label']
# We do not just do the following, as Sage's installed database
# might not have all the curves in the LMFDB database.
# minq_label = E.minimal_quadratic_twist()[0].label()

    if 'gens' in data:
        generator = parse_gens(data['gens'])
    if len(G) == 0:
        tor_struct = '\mathrm{Trivial}'
        tor_group = '\mathrm{Trivial}'
    else:
        tor_group = ' \\times '.join(['\Z/{%s}\Z' % a.order() for a in G])
    if 'torsion_structure' in data:
        info['tor_structure'] = ' \\times '.join(['\Z/{%s}\Z' % int(a) for a in data['torsion_structure']])
    else:
        info['tor_structure'] = tor_group

    info.update(data)
    if rank >= 2:
        lder_tex = "L%s(E,1)" % ("^{(" + str(rank) + ")}")
    elif rank == 1:
        lder_tex = "L%s(E,1)" % ("'" * rank)
    else:
        assert rank == 0
        lder_tex = "L(E,1)"
    info['Gamma0optimal'] = (
        cremona_label[-1] == '1' if cremona_iso_class != '990h' else cremona_label[-1] == '3')
    info['modular_degree'] = modular_degree
    p_adic_data_exists = (C.elliptic_curves.padic_db.find(
        {'lmfdb_iso': lmfdb_iso_class}).count()) > 0 and info['Gamma0optimal']

    # Local data
    local_data = []
    for p in N.prime_factors():
        local_info = E.local_data(p)
        local_data.append({'p': p,
                           'tamagawa_number': local_info.tamagawa_number(),
                           'kodaira_symbol': web_latex(local_info.kodaira_symbol()).replace('$', ''),
                           'reduction_type': local_info.bad_reduction_type()
                           })

    mod_form_iso = lmfdb_label_regex.match(lmfdb_iso_class).groups()[1]

#.........这里部分代码省略.........

# 需要导入模块: from sage.all import ZZ [as 别名]
# 或者: from sage.all.ZZ import factor [as 别名]
def render_curve_webpage_by_label(label):
    C = lmfdb.base.getDBConnection()
    data = C.elliptic_curves.curves.find_one({"lmfdb_label": label})
    if data is None:
        return elliptic_curve_jump_error(label, {})
    info = {}
    ainvs = [int(a) for a in data["ainvs"]]
    E = EllipticCurve(ainvs)
    cremona_label = data["label"]
    lmfdb_label = data["lmfdb_label"]
    N = ZZ(data["conductor"])
    cremona_iso_class = data["iso"]  # eg '37a'
    lmfdb_iso_class = data["lmfdb_iso"]  # eg '37.a'
    rank = data["rank"]
    try:
        j_invariant = QQ(str(data["jinv"]))
    except KeyError:
        j_invariant = E.j_invariant()
    if j_invariant == 0:
        j_inv_factored = latex(0)
    else:
        j_inv_factored = latex(j_invariant.factor())
    jinv = unicode(str(j_invariant))
    CMD = 0
    CM = "no"
    EndE = "\(\Z\)"
    if E.has_cm():
        CMD = E.cm_discriminant()
        CM = "yes (\(%s\))" % CMD
        if CMD % 4 == 0:
            d4 = ZZ(CMD) // 4
            # r = d4.squarefree_part()
            # f = (d4//r).isqrt()
            # f="" if f==1 else str(f)
            # EndE = "\(\Z[%s\sqrt{%s}]\)"%(f,r)
            EndE = "\(\Z[\sqrt{%s}]\)" % (d4)
        else:
            EndE = "\(\Z[(1+\sqrt{%s})/2]\)" % CMD

    # plot=E.plot()
    discriminant = E.discriminant()
    xintpoints_projective = [E.lift_x(x) for x in xintegral_point(data["x-coordinates_of_integral_points"])]
    xintpoints = proj_to_aff(xintpoints_projective)
    if "degree" in data:
        modular_degree = data["degree"]
    else:
        try:
            modular_degree = E.modular_degree()
        except RuntimeError:
            modular_degree = 0  # invalid, will be displayed nicely

    G = E.torsion_subgroup().gens()
    E_pari = E.pari_curve(prec=200)
    from sage.libs.pari.all import PariError

    try:
        minq = E.minimal_quadratic_twist()[0]
    except PariError:  # this does occur with 164411a1
        print "PariError computing minimal quadratic twist of elliptic curve %s" % lmfdb_label
        minq = E
    if E == minq:
        minq_label = lmfdb_label
    else:
        minq_ainvs = [str(c) for c in minq.ainvs()]
        minq_label = C.elliptic_curves.curves.find_one({"ainvs": minq_ainvs})["lmfdb_label"]
    # We do not just do the following, as Sage's installed database
    # might not have all the curves in the LMFDB database.
    # minq_label = E.minimal_quadratic_twist()[0].label()

    if "gens" in data:
        generator = parse_gens(data["gens"])
    if len(G) == 0:
        tor_struct = "\mathrm{Trivial}"
        tor_group = "\mathrm{Trivial}"
    else:
        tor_group = " \\times ".join(["\Z/{%s}\Z" % a.order() for a in G])
    if "torsion_structure" in data:
        info["tor_structure"] = " \\times ".join(["\Z/{%s}\Z" % int(a) for a in data["torsion_structure"]])
    else:
        info["tor_structure"] = tor_group

    info.update(data)
    if rank >= 2:
        lder_tex = "L%s(E,1)" % ("^{(" + str(rank) + ")}")
    elif rank == 1:
        lder_tex = "L%s(E,1)" % ("'" * rank)
    else:
        assert rank == 0
        lder_tex = "L(E,1)"
    info["Gamma0optimal"] = cremona_label[-1] == "1" if cremona_iso_class != "990h" else cremona_label[-1] == "3"
    info["modular_degree"] = modular_degree
    p_adic_data_exists = (C.elliptic_curves.padic_db.find({"lmfdb_iso": lmfdb_iso_class}).count()) > 0 and info[
        "Gamma0optimal"
    ]

    # Local data
    local_data = []
    for p in N.prime_factors():
        local_info = E.local_data(p, algorithm="generic")
        local_data.append(
#.........这里部分代码省略.........