test_numberfield.py

from lmfdb.tests import LmfdbTest

class NumberFieldTest(LmfdbTest):
    # All tests should pass
    def test_Q(self):
        self.check_args('/NumberField/Q', r'\chi_{1}')
        self.check_args('/NumberField/1.1.1.1', r'\chi_{1}')

    def test_hard_degree10(self):
        self.check_args('/NumberField/10.10.1107649855354064.1', '10T36')
        self.check_args('/NumberField/10.10.138420300533025695415730492558689.1', '10T38')

    def test_hard_degree16(self):
        self.check_args('/NumberField/16.0.13307764731675384304522756096.1', '16T1535')

    def test_search_ramif_cl_deg(self):
        self.check_args('/NumberField/?degree=5&class_group=[2%2C2]&ur_primes=7&discriminant=&ram_quantifier=exactly&ram_primes=2%2C3%2C5', '5.1.27000000000.8')

    def test_abelian_conductor(self):
        self.check_args('/NumberField/5.5.5719140625.2', '275') # conductor

    def test_stuff_not_computed(self):
        self.check_args('/NumberField/23.23.931347256889446325436632107655346061164193665348344821578377438399536607931200329.1', 'ot computed')

    def test_search_poly_mean2parser(self):
        # X^3-4x+2
        self.check_args('/NumberField/?jump=X**3-4x%2B2&search=Go', '3.3.148.1') # label
        # z^3 - 4*z + 2
        self.check_args('/NumberField/?jump=z%5E3+-+4*z%2B2', '3.3.148.1') # label

    def test_search_zeta(self):
        self.check_args('/NumberField/?jump=Qzeta23&search=Go', '[3]') # class group
        self.check_args('/NumberField/?jump=Qzeta_23&search=Go', '[3]') # class group
        self.check_args('/NumberField/?jump=qzeta23%2B&search=Go', '1014.3133') # regulator
        self.check_args('/NumberField/?jump=qzeta_23%2B&search=Go', '1014.3133') # regulator

    def test_search_sqrt(self):
        self.check_args('/NumberField/?jump=Qsqrt-163&search=Go', '41') # minpoly
        self.check_args('/NumberField/?jump=q(sqrt-163)&search=Go', '41') # minpoly

    def test_search_multiple_fields(self):
        # Test comma-separated list of field labels
        self.check_args('/NumberField/?jump=2.2.5.1%2c+3.3.49.1&search=Go', '2.2.5.1')
        self.check_args('/NumberField/?jump=2.2.5.1%2c+3.3.49.1&search=Go', '3.3.49.1')
        # Test comma-separated list with different input formats
        self.check_args('/NumberField/?jump=Qsqrt5%2c+x%5E2-3&search=Go', '2.2.5.1')
        self.check_args('/NumberField/?jump=Qsqrt5%2c+x%5E2-3&search=Go', '2.2.12.1')

    def test_search_abelian_jump(self):
        # Abelian fields entered by a non-reduced defining polynomial are
        # identified without running polredabs (issue #5471).
        from sage.all import pari
        from urllib.parse import quote
        # the degree 47 field of conductor 283 from the issue, entered via
        # the minimal polynomial of z + z^2 for z a root of the stored
        # polynomial
        label = "47.47.60558628944427886416035618894711378994697503545758179730765967261479053047453845062877188530544503628007178201769.1"
        coeffs = self.db.nf_fields.lookup(label, "coeffs")
        g = pari([int(c) for c in coeffs]).Polrev()
        T = pari("x + x^2").Mod(g).charpoly()
        self.check_args('/NumberField/?jump=' + quote(str(T)),
                        [label, 'uses a different defining polynomial'])
        # same for a moderate degree: Q(zeta_32), degree 16
        T = pari("x + x^2").Mod(pari("polcyclo(32)")).charpoly()
        self.check_args('/NumberField/?jump=' + quote(str(T)),
                        '16.0.18446744073709551616.1')

    def test_abelian_nf_label(self):
        # the underlying fast path for issue #5471
        from sage.all import pari
        from lmfdb.number_fields.web_number_field import abelian_nf_label
        # degree 8: Q(zeta_20), entered via the minimal polynomial of z + 3z^3
        T = (pari("x") + 3 * pari("x^3")).Mod(pari("polcyclo(20)")).charpoly()
        assert abelian_nf_label(T) == "8.0.4000000.1"
        # non-Galois and Galois-but-non-abelian inputs are left to the
        # polredabs path
        assert abelian_nf_label(pari("x^8 - 2")) is None
        assert abelian_nf_label(pari("polcompositum(x^4 - 2, x^2 + 1)[1]")) is None
        # same field, but entered so that the order we can certify maximal is
        # very far from maximal: the index is divisible by two primes above
        # 10^5, which stay out of S, so nfroots is called with a conditional
        # structure (hence gets the defining polynomial, not the nf)
        m = 100003 * 100019
        T = (m * pari("x")).Mod(pari("polcyclo(20)")).charpoly()
        assert abelian_nf_label(T) == "8.0.4000000.1"

    def test_known_discriminant_primes(self):
        # a large ramified prime shows up in the discriminant as a prime
        # power, not as a prime (issue #5471)
        from sage.all import ZZ
        from lmfdb.number_fields.web_number_field import _known_discriminant_primes
        q = ZZ(100003)
        S = _known_discriminant_primes(ZZ(2)**20 * ZZ(5)**10 * q**7)
        assert S == [ZZ(2), ZZ(5), q]
        # a cofactor with two large prime factors is left unfactored
        assert _known_discriminant_primes(ZZ(2)**20 * q * ZZ(100019)) == [ZZ(2)]

    def test_jump_degree_too_large(self):
        # for degrees beyond anything in the database the jump returns
        # quickly instead of attempting polredabs (issue #5471): here a
        # degree 94 subfield of Q(zeta_283), for which polredabs takes
        # more than five minutes
        from sage.all import pari
        from urllib.parse import quote
        T = pari.polsubcyclo(283, 94)
        if T.type() == 't_VEC':
            T = T[0]
        self.check_args('/NumberField/?jump=' + quote(str(T)),
                        'does not define a number field in the database')

    def test_search_disc(self):
        self.check_args('/NumberField/?discriminant=1988-2014', '401') # factor of one of the discriminants

    def test_url_label(self):
        self.check_args('/NumberField/2.2.5.1', '0.481211825') # regulator

    def test_url_naturallabel(self):
        self.check_args('/NumberField/Qsqrt5', '0.481211825') # regulator

    def test_url_naturallabel_custom(self):
        # Test various different custom nicknames for number fields
        self.check_args('/NumberField/Qi', '2.0.4.1')
        self.check_args('/NumberField/Qphi', '2.2.5.1')
        self.check_args('/NumberField/Qcbrt2', '3.1.108.1')
        self.check_args('/NumberField/Q(sqrt2+sqrt3)', '4.4.2304.1')
        self.check_args('/NumberField/Q(sqrt2,sqrt3)', '4.4.2304.1')
        self.check_args('/NumberField/Q(sqrt(1 + sqrt2))', '4.2.1024.1')
        self.check_args('/NumberField/Q(sqrt2,sqrt3,cbrt2)', '12.4.320979616137216.3')
        self.check_args('/NumberField/Q(sqrt2,-sqrt2)', '2.2.8.1')
        self.check_args('/NumberField/Q(sqrt2-sqrt2)', '1.1.1.1')

    def test_arith_equiv(self):
        self.check_args('/NumberField/7.3.6431296.1', '7.3.6431296.2') # arith equiv field

    def test_sextic_twin(self):
        self.check_args('/NumberField/6.0.10816.1', 'Twin sextic algebra')

    def test_how_computed(self):
        self.check_args('/NumberField/Source', 'Hunter searches')

    def test_galois_group_page(self):
        self.check_args('/NumberField/GaloisGroups', 'abstract group may have')

    def test_imaginary_quadratic_page(self):
        self.check_args('/NumberField/QuadraticImaginaryClassGroups', 'extensive computations')

    def test_discriminants_page(self):
        self.check_args('/NumberField/Source', 'Jones-David Roberts')

    def test_field_labels_page(self):
        self.check_args('/NumberField/FieldLabels', 'with the same signature and absolute value of the')

    def test_url_bad(self):
        self.check_args('/NumberField/junk', 'Error')  # error message

    def test_random_field(self):
        self.check_args('/NumberField/random', 'Discriminant')

    def test_statistics(self):
        self.check_args('/NumberField/stats', 'Class number')

    def test_pretty_labels(self):
        # Test "prettified" latex labels for number fields
        self.check_args('/NumberField/1.1.1.1', r'\Q')
        self.check_args('/NumberField/2.0.4.1', r'\Q(\sqrt{-1})')
        self.check_args('/NumberField/4.4.1600.1', r'\Q(\sqrt{2}, \sqrt{5})')
        self.check_args('/NumberField/6.0.16807.1', r'\Q(\zeta_{7})')
        self.check_args('/NumberField/3.3.49.1', r'\Q(\zeta_{7})^+')
        self.check_args('/NumberField/3.1.300.1', r'\Q(\sqrt[3]{10})')
        self.check_args('/NumberField/4.2.2048.1', r'\Q(\sqrt[4]{2})')
        self.check_args('/NumberField/4.0.512.1', r'\Q(\sqrt{1 + i})')
        self.check_args('/NumberField/4.2.1024.1', r'\Q(\sqrt{1 + \sqrt{2}})')
        self.check_args('/NumberField/4.0.2048.2', r'\Q(\sqrt{-2 + \sqrt{2}})')
        self.check_args('/NumberField/8.8.3317760000.1', r'\Q(\sqrt{2}, \sqrt{3}, \sqrt{5})')
        self.check_args('/NumberField/16.0.11007531417600000000.1', r'\Q(i, \sqrt{2}, \sqrt{3}, \sqrt{5})')
        self.check_args('/NumberField/32.0.4026692887688564776141139207792885760000000000000000.1', r'\Q(i, \sqrt{2}, \sqrt{3}, \sqrt{5}, \sqrt{7})')

    def test_signature_search(self):
        # Square brackets
        self.check_args('/NumberField/?start=0&degree=6&signature=%5B0%2C3%5D&count=100', '6.0.61131.1')
        self.check_args('/NumberField/?start=0&degree=7&signature=%5B3%2C2%5D&count=100', '7.3.1420409.1')
        # Round brackets
        self.check_args('/NumberField/?start=0&degree=6&signature=%280%2C3%29&count=100', '6.0.61131.1')
        self.check_args('/NumberField/?start=0&degree=7&signature=%283%2C2%29&count=100', '7.3.1420409.1')

    def test_signature_display(self):
        # Verify that signatures are displayed with parentheses, not square brackets
        self.check_args('/NumberField/6.0.61131.1', '(0, 3)')  # degree 6 field with signature (0, 3)
        self.check_args('/NumberField/7.3.1420409.1', '(3, 2)')  # degree 7 field with signature (3, 2)

    def test_relative_class_number(self):
        self.check_args('/NumberField/4.0.1327873600.2', '2108')

    def test_fundamental_units(self):
        self.check_args('NumberField/2.2.10069.1', '43388173')
        self.check_args('NumberField/3.3.10004569.1', '22153437467081345')

    def test_split_ors(self):
        self.check_args('/NumberField/?signature=%5B0%2C3%5D&galois_group=S3', '6.0.177147.2')
        self.check_args('/NumberField/?signature=%5B3%2C0%5D&galois_group=S3', '3.3.229.1')
        self.check_args('/NumberField/?signature=[4%2C0]&galois_group=C2xC2&class_number=3%2C6','4.4.1311025.1')
        self.check_args('/NumberField/?signature=[4%2C0]&galois_group=C2xC2&class_number=6%2C3','4.4.1311025.1')
        self.check_args('/NumberField/?signature=[4%2C0]&galois_group=C2xC2&class_number=5-6%2C3','4.4.485809.1')

    def test_underlying_data(self):
        self.check_args('NumberField/2.2.10069.1', ['Underlying data', 'data/2.2.10069.1'])

    def test_diagram_search(self):
        # The Diagram button should be offered on the search page
        self.check_args('/NumberField/?degree=2', 'Diagram search')
        # Default axes use the computed signed discriminant (disc = disc_sign * disc_abs),
        # which is not a stored column; this used to 500 with KeyError: 'disc'.
        page = self.tc.get('/NumberField/?search_type=Diagram&degree=2&count=100').get_data(as_text=True)
        assert 'my_dataviz' in page
        assert 'discriminant' in page
        # 2.0.3.1 is the field of discriminant -3, so its x-coordinate is -3
        assert '"label": "2.0.3.1"' in page and '"x": "-3"' in page
        # The computed absolute-discriminant axis also works
        page2 = self.tc.get('/NumberField/?hst=Diagram&x-axis=disc_abs&y-axis=rd&degree=3&count=50').get_data(as_text=True)
        assert 'my_dataviz' in page2
        assert 'absolute discriminant' in page2

    def test_errors(self):
        self.check_args('NumberField/18.0.10490638424...4432.1/download/sage', 'Invalid label')
        self.check_args('NumberField/4.3.2.1/download/sage', 'There is no number field with label 4.3.2.1')

    def test_signature_download(self):
        # Test that signature is downloaded as [r1, r2] not [r2, degree]
        # For degree 2 fields with negative discriminant: signature is [0, 1] (complex)
        # For degree 2 fields with positive discriminant: signature is [2, 0] (real)
        url = ('/NumberField/?download=1'
               '&query=%7B%27degree%27%3A+2%2C+%27%24or%27%3A+%5B%7B%27disc_sign%27%3A+'
               '-1%2C+%27disc_abs%27%3A+%7B%27%24gte%27%3A+1%2C+%27%24lte%27%3A+3%7D%2C+'
               '%27degree%27%3A+2%7D%2C+%7B%27disc_sign%27%3A+1%2C+%27disc_abs%27%3A+'
               '%7B%27%24lte%27%3A+5%2C+%27%24gte%27%3A+1%7D%2C+%27degree%27%3A+2%7D%5D%7D'
               '&degree=2&discriminant=-3-5&showcol=signature&Submit=text')
        page = self.tc.get(url).get_data(as_text=True)
        # Check that signature format is [r1, r2] where:
        # - For imaginary quadratic fields (disc < 0, degree=2): r1=0, r2=1, so signature=[0, 1]
        # - For real quadratic fields (disc > 0, degree=2): r1=2, r2=0, so signature=[2, 0]
        # The bug was that it showed [r2, degree] = [1, 2] or [0, 2] instead
        assert '[0, 1]' in page  # imaginary quadratic field (with space, no quotes)
        assert '[2, 0]' in page  # real quadratic field (with space, no quotes)
        # Make sure we're NOT getting the buggy format [r2, degree]
        assert '[1, 2]' not in page  # wrong format for imaginary quadratic
        assert '[0, 2]' not in page  # wrong format for real quadratic
        # Also ensure we're not getting quoted strings
        assert '"[0, 1]"' not in page
        assert '"[2, 0]"' not in page