Testing whether elliptic curves over number fields are \(\QQ\)-curves#

AUTHORS:

  • John Cremona (February 2021)

The code here implements the algorithm of Cremona and Najman presented in [CrNa2020].

sage.schemes.elliptic_curves.Qcurves.Step4Test(E, B, oldB=0, verbose=False)[source]#

Apply local Q-curve test to E at all primes up to B.

INPUT:

  • \(E\) (elliptic curve): an elliptic curve defined over a number field

  • \(B\) (integer): upper bound on primes to test

  • oldB (integer, default 0): lower bound on primes to test

  • verbose (boolean, default False): verbosity flag

OUTPUT:

Either (False, \(p\)), if the local test at \(p\) proves that \(E\) is not a \(\QQ\)-curve, or (True, \(0\)) if all local tests at primes between oldB and B fail to prove that \(E\) is not a \(\QQ\)-curve.

ALGORITHM (see [CrNa2020] for details):

This local test at \(p\) only applies if \(E\) has good reduction at all of the primes lying above \(p\) in the base field \(K\) of \(E\). It tests whether (1) \(E\) is either ordinary at all \(P\mid p\), or supersingular at all; (2) if ordinary at all, it tests that the squarefree part of \(a_P^2-4N(P)\) is the same for all \(P\mid p\).

EXAMPLES:

A non-\(\QQ\)-curve over a quartic field (with LMFDB label ‘4.4.8112.1-12.1-a1’) fails this test at \(p=13\):

sage: from sage.schemes.elliptic_curves.Qcurves import Step4Test
sage: R.<x> = PolynomialRing(QQ)
sage: K.<a> = NumberField(R([3, 0, -5, 0, 1]))                                  # needs sage.rings.number_field
sage: E = EllipticCurve([K([-3,-4,1,1]), K([4,-1,-1,0]), K([-2,0,1,0]),         # needs sage.rings.number_field
....:                    K([-621,778,138,-178]), K([9509,2046,-24728,10380])])
sage: Step4Test(E, 100, verbose=True)                                           # needs sage.rings.number_field
No: inconsistency at the 2 ordinary primes dividing 13
- Frobenius discriminants mod squares: [-3, -1]
(False, 13)

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.Qcurves import Step4Test
>>> R = PolynomialRing(QQ, names=('x',)); (x,) = R._first_ngens(1)
>>> K = NumberField(R([Integer(3), Integer(0), -Integer(5), Integer(0), Integer(1)]), names=('a',)); (a,) = K._first_ngens(1)# needs sage.rings.number_field
>>> E = EllipticCurve([K([-Integer(3),-Integer(4),Integer(1),Integer(1)]), K([Integer(4),-Integer(1),-Integer(1),Integer(0)]), K([-Integer(2),Integer(0),Integer(1),Integer(0)]),         # needs sage.rings.number_field
...                    K([-Integer(621),Integer(778),Integer(138),-Integer(178)]), K([Integer(9509),Integer(2046),-Integer(24728),Integer(10380)])])
>>> Step4Test(E, Integer(100), verbose=True)                                           # needs sage.rings.number_field
No: inconsistency at the 2 ordinary primes dividing 13
- Frobenius discriminants mod squares: [-3, -1]
(False, 13)

A \(\QQ\)-curve over a sextic field (with LMFDB label ‘6.6.1259712.1-64.1-a6’) passes this test for all \(p<100\):

sage: from sage.schemes.elliptic_curves.Qcurves import Step4Test
sage: R.<x> = PolynomialRing(QQ)
sage: K.<a> = NumberField(R([-3, 0, 9, 0, -6, 0, 1]))                           # needs sage.rings.number_field
sage: E = EllipticCurve([K([1,-3,0,1,0,0]), K([5,-3,-6,1,1,0]),                 # needs sage.rings.number_field
....:                    K([1,-3,0,1,0,0]), K([-139,-129,331,277,-76,-63]),
....:                    K([2466,1898,-5916,-4582,1361,1055])])
sage: Step4Test(E, 100, verbose=True)                                           # needs sage.rings.number_field
(True, 0)

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.Qcurves import Step4Test
>>> R = PolynomialRing(QQ, names=('x',)); (x,) = R._first_ngens(1)
>>> K = NumberField(R([-Integer(3), Integer(0), Integer(9), Integer(0), -Integer(6), Integer(0), Integer(1)]), names=('a',)); (a,) = K._first_ngens(1)# needs sage.rings.number_field
>>> E = EllipticCurve([K([Integer(1),-Integer(3),Integer(0),Integer(1),Integer(0),Integer(0)]), K([Integer(5),-Integer(3),-Integer(6),Integer(1),Integer(1),Integer(0)]),                 # needs sage.rings.number_field
...                    K([Integer(1),-Integer(3),Integer(0),Integer(1),Integer(0),Integer(0)]), K([-Integer(139),-Integer(129),Integer(331),Integer(277),-Integer(76),-Integer(63)]),
...                    K([Integer(2466),Integer(1898),-Integer(5916),-Integer(4582),Integer(1361),Integer(1055)])])
>>> Step4Test(E, Integer(100), verbose=True)                                           # needs sage.rings.number_field
(True, 0)
sage.schemes.elliptic_curves.Qcurves.conjugacy_test(jlist, verbose=False)[source]#

Test whether a list of algebraic numbers contains a complete conjugacy class of 2-power degree.

INPUT:

  • jlist (list): a list of algebraic numbers in the same field

  • verbose (boolean, default False): verbosity flag

OUTPUT:

A possibly empty list of irreducible polynomials over \(\QQ\) of 2-power degree all of whose roots are in the list.

EXAMPLES:

sage: # needs sage.rings.number_field
sage: from sage.schemes.elliptic_curves.Qcurves import conjugacy_test
sage: conjugacy_test([3])
[x - 3]
sage: K.<a> = QuadraticField(2)
sage: conjugacy_test([K(3), a])
[x - 3]
sage: conjugacy_test([K(3), 3 + a])
[x - 3]
sage: conjugacy_test([3 + a])
[]
sage: conjugacy_test([3 + a, 3 - a])
[x^2 - 6*x + 7]
sage: x = polygen(QQ)
sage: f = x^3 - 3
sage: K.<a> = f.splitting_field()
sage: js = f.roots(K, multiplicities=False)
sage: conjugacy_test(js)
[]
sage: f = x^4 - 3
sage: K.<a> = NumberField(f)
sage: js = f.roots(K, multiplicities=False)
sage: conjugacy_test(js)
[]
sage: K.<a> = f.splitting_field()
sage: js = f.roots(K, multiplicities=False)
sage: conjugacy_test(js)
[x^4 - 3]

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> from sage.schemes.elliptic_curves.Qcurves import conjugacy_test
>>> conjugacy_test([Integer(3)])
[x - 3]
>>> K = QuadraticField(Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> conjugacy_test([K(Integer(3)), a])
[x - 3]
>>> conjugacy_test([K(Integer(3)), Integer(3) + a])
[x - 3]
>>> conjugacy_test([Integer(3) + a])
[]
>>> conjugacy_test([Integer(3) + a, Integer(3) - a])
[x^2 - 6*x + 7]
>>> x = polygen(QQ)
>>> f = x**Integer(3) - Integer(3)
>>> K = f.splitting_field(names=('a',)); (a,) = K._first_ngens(1)
>>> js = f.roots(K, multiplicities=False)
>>> conjugacy_test(js)
[]
>>> f = x**Integer(4) - Integer(3)
>>> K = NumberField(f, names=('a',)); (a,) = K._first_ngens(1)
>>> js = f.roots(K, multiplicities=False)
>>> conjugacy_test(js)
[]
>>> K = f.splitting_field(names=('a',)); (a,) = K._first_ngens(1)
>>> js = f.roots(K, multiplicities=False)
>>> conjugacy_test(js)
[x^4 - 3]
sage.schemes.elliptic_curves.Qcurves.is_Q_curve(E, maxp=100, certificate=False, verbose=False)[source]#

Return whether E is a \(\QQ\)-curve, with optional certificate.

INPUT:

  • E (elliptic curve) – an elliptic curve over a number field.

  • maxp (int, default 100): bound on primes used for checking necessary local conditions. The result will not depend on this, but using a larger value may return False faster.

  • certificate (bool, default False): if True then a second value is returned giving a certificate for the \(\QQ\)-curve property.

OUTPUT:

If certificate is False: either True (if \(E\) is a \(\QQ\)-curve), or False.

If certificate is True: a tuple consisting of a boolean flag as before and a certificate, defined as follows:

  • when the flag is True, so \(E\) is a \(\QQ\)-curve:

    • either {‘CM’:\(D\)} where \(D\) is a negative discriminant, when \(E\) has potential CM with discriminant \(D\);

    • otherwise {‘CM’: \(0\), ‘core_poly’: \(f\), ‘rho’: \(\rho\), ‘r’: \(r\), ‘N’: \(N\)}, when \(E\) is a non-CM \(\QQ\)-curve, where the core polynomial \(f\) is an irreducible monic polynomial over \(QQ\) of degree \(2^\rho\), all of whose roots are \(j\)-invariants of curves isogenous to \(E\), the core level \(N\) is a square-free integer with \(r\) prime factors which is the LCM of the degrees of the isogenies between these conjugates. For example, if there exists a curve \(E'\) isogenous to \(E\) with \(j(E')=j\in\QQ\), then the certificate is {‘CM’:0, ‘r’:0, ‘rho’:0, ‘core_poly’: x-j, ‘N’:1}.

  • when the flag is False, so \(E\) is not a \(\QQ\)-curve, the certificate is a prime \(p\) such that the reductions of \(E\) at the primes dividing \(p\) are inconsistent with the property of being a \(\QQ\)-curve. See the ALGORITHM section for details.

ALGORITHM:

See [CrNa2020] for details.

1. If \(E\) has rational \(j\)-invariant, or has CM, then return True.

2. Replace \(E\) by a curve defined over \(K=\QQ(j(E))\). Let \(N\) be the conductor norm.

3. For all primes \(p\mid N\) check that the valuations of \(j\) at all \(P\mid p\) are either all negative or all non-negative; if not, return False.

4. For \(p\le maxp\), \(p\not\mid N\), check that either \(E\) is ordinary mod \(P\) for all \(P\mid p\), or \(E\) is supersingular mod \(P\) for all \(P\mid p\); if neither, return False. If all are ordinary, check that the integers \(a_P(E)^2-4N(P)\) have the same square-free part; if not, return False.

5. Compute the \(K\)-isogeny class of \(E\) using the “heuristic” option (which is faster, but not guaranteed to be complete). Check whether the set of \(j\)-invariants of curves in the class of \(2\)-power degree contains a complete Galois orbit. If so, return True.

6. Otherwise repeat step 4 for more primes, and if still undecided, repeat Step 5 without the “heuristic” option, to get the complete \(K\)-isogeny class (which will probably be no bigger than before). Now return True if the set of \(j\)-invariants of curves in the class contains a complete Galois orbit, otherwise return False.

EXAMPLES:

A non-CM curve over \(\QQ\) and a CM curve over \(\QQ\) are both trivially \(\QQ\)-curves:

sage: from sage.schemes.elliptic_curves.Qcurves import is_Q_curve
sage: E = EllipticCurve([1,2,3,4,5])
sage: flag, cert = is_Q_curve(E, certificate=True)
sage: flag
True
sage: cert
{'CM': 0, 'N': 1, 'core_poly': x, 'r': 0, 'rho': 0}

sage: E = EllipticCurve(j=8000)
sage: flag, cert = is_Q_curve(E, certificate=True)
sage: flag
True
sage: cert
{'CM': -8}

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.Qcurves import is_Q_curve
>>> E = EllipticCurve([Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)])
>>> flag, cert = is_Q_curve(E, certificate=True)
>>> flag
True
>>> cert
{'CM': 0, 'N': 1, 'core_poly': x, 'r': 0, 'rho': 0}

>>> E = EllipticCurve(j=Integer(8000))
>>> flag, cert = is_Q_curve(E, certificate=True)
>>> flag
True
>>> cert
{'CM': -8}

A non-\(\QQ\)-curve over a quartic field. The local data at bad primes above \(3\) is inconsistent:

sage: from sage.schemes.elliptic_curves.Qcurves import is_Q_curve
sage: R.<x> = PolynomialRing(QQ)
sage: K.<a> = NumberField(R([3, 0, -5, 0, 1]))                                  # needs sage.rings.number_field
sage: E = EllipticCurve([K([-3,-4,1,1]), K([4,-1,-1,0]), K([-2,0,1,0]),         # needs sage.rings.number_field
....:                    K([-621,778,138,-178]), K([9509,2046,-24728,10380])])
sage: is_Q_curve(E, certificate=True, verbose=True)                             # needs sage.rings.number_field
Checking whether Elliptic Curve defined by y^2 + (a^3+a^2-4*a-3)*x*y + (a^2-2)*y = x^3 + (-a^2-a+4)*x^2 + (-178*a^3+138*a^2+778*a-621)*x + (10380*a^3-24728*a^2+2046*a+9509) over Number Field in a with defining polynomial x^4 - 5*x^2 + 3 is a Q-curve
No: inconsistency at the 2 primes dividing 3
- potentially multiplicative: [True, False]
(False, 3)

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.Qcurves import is_Q_curve
>>> R = PolynomialRing(QQ, names=('x',)); (x,) = R._first_ngens(1)
>>> K = NumberField(R([Integer(3), Integer(0), -Integer(5), Integer(0), Integer(1)]), names=('a',)); (a,) = K._first_ngens(1)# needs sage.rings.number_field
>>> E = EllipticCurve([K([-Integer(3),-Integer(4),Integer(1),Integer(1)]), K([Integer(4),-Integer(1),-Integer(1),Integer(0)]), K([-Integer(2),Integer(0),Integer(1),Integer(0)]),         # needs sage.rings.number_field
...                    K([-Integer(621),Integer(778),Integer(138),-Integer(178)]), K([Integer(9509),Integer(2046),-Integer(24728),Integer(10380)])])
>>> is_Q_curve(E, certificate=True, verbose=True)                             # needs sage.rings.number_field
Checking whether Elliptic Curve defined by y^2 + (a^3+a^2-4*a-3)*x*y + (a^2-2)*y = x^3 + (-a^2-a+4)*x^2 + (-178*a^3+138*a^2+778*a-621)*x + (10380*a^3-24728*a^2+2046*a+9509) over Number Field in a with defining polynomial x^4 - 5*x^2 + 3 is a Q-curve
No: inconsistency at the 2 primes dividing 3
- potentially multiplicative: [True, False]
(False, 3)

A non-\(\QQ\)-curve over a quadratic field. The local data at bad primes is consistent, but the local test at good primes above \(13\) is not:

sage: K.<a> = NumberField(R([-10, 0, 1]))                                       # needs sage.rings.number_field
sage: E = EllipticCurve([K([0,1]), K([-1,-1]), K([0,0]),                        # needs sage.rings.number_field
....:                    K([-236,40]), K([-1840,464])])
sage: is_Q_curve(E, certificate=True, verbose=True)                             # needs sage.rings.number_field
Checking whether Elliptic Curve defined by y^2 + a*x*y = x^3 + (-a-1)*x^2 + (40*a-236)*x + (464*a-1840) over Number Field in a with defining polynomial x^2 - 10 is a Q-curve
Applying local tests at good primes above p<=100
No: inconsistency at the 2 ordinary primes dividing 13
- Frobenius discriminants mod squares: [-1, -3]
No: local test at p=13 failed
(False, 13)

>>> from sage.all import *
>>> K = NumberField(R([-Integer(10), Integer(0), Integer(1)]), names=('a',)); (a,) = K._first_ngens(1)# needs sage.rings.number_field
>>> E = EllipticCurve([K([Integer(0),Integer(1)]), K([-Integer(1),-Integer(1)]), K([Integer(0),Integer(0)]),                        # needs sage.rings.number_field
...                    K([-Integer(236),Integer(40)]), K([-Integer(1840),Integer(464)])])
>>> is_Q_curve(E, certificate=True, verbose=True)                             # needs sage.rings.number_field
Checking whether Elliptic Curve defined by y^2 + a*x*y = x^3 + (-a-1)*x^2 + (40*a-236)*x + (464*a-1840) over Number Field in a with defining polynomial x^2 - 10 is a Q-curve
Applying local tests at good primes above p<=100
No: inconsistency at the 2 ordinary primes dividing 13
- Frobenius discriminants mod squares: [-1, -3]
No: local test at p=13 failed
(False, 13)

A quadratic \(\QQ\)-curve with CM discriminant \(-15\) (\(j\)-invariant not in \(\QQ\)):

sage: from sage.schemes.elliptic_curves.Qcurves import is_Q_curve
sage: R.<x> = PolynomialRing(QQ)
sage: K.<a> = NumberField(R([-1, -1, 1]))                                       # needs sage.rings.number_field
sage: E = EllipticCurve([K([1,0]), K([-1,0]), K([0,1]), K([0,-2]), K([0,1])])   # needs sage.rings.number_field
sage: is_Q_curve(E, certificate=True, verbose=True)                             # needs sage.rings.number_field
Checking whether Elliptic Curve defined by y^2 + x*y + a*y = x^3 + (-1)*x^2 + (-2*a)*x + a over Number Field in a with defining polynomial x^2 - x - 1 is a Q-curve
Yes: E is CM (discriminant -15)
(True, {'CM': -15})

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.Qcurves import is_Q_curve
>>> R = PolynomialRing(QQ, names=('x',)); (x,) = R._first_ngens(1)
>>> K = NumberField(R([-Integer(1), -Integer(1), Integer(1)]), names=('a',)); (a,) = K._first_ngens(1)# needs sage.rings.number_field
>>> E = EllipticCurve([K([Integer(1),Integer(0)]), K([-Integer(1),Integer(0)]), K([Integer(0),Integer(1)]), K([Integer(0),-Integer(2)]), K([Integer(0),Integer(1)])])   # needs sage.rings.number_field
>>> is_Q_curve(E, certificate=True, verbose=True)                             # needs sage.rings.number_field
Checking whether Elliptic Curve defined by y^2 + x*y + a*y = x^3 + (-1)*x^2 + (-2*a)*x + a over Number Field in a with defining polynomial x^2 - x - 1 is a Q-curve
Yes: E is CM (discriminant -15)
(True, {'CM': -15})

An example over \(\QQ(\sqrt{2},\sqrt{3})\). The \(j\)-invariant is in \(\QQ(\sqrt{6})\), so computations will be done over that field, and in fact there is an isogenous curve with rational \(j\), so we have a so-called rational \(\QQ\)-curve:

sage: # needs sage.rings.number_field
sage: K.<a> = NumberField(R([1, 0, -4, 0, 1]))
sage: E = EllipticCurve([K([-2,-4,1,1]), K([0,1,0,0]), K([0,1,0,0]),
....:                    K([-4780,9170,1265,-2463]),
....:                    K([163923,-316598,-43876,84852])])
sage: flag, cert = is_Q_curve(E, certificate=True)
sage: flag
True
sage: cert
{'CM': 0, 'N': 1, 'core_degs': [1], 'core_poly': x - 85184/3, 'r': 0, 'rho': 0}

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> K = NumberField(R([Integer(1), Integer(0), -Integer(4), Integer(0), Integer(1)]), names=('a',)); (a,) = K._first_ngens(1)
>>> E = EllipticCurve([K([-Integer(2),-Integer(4),Integer(1),Integer(1)]), K([Integer(0),Integer(1),Integer(0),Integer(0)]), K([Integer(0),Integer(1),Integer(0),Integer(0)]),
...                    K([-Integer(4780),Integer(9170),Integer(1265),-Integer(2463)]),
...                    K([Integer(163923),-Integer(316598),-Integer(43876),Integer(84852)])])
>>> flag, cert = is_Q_curve(E, certificate=True)
>>> flag
True
>>> cert
{'CM': 0, 'N': 1, 'core_degs': [1], 'core_poly': x - 85184/3, 'r': 0, 'rho': 0}

Over the same field, a so-called strict \(\QQ\)-curve which is not isogenous to one with rational \(j\), but whose core field is quadratic. In fact the isogeny class over \(K\) consists of \(6\) curves, four with conjugate quartic \(j\)-invariants and \(2\) with quadratic conjugate \(j\)-invariants in \(\QQ(\sqrt{3})\) (but which are not base-changes from the quadratic subfield):

sage: # needs sage.rings.number_field
sage: E = EllipticCurve([K([0,-3,0,1]), K([1,4,0,-1]), K([0,0,0,0]),
....:                    K([-2,-16,0,4]), K([-19,-32,4,8])])
sage: flag, cert = is_Q_curve(E, certificate=True)
sage: flag
True
sage: cert
{'CM': 0,
 'N': 2,
 'core_degs': [1, 2],
 'core_poly': x^2 - 840064*x + 1593413632,
 'r': 1,
 'rho': 1}

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> E = EllipticCurve([K([Integer(0),-Integer(3),Integer(0),Integer(1)]), K([Integer(1),Integer(4),Integer(0),-Integer(1)]), K([Integer(0),Integer(0),Integer(0),Integer(0)]),
...                    K([-Integer(2),-Integer(16),Integer(0),Integer(4)]), K([-Integer(19),-Integer(32),Integer(4),Integer(8)])])
>>> flag, cert = is_Q_curve(E, certificate=True)
>>> flag
True
>>> cert
{'CM': 0,
 'N': 2,
 'core_degs': [1, 2],
 'core_poly': x^2 - 840064*x + 1593413632,
 'r': 1,
 'rho': 1}