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)#
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 testverbose(boolean, defaultFalse): 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 betweenoldBandBfail 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)
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)
- sage.schemes.elliptic_curves.Qcurves.conjugacy_test(jlist, verbose=False)#
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 fieldverbose(boolean, defaultFalse): 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]
- sage.schemes.elliptic_curves.Qcurves.is_Q_curve(E, maxp=100, certificate=False, verbose=False)#
Return whether
Eis 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 returnFalsefaster.certificate(bool, defaultFalse): ifTruethen a second value is returned giving a certificate for the \(\QQ\)-curve property.
OUTPUT:
If
certificateisFalse: eitherTrue(if \(E\) is a \(\QQ\)-curve), orFalse.If
certificateisTrue: 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, returnFalse.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
Trueif the set of \(j\)-invariants of curves in the class contains a complete Galois orbit, otherwise returnFalse.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}
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)
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)
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})
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}
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}