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 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]))
sage: E = EllipticCurve([K([-3,-4,1,1]),K([4,-1,-1,0]),K([-2,0,1,0]),K([-621,778,138,-178]),K([9509,2046,-24728,10380])])
sage: Step4Test(E, 100, verbose=True)
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]))
sage: E = EllipticCurve([K([1,-3,0,1,0,0]),K([5,-3,-6,1,1,0]),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)
(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 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: 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 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}

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]))
sage: E = EllipticCurve([K([-3,-4,1,1]),K([4,-1,-1,0]),K([-2,0,1,0]),K([-621,778,138,-178]),K([9509,2046,-24728,10380])])
sage: is_Q_curve(E, certificate=True, verbose=True)
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]))
sage: E = EllipticCurve([K([0,1]),K([-1,-1]),K([0,0]),K([-236,40]),K([-1840,464])])
sage: is_Q_curve(E, certificate=True, verbose=True)
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]))
sage: E = EllipticCurve([K([1,0]),K([-1,0]),K([0,1]),K([0,-2]),K([0,1])])
sage: is_Q_curve(E, certificate=True, verbose=True)
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: 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: 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}