Testing whether elliptic curves over number fields are $\mathbf{Q}$-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 $\mathbf{Q}$-curve, or (True, $0$) if all local tests at primes between oldB and B fail to prove that $E$ is not a $\mathbf{Q}$-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-$\mathbf{Q}$-curve over a quartic field (with LMFDB label ‘4.4.8112.1-12.1-a1’) fails this test at $p=13$:

Sage

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)

Sage Live

from sage.schemes.elliptic_curves.Qcurves import Step4Test
R.<x> = PolynomialRing(QQ)
K.<a> = NumberField(R([3, 0, -5, 0, 1]))                                  # needs sage.rings.number_field
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])])
Step4Test(E, 100, verbose=True)                                           # needs sage.rings.number_field

Python

>>> 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 $\mathbf{Q}$-curve over a sextic field (with LMFDB label ‘6.6.1259712.1-64.1-a6’) passes this test for all $p<100$:

Sage

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 Live

from sage.schemes.elliptic_curves.Qcurves import Step4Test
R.<x> = PolynomialRing(QQ)
K.<a> = NumberField(R([-3, 0, 9, 0, -6, 0, 1]))                           # needs sage.rings.number_field
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])])
Step4Test(E, 100, verbose=True)                                           # needs sage.rings.number_field

Python

>>> 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)

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 $\mathbf{Q}$ of 2-power degree all of whose roots are in the list.

EXAMPLES:

Sage

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 Live

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

Python

>>> 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)

Return whether E is a $\mathbf{Q}$-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 $\mathbf{Q}$-curve property.

OUTPUT:

If certificate is False: either True (if $E$ is a $\mathbf{Q}$-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 $\mathbf{Q}$-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 $\mathbf{Q}$-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 $E^{\prime }$ isogenous to $E$ with $j(E^{\prime })=j\in \mathbf{Q}$, 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 $\mathbf{Q}$-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 $\mathbf{Q}$-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 $K=\mathbf{Q}(j(E))$. 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 $p\le maxp$, $p\nmid N$, 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 $\mathbf{Q}$ and a CM curve over $\mathbf{Q}$ are both trivially $\mathbf{Q}$-curves:

Sage

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}

Sage Live

from sage.schemes.elliptic_curves.Qcurves import is_Q_curve
E = EllipticCurve([1,2,3,4,5])
flag, cert = is_Q_curve(E, certificate=True)
flag
cert
E = EllipticCurve(j=8000)
flag, cert = is_Q_curve(E, certificate=True)
flag
cert

Python

>>> 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-$\mathbf{Q}$-curve over a quartic field. The local data at bad primes above $3$ is inconsistent:

Sage

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)

Sage Live

from sage.schemes.elliptic_curves.Qcurves import is_Q_curve
R.<x> = PolynomialRing(QQ)
K.<a> = NumberField(R([3, 0, -5, 0, 1]))                                  # needs sage.rings.number_field
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])])
is_Q_curve(E, certificate=True, verbose=True)                             # needs sage.rings.number_field

Python

>>> 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-$\mathbf{Q}$-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

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)

Sage Live

K.<a> = NumberField(R([-10, 0, 1]))                                       # needs sage.rings.number_field
E = EllipticCurve([K([0,1]), K([-1,-1]), K([0,0]),                        # needs sage.rings.number_field
                   K([-236,40]), K([-1840,464])])
is_Q_curve(E, certificate=True, verbose=True)                             # needs sage.rings.number_field

Python

>>> 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 $\mathbf{Q}$-curve with CM discriminant $-15$ ($j$-invariant not in $\mathbf{Q}$):

Sage

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})

Sage Live

from sage.schemes.elliptic_curves.Qcurves import is_Q_curve
R.<x> = PolynomialRing(QQ)
K.<a> = NumberField(R([-1, -1, 1]))                                       # needs sage.rings.number_field
E = EllipticCurve([K([1,0]), K([-1,0]), K([0,1]), K([0,-2]), K([0,1])])   # needs sage.rings.number_field
is_Q_curve(E, certificate=True, verbose=True)                             # needs sage.rings.number_field

Python

>>> 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 $\mathbf{Q}(\sqrt{2},\sqrt{3})$. The $j$-invariant is in $\mathbf{Q}(\sqrt{6})$, 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 $\mathbf{Q}$-curve:

Sage

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}

Sage Live

# needs sage.rings.number_field
K.<a> = NumberField(R([1, 0, -4, 0, 1]))
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])])
flag, cert = is_Q_curve(E, certificate=True)
flag
cert

Python

>>> 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 $\mathbf{Q}$-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 $\mathbf{Q}(\sqrt{3})$ (but which are not base-changes from the quadratic subfield):

Sage

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}

Sage Live

# needs sage.rings.number_field
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])])
flag, cert = is_Q_curve(E, certificate=True)
flag
cert

Python

>>> 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}

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