def is_Q_curve(E, maxp=100, certificate=False, verbose=False):
r"""
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}
"""
from sage.rings.number_field.number_field_base import is_NumberField
if verbose:
print("Checking whether {} is a Q-curve".format(E))
try:
assert is_NumberField(E.base_field())
except (AttributeError, AssertionError):
raise TypeError(
"{} must be an elliptic curve defined over a number field in is_Q_curve()"
)
from sage.rings.integer_ring import ZZ
from sage.arith.functions import lcm
from sage.libs.pari import pari
from sage.rings.number_field.number_field import NumberField
from sage.schemes.elliptic_curves.constructor import EllipticCurve
from sage.schemes.elliptic_curves.cm import cm_j_invariants_and_orders, is_cm_j_invariant
# Step 1
# all curves with rational j-invariant are Q-curves:
jE = E.j_invariant()
if jE in QQ:
if verbose:
print("Yes: j(E) is in QQ")
if certificate:
# test for CM
for d, f, j in cm_j_invariants_and_orders(QQ):
if jE == j:
return True, {'CM': d * f**2}
# else not CM
return True, {
'CM': ZZ(0),
'r': ZZ(0),
'rho': ZZ(0),
'N': ZZ(1),
'core_poly': polygen(QQ)
}
else:
return True
# CM curves are Q-curves:
flag, df = is_cm_j_invariant(jE)
if flag:
d, f = df
D = d * f**2
if verbose:
print("Yes: E is CM (discriminant {})".format(D))
if certificate:
return True, {'CM': D}
else:
return True
# Step 2: replace E by a curve defined over Q(j(E)):
K = E.base_field()
jpoly = jE.minpoly()
if jpoly.degree() < K.degree():
if verbose:
print("switching to smaller base field: j's minpoly is {}".format(
jpoly))
f = pari(jpoly).polredbest().sage({'x': jpoly.parent().gen()})
K2 = NumberField(f, 'b')
jE = jpoly.roots(K2)[0][0]
if verbose:
print("New j is {} over {}, with minpoly {}".format(
jE, K2, jE.minpoly()))
#assert jE.minpoly()==jpoly
E = EllipticCurve(j=jE)
K = K2
if verbose:
print("New test curve is {}".format(E))
# Step 3: check primes of bad reduction
NN = E.conductor().norm()
for p in NN.support():
Plist = K.primes_above(p)
if len(Plist) < 2:
continue
# pot_mult = potential multiplicative reduction
pot_mult = [jE.valuation(P) < 0 for P in Plist]
consistent = all(pot_mult) or not any(pot_mult)
if not consistent:
if verbose:
print("No: inconsistency at the {} primes dividing {}".format(
len(Plist), p))
print(" - potentially multiplicative: {}".format(pot_mult))
if certificate:
return False, p
else:
return False
# Step 4 check: primes P of good reduction above p<=B:
if verbose:
print("Applying local tests at good primes above p<={}".format(maxp))
res4, p = Step4Test(E, B=maxp, oldB=0, verbose=verbose)
if not res4:
if verbose:
print("No: local test at p={} failed".format(p))
if certificate:
return False, p
else:
return False
if verbose:
print("...all local tests pass for p<={}".format(maxp))
# Step 5: compute the (partial) K-isogeny class of E and test the
# set of j-invariants in the class:
C = E.isogeny_class(algorithm='heuristic', minimal_models=False)
jC = [E2.j_invariant() for E2 in C]
centrejpols = conjugacy_test(jC, verbose=verbose)
if centrejpols:
if verbose:
print(
"Yes: the isogeny class contains a complete conjugacy class of j-invariants"
)
if certificate:
for f in centrejpols:
rho = f.degree().valuation(2)
centre_indices = [i for i, j in enumerate(jC) if f(j) == 0]
M = C.matrix()
core_degs = [M[centre_indices[0], i] for i in centre_indices]
level = lcm(core_degs)
if level.is_squarefree():
r = len(level.prime_divisors())
cert = {
'CM': ZZ(0),
'core_poly': f,
'rho': rho,
'r': r,
'N': level,
'core_degs': core_degs
}
return True, cert
print("No central curve found")
else:
return True
# Now we are undecided. This can happen if either (1) E is not a
# Q-curve but we did not use enough primes in Step 4 to detect
# this, or (2) E is a Q-curve but in Step 5 we did not compute the
# complete isogeny class. Case (2) is most unlikely since the
# heuristic bound used in computing isogeny classes means that we
# have all isogenous curves linked to E by an isogeny of degree
# supported on primes<1000.
# We first rerun Step 4 with a larger bound.
xmaxp = 10 * maxp
if verbose:
print(
"Undecided after first round, so we apply more local tests, up to {}"
.format(xmaxp))
res4, p = Step4Test(E, B=xmaxp, oldB=maxp, verbose=verbose)
if not res4:
if verbose:
print("No: local test at p={} failed".format(p))
if certificate:
return False, p
else:
return False
# Now we rerun Step 5 using a rigorous computation of the complete
# isogeny class. This will probably contain no more curves than
# before, in which case -- since we already tested that the set of
# j-invariants does not contain a complete Galois conjugacy class
# -- we can deduce that E is not a Q-curve.
if verbose:
print("...all local tests pass for p<={}".format(xmaxp))
print("We now compute the complete isogeny class...")
Cfull = E.isogeny_class(minimal_models=False)
jCfull = [E2.j_invariant() for E2 in Cfull]
if len(jC) == len(jCfull):
if verbose:
print("...and find that we already had the complete class:so No")
if certificate:
return False, 0
else:
return False
if verbose:
print(
"...and find that the class contains {} curves, not just the {} we computed originally"
.format(len(jCfull), len(jC)))
centrejpols = conjugacy_test(jCfull, verbose=verbose)
if cert:
if verbose:
print(
"Yes: the isogeny class contains a complete conjugacy class of j-invariants"
)
if certificate:
return True, centrejpols
else:
return True
if verbose:
print(
"No: the isogeny class does *not* contain a complete conjugacy class of j-invariants"
)
if certificate:
return False, 0
else:
return False
