def is_cm_j_invariant(j):
"""
Return whether or not this is a CM `j`-invariant.
INPUT:
- ``j`` -- an element of a number field `K`
OUTPUT:
A pair (bool, (d,f)) which is either (False, None) if `j` is not a
CM j-invariant or (True, (d,f)) if `j` is the `j`-invariant of the
imaginary quadratic order of discriminant `D=df^2` where `d` is
the associated fundamental discriminant and `f` the index.
.. note::
The current implementation makes use of the classification of
all orders of class number up to 100, and hence will raise an
error if `j` is an algebraic integer of degree greater than
this. It would be possible to implement a more general
version, using the fact that `d` must be supported on the
primes dividing the discriminant of the minimal polynomial of
`j`.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.cm import is_cm_j_invariant
sage: is_cm_j_invariant(0)
(True, (-3, 1))
sage: is_cm_j_invariant(8000)
(True, (-8, 1))
sage: K.<a> = QuadraticField(5)
sage: is_cm_j_invariant(282880*a + 632000)
(True, (-20, 1))
sage: K.<a> = NumberField(x^3 - 2)
sage: is_cm_j_invariant(31710790944000*a^2 + 39953093016000*a + 50337742902000)
(True, (-3, 6))
TESTS::
sage: from sage.schemes.elliptic_curves.cm import is_cm_j_invariant
sage: all([is_cm_j_invariant(j) == (True, (d,f)) for d,f,j in cm_j_invariants_and_orders(QQ)])
True
"""
from sage.rings.all import NumberFieldElement
if not isinstance(j, NumberFieldElement) and not j in QQ:
raise NotImplementedError("is_cm_j_invariant() is only implemented for number field elements")
if not j.is_integral():
return False, None
jpol = PolynomialRing(QQ,'x')([-j,1]) if j in QQ else j.absolute_minpoly()
h = jpol.degree()
if h>100:
raise NotImplementedError("CM data only available for class numbers up to 100")
for d,f in cm_orders(h):
if jpol == hilbert_class_polynomial(d*f**2):
return True, (d,f)
return False, None
def is_cm_j_invariant(j, method='new'):
"""
Return whether or not this is a CM `j`-invariant.
INPUT:
- ``j`` -- an element of a number field `K`
OUTPUT:
A pair (bool, (d,f)) which is either (False, None) if `j` is not a
CM j-invariant or (True, (d,f)) if `j` is the `j`-invariant of the
imaginary quadratic order of discriminant `D=df^2` where `d` is
the associated fundamental discriminant and `f` the index.
.. note::
The current implementation makes use of the classification of
all orders of class number up to 100, and hence will raise an
error if `j` is an algebraic integer of degree greater than
this. It would be possible to implement a more general
version, using the fact that `d` must be supported on the
primes dividing the discriminant of the minimal polynomial of
`j`.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.cm import is_cm_j_invariant
sage: is_cm_j_invariant(0)
(True, (-3, 1))
sage: is_cm_j_invariant(8000)
(True, (-8, 1))
sage: K.<a> = QuadraticField(5)
sage: is_cm_j_invariant(282880*a + 632000)
(True, (-20, 1))
sage: K.<a> = NumberField(x^3 - 2)
sage: is_cm_j_invariant(31710790944000*a^2 + 39953093016000*a + 50337742902000)
(True, (-3, 6))
TESTS::
sage: from sage.schemes.elliptic_curves.cm import is_cm_j_invariant
sage: all([is_cm_j_invariant(j) == (True, (d,f)) for d,f,j in cm_j_invariants_and_orders(QQ)])
True
"""
# First we check that j is an algebraic number:
from sage.rings.all import NumberFieldElement, NumberField
if not isinstance(j, NumberFieldElement) and not j in QQ:
raise NotImplementedError(
"is_cm_j_invariant() is only implemented for number field elements"
)
# for j in ZZ we have a lookup-table:
if j in ZZ:
j = ZZ(j)
table = dict([(jj, (d, f))
for d, f, jj in cm_j_invariants_and_orders(QQ)])
if j in table:
return True, table[j]
return False, None
# Otherwise if j is in Q then it is not integral so is not CM:
if j in QQ:
return False, None
# Now j has degree at least 2. If it is not integral so is not CM:
if not j.is_integral():
return False, None
# Next we find its minimal polynomial and degree h, and if h is
# less than the degree of j.parent() we recreate j as an element
# of Q(j):
jpol = PolynomialRing(QQ, 'x')([-j, 1
]) if j in QQ else j.absolute_minpoly()
h = jpol.degree()
# This will be used as a fall-back if we cannot determine the
# result using local data. For this to be necessary there would
# have to be very few primes of degree 1 and norm under 1000,
# since we only need to find one prime of degree 1, good
# reduction for which a_P is nonzero.
if method == 'old':
if h > 100:
raise NotImplementedError(
"CM data only available for class numbers up to 100")
for d, f in cm_orders(h):
if jpol == hilbert_class_polynomial(d * f**2):
return True, (d, f)
return False, None
# replace j by a clone whose parent is Q(j), if necessary:
K = j.parent()
if h < K.absolute_degree():
K = NumberField(jpol, 'j')
j = K.gen()
# Construct an elliptic curve with j-invariant j, with
# integral model:
from sage.schemes.elliptic_curves.all import EllipticCurve
E = EllipticCurve(j=j).integral_model()
D = E.discriminant()
prime_bound = 1000 # test primes of degree 1 up to this norm
max_primes = 20 # test at most this many primes
num_prime = 0
cmd = 0
cmf = 0
# Test primes of good reduction. If E has CM then for half the
# primes P we will have a_P=0, and for all other prime P the CM
# field is Q(sqrt(a_P^2-4N(P))). Hence if these fields are
# different for two primes then E does not have CM. If they are
# all equal for the primes tested, then we have a candidate CM
# field. Moreover the discriminant of the endomorphism ring
# divides all the values a_P^2-4N(P), since that is the
# discriminant of the order containing the Frobenius at P. So we
# end up with a finite number (usually one) of candidate
# discriminants to test. Each is tested by checking that its class
# number is h, and if so then that j is a root of its Hilbert
# class polynomial. In practice non CM curves will be eliminated
# by the local test at a small number of primes (probably just 2).
for P in K.primes_of_degree_one_iter(prime_bound):
if num_prime > max_primes:
if cmd: # we have a candidate CM field already
break
else: # we need to try more primes
max_primes *= 2
if D.valuation(P) > 0: # skip bad primes
continue
aP = E.reduction(P).trace_of_frobenius()
if aP == 0: # skip supersingular primes
continue
num_prime += 1
DP = aP**2 - 4 * P.norm()
dP = DP.squarefree_part()
fP = ZZ(DP // dP).isqrt()
if cmd == 0: # first one, so store d and f
cmd = dP
cmf = fP
elif cmd != dP: # inconsistent with previous
return False, None
else: # consistent d, so update f
cmf = cmf.gcd(fP)
if cmd == 0: # no conclusion, we found no degree 1 primes, revert to old method
return is_cm_j_invariant(j, method='old')
# it looks like cm by disc cmd * f**2 where f divides cmf
if cmd % 4 != 1:
cmd = cmd * 4
cmf = cmf // 2
# Now we must check if h(cmd*f**2)==h for f|cmf; if so we check
# whether j is a root of the associated Hilbert class polynomial.
for f in cmf.divisors(): # only positive divisors
d = cmd * f**2
if h != d.class_number():
continue
pol = hilbert_class_polynomial(d)
if pol(j) == 0:
return True, (cmd, f)
return False, None
def is_cm_j_invariant(j):
"""
Return whether or not this is a CM `j`-invariant.
INPUT:
- ``j`` -- an element of a number field `K`
OUTPUT:
A pair (bool, (d,f)) which is either (False, None) if `j` is not a
CM j-invariant or (True, (d,f)) if `j` is the `j`-invariant of the
imaginary quadratic order of discriminant `D=df^2` where `d` is
the associated fundamental discriminant and `f` the index.
.. note::
The current implementation makes use of the classification of
all orders of class number up to 100, and hence will raise an
error if `j` is an algebraic integer of degree greater than
this. It would be possible to implement a more general
version, using the fact that `d` must be supported on the
primes dividing the discriminant of the minimal polynomial of
`j`.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.cm import is_cm_j_invariant
sage: is_cm_j_invariant(0)
(True, (-3, 1))
sage: is_cm_j_invariant(8000)
(True, (-8, 1))
sage: K.<a> = QuadraticField(5)
sage: is_cm_j_invariant(282880*a + 632000)
(True, (-20, 1))
sage: K.<a> = NumberField(x^3 - 2)
sage: is_cm_j_invariant(31710790944000*a^2 + 39953093016000*a + 50337742902000)
(True, (-3, 6))
TESTS::
sage: from sage.schemes.elliptic_curves.cm import is_cm_j_invariant
sage: all([is_cm_j_invariant(j) == (True, (d,f)) for d,f,j in cm_j_invariants_and_orders(QQ)])
True
"""
from sage.rings.all import NumberFieldElement
if not isinstance(j, NumberFieldElement) and not j in QQ:
raise NotImplementedError(
"is_cm_j_invariant() is only implemented for number field elements"
)
if not j.is_integral():
return False, None
jpol = PolynomialRing(QQ, 'x')([-j, 1
]) if j in QQ else j.absolute_minpoly()
h = jpol.degree()
if h > 100:
raise NotImplementedError(
"CM data only available for class numbers up to 100")
for d, f in cm_orders(h):
if jpol == hilbert_class_polynomial(d * f**2):
return True, (d, f)
return False, None
def is_cm_j_invariant(j, method='new'):
"""
Return whether or not this is a CM `j`-invariant.
INPUT:
- ``j`` -- an element of a number field `K`
OUTPUT:
A pair (bool, (d,f)) which is either (False, None) if `j` is not a
CM j-invariant or (True, (d,f)) if `j` is the `j`-invariant of the
imaginary quadratic order of discriminant `D=df^2` where `d` is
the associated fundamental discriminant and `f` the index.
.. note::
The current implementation makes use of the classification of
all orders of class number up to 100, and hence will raise an
error if `j` is an algebraic integer of degree greater than
this. It would be possible to implement a more general
version, using the fact that `d` must be supported on the
primes dividing the discriminant of the minimal polynomial of
`j`.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.cm import is_cm_j_invariant
sage: is_cm_j_invariant(0)
(True, (-3, 1))
sage: is_cm_j_invariant(8000)
(True, (-8, 1))
sage: K.<a> = QuadraticField(5)
sage: is_cm_j_invariant(282880*a + 632000)
(True, (-20, 1))
sage: K.<a> = NumberField(x^3 - 2)
sage: is_cm_j_invariant(31710790944000*a^2 + 39953093016000*a + 50337742902000)
(True, (-3, 6))
TESTS::
sage: from sage.schemes.elliptic_curves.cm import is_cm_j_invariant
sage: all([is_cm_j_invariant(j) == (True, (d,f)) for d,f,j in cm_j_invariants_and_orders(QQ)])
True
"""
# First we check that j is an algebraic number:
from sage.rings.all import NumberFieldElement, NumberField
if not isinstance(j, NumberFieldElement) and not j in QQ:
raise NotImplementedError("is_cm_j_invariant() is only implemented for number field elements")
# for j in ZZ we have a lookup-table:
if j in ZZ:
j = ZZ(j)
table = dict([(jj,(d,f)) for d,f,jj in cm_j_invariants_and_orders(QQ)])
if j in table:
return True, table[j]
return False, None
# Otherwise if j is in Q then it is not integral so is not CM:
if j in QQ:
return False, None
# Now j has degree at least 2. If it is not integral so is not CM:
if not j.is_integral():
return False, None
# Next we find its minimal polynomial and degree h, and if h is
# less than the degree of j.parent() we recreate j as an element
# of Q(j):
jpol = PolynomialRing(QQ,'x')([-j,1]) if j in QQ else j.absolute_minpoly()
h = jpol.degree()
# This will be used as a fall-back if we cannot determine the
# result using local data. For this to be necessary there would
# have to be very few primes of degree 1 and norm under 1000,
# since we only need to find one prime of degree 1, good
# reduction for which a_P is nonzero.
if method=='old':
if h>100:
raise NotImplementedError("CM data only available for class numbers up to 100")
for d,f in cm_orders(h):
if jpol == hilbert_class_polynomial(d*f**2):
return True, (d,f)
return False, None
# replace j by a clone whose parent is Q(j), if necessary:
K = j.parent()
if h < K.absolute_degree():
K = NumberField(jpol, 'j')
j = K.gen()
# Construct an elliptic curve with j-invariant j, with
# integral model:
from sage.schemes.elliptic_curves.all import EllipticCurve
E = EllipticCurve(j=j).integral_model()
D = E.discriminant()
prime_bound = 1000 # test primes of degree 1 up to this norm
max_primes = 20 # test at most this many primes
num_prime = 0
cmd = 0
cmf = 0
# Test primes of good reduction. If E has CM then for half the
# primes P we will have a_P=0, and for all other prime P the CM
# field is Q(sqrt(a_P^2-4N(P))). Hence if these fields are
# different for two primes then E does not have CM. If they are
# all equal for the primes tested, then we have a candidate CM
# field. Moreover the discriminant of the endomorphism ring
# divides all the values a_P^2-4N(P), since that is the
# discriminant of the order containing the Frobenius at P. So we
# end up with a finite number (usually one) of candidate
# discriminats to test. Each is tested by checking that its class
# number is h, and if so then that j is a root of its Hilbert
# class polynomial. In practice non CM curves will be eliminated
# by the local test at a small number of primes (probably just 2).
for P in K.primes_of_degree_one_iter(prime_bound):
if num_prime > max_primes:
if cmd: # we have a candidate CM field already
break
else: # we need to try more primes
max_primes *=2
if D.valuation(P)>0: # skip bad primes
continue
aP = E.reduction(P).trace_of_frobenius()
if aP == 0: # skip supersingular primes
continue
num_prime += 1
DP = aP**2 - 4*P.norm()
dP = DP.squarefree_part()
fP = ZZ(DP//dP).isqrt()
if cmd==0: # first one, so store d and f
cmd = dP
cmf = fP
elif cmd != dP: # inconsistent with previous
return False, None
else: # consistent d, so update f
cmf = cmf.gcd(fP)
if cmd==0: # no conclusion, we found no degree 1 primes, revert to old method
return is_cm_j_invariant(j, method='old')
# it looks like cm by disc cmd * f**2 where f divides cmf
if cmd%4!=1:
cmd = cmd*4
cmf = cmf//2
# Now we must check if h(cmd*f**2)==h for f|cmf; if so we check
# whether j is a root of the associated Hilbert class polynomial.
for f in cmf.divisors(): # only positive divisors
d = cmd*f**2
if h != d.class_number():
continue
pol = hilbert_class_polynomial(d)
if pol(j)==0:
return True, (cmd,f)
return False, None
