def has_rational_point(self, point = False,
algorithm = 'default', read_cache = True):
r"""
Returns True if and only if the conic ``self``
has a point over its base field `B`.
If ``point`` is True, then returns a second output, which is
a rational point if one exists.
Points are cached whenever they are found. Cached information
is used if and only if ``read_cache`` is True.
EXAMPLES:
sage: Conic(RR, [1, 1, 1]).has_rational_point()
False
sage: Conic(CC, [1, 1, 1]).has_rational_point()
True
sage: Conic(RR, [1, 2, -3]).has_rational_point(point = True)
(True, (1.73205080756888 : 0.000000000000000 : 1.00000000000000))
"""
if read_cache:
if self._rational_point is not None:
if point:
return True, self._rational_point
else:
return True
B = self.base_ring()
if is_ComplexField(B):
if point:
[_,_,_,d,e,f] = self._coefficients
if d == 0:
return True, self.point([0,1,0])
return True, self.point([0, ((e**2-4*d*f).sqrt()-e)/(2*d), 1],
check = False)
return True
if is_RealField(B):
D, T = self.diagonal_matrix()
[a, b, c] = [D[0,0], D[1,1], D[2,2]]
if a == 0:
ret = True, self.point(T*vector([1,0,0]), check = False)
elif a*c <= 0:
ret = True, self.point(T*vector([(-c/a).sqrt(),0,1]),
check = False)
elif b == 0:
ret = True, self.point(T*vector([0,1,0]), check = False)
elif b*c <= 0:
ret = True, self.point(T*vector([0,(-c/b).sqrt(),0,1]),
check = False)
else:
ret = False, None
if point:
return ret
return ret[0]
raise NotImplementedError, "has_rational_point not implemented for " \
"conics over base field %s" % B
def has_rational_point(self, point = False,
algorithm = 'default', read_cache = True):
r"""
Returns True if and only if the conic ``self``
has a point over its base field `B`.
If ``point`` is True, then returns a second output, which is
a rational point if one exists.
Points are cached whenever they are found. Cached information
is used if and only if ``read_cache`` is True.
ALGORITHM:
The parameter ``algorithm`` specifies the algorithm
to be used:
- ``'default'`` -- If the base field is real or complex,
use an elementary native Sage implementation.
- ``'magma'`` (requires Magma to be installed) --
delegates the task to the Magma computer algebra
system.
EXAMPLES:
sage: Conic(RR, [1, 1, 1]).has_rational_point()
False
sage: Conic(CC, [1, 1, 1]).has_rational_point()
True
sage: Conic(RR, [1, 2, -3]).has_rational_point(point = True)
(True, (1.73205080756888 : 0.000000000000000 : 1.00000000000000))
Conics over polynomial rings can not be solved yet without Magma::
sage: R.<t> = QQ[]
sage: C = Conic([-2,t^2+1,t^2-1])
sage: C.has_rational_point()
Traceback (most recent call last):
...
NotImplementedError: has_rational_point not implemented for conics over base field Fraction Field of Univariate Polynomial Ring in t over Rational Field
But they can be solved with Magma::
sage: C.has_rational_point(algorithm='magma') # optional - magma
True
sage: C.has_rational_point(algorithm='magma', point=True) # optional - magma
(True, (t : 1 : 1))
sage: D = Conic([t,1,t^2])
sage: D.has_rational_point(algorithm='magma') # optional - magma
False
TESTS:
One of the following fields comes with an embedding into the complex
numbers, one does not. Check that they are both handled correctly by
the Magma interface.::
sage: K.<i> = QuadraticField(-1)
sage: K.coerce_embedding()
Generic morphism:
From: Number Field in i with defining polynomial x^2 + 1
To: Complex Lazy Field
Defn: i -> 1*I
sage: Conic(K, [1,1,1]).rational_point(algorithm='magma') # optional - magma
(-i : 1 : 0)
sage: x = QQ['x'].gen()
sage: L.<i> = NumberField(x^2+1, embedding=None)
sage: Conic(L, [1,1,1]).rational_point(algorithm='magma') # optional - magma
(-i : 1 : 0)
sage: L == K
False
"""
if read_cache:
if self._rational_point is not None:
if point:
return True, self._rational_point
else:
return True
B = self.base_ring()
if algorithm == 'magma':
from sage.interfaces.magma import magma
M = magma(self)
b = M.HasRationalPoint().sage()
if not point:
return b
if not b:
return False, None
M_pt = M.HasRationalPoint(nvals=2)[1]
# Various attempts will be made to convert `pt` to
# a Sage object. The end result will always be checked
# by self.point().
pt = [M_pt[1], M_pt[2], M_pt[3]]
# The first attempt is to use sequences. This is efficient and
# succeeds in cases where the Magma interface fails to convert
# number field elements, because embeddings between number fields
# may be lost on conversion to and from Magma.
# This should deal with all absolute number fields.
try:
return True, self.point([B(c.Eltseq().sage()) for c in pt])
except TypeError:
pass
# The second attempt tries to split Magma elements into
# numerators and denominators first. This is neccessary
# for the field of rational functions, because (at the moment of
# writing) fraction field elements are not converted automatically
# from Magma to Sage.
try:
return True, self.point( \
[B(c.Numerator().sage()/c.Denominator().sage()) for c in pt])
except (TypeError, NameError):
pass
# Finally, let the Magma interface handle conversion.
try:
return True, self.point([B(c.sage()) for c in pt])
except (TypeError, NameError):
pass
raise NotImplementedError, "No correct conversion implemented for converting the Magma point %s on %s to a correct Sage point on self (=%s)" % (M_pt, M, self)
if algorithm != 'default':
raise ValueError, "Unknown algorithm: %s" % algorithm
if is_ComplexField(B):
if point:
[_,_,_,d,e,f] = self._coefficients
if d == 0:
return True, self.point([0,1,0])
return True, self.point([0, ((e**2-4*d*f).sqrt()-e)/(2*d), 1],
check = False)
return True
if is_RealField(B):
D, T = self.diagonal_matrix()
[a, b, c] = [D[0,0], D[1,1], D[2,2]]
if a == 0:
ret = True, self.point(T*vector([1,0,0]), check = False)
elif a*c <= 0:
ret = True, self.point(T*vector([(-c/a).sqrt(),0,1]),
check = False)
elif b == 0:
ret = True, self.point(T*vector([0,1,0]), check = False)
elif b*c <= 0:
ret = True, self.point(T*vector([0,(-c/b).sqrt(),0,1]),
check = False)
else:
ret = False, None
if point:
return ret
return ret[0]
raise NotImplementedError, "has_rational_point not implemented for " \
"conics over base field %s" % B
def isogenies_prime_degree(self, l=None):
"""
Generic code, valid for all fields, for those l for which the
modular curve has genus 0.
INPUT:
- ``l`` -- either None, a prime or a list of primes, from [2,3,5,7,13].
OUTPUT:
(list) All `l`-isogenies for the given `l` with domain self.
METHOD:
Calls the generic function
``isogenies_prime_degree_genus_0()``. This requires that
certain operations have been implemented over the base field,
such as root-finding for univariate polynomials.
EXAMPLES::
sage: F = QQbar
sage: E = EllipticCurve(F, [1,18]); E
Elliptic Curve defined by y^2 = x^3 + x + 18 over Algebraic Field
sage: F = CC
sage: E = EllipticCurve(F, [1,18]); E
Elliptic Curve defined by y^2 = x^3 + 1.00000000000000*x + 18.0000000000000 over Complex Field with 53 bits of precision
sage: E.isogenies_prime_degree(11)
Traceback (most recent call last):
...
NotImplementedError: This code could be implemented for general complex fields, but has not been yet.
Examples over finite fields::
sage: E = EllipticCurve(GF(next_prime(1000000)), [7,8])
sage: E.isogenies_prime_degree()
[Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 970389*x + 794257 over Finite Field of size 1000003, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 29783*x + 206196 over Finite Field of size 1000003, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 999960*x + 78 over Finite Field of size 1000003, Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 878063*x + 845666 over Finite Field of size 1000003, Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 375648*x + 342776 over Finite Field of size 1000003]
sage: E.isogenies_prime_degree(2)
[Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 970389*x + 794257 over Finite Field of size 1000003, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 29783*x + 206196 over Finite Field of size 1000003, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 999960*x + 78 over Finite Field of size 1000003]
sage: E.isogenies_prime_degree(3)
[]
sage: E.isogenies_prime_degree(5)
[]
sage: E.isogenies_prime_degree(7)
[]
sage: E.isogenies_prime_degree(13)
[Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 878063*x + 845666 over Finite Field of size 1000003,
Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 375648*x + 342776 over Finite Field of size 1000003]
sage: E.isogenies_prime_degree([2, 3, 5, 7, 13])
[Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 970389*x + 794257 over Finite Field of size 1000003, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 29783*x + 206196 over Finite Field of size 1000003, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 999960*x + 78 over Finite Field of size 1000003, Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 878063*x + 845666 over Finite Field of size 1000003, Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 375648*x + 342776 over Finite Field of size 1000003]
sage: E.isogenies_prime_degree([2, 4])
Traceback (most recent call last):
...
ValueError: 4 is not prime.
sage: E.isogenies_prime_degree([2, 29])
Traceback (most recent call last):
...
NotImplementedError: Over general fields, only isogenies of degree 2, 3, 5, 7 or 13 have been implemented.
sage: E.isogenies_prime_degree(4)
Traceback (most recent call last):
...
ValueError: 4 is not prime.
sage: E.isogenies_prime_degree(11)
Traceback (most recent call last):
...
NotImplementedError: Over general fields, only isogenies of degree 2, 3, 5, 7 or 13 have been implemented.
sage: E = EllipticCurve(GF(17),[2,0])
sage: E.isogenies_prime_degree(3)
[]
sage: E.isogenies_prime_degree(2)
[Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 2*x over Finite Field of size 17 to Elliptic Curve defined by y^2 = x^3 + 9*x over Finite Field of size 17, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 2*x over Finite Field of size 17 to Elliptic Curve defined by y^2 = x^3 + 5*x + 9 over Finite Field of size 17, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 2*x over Finite Field of size 17 to Elliptic Curve defined by y^2 = x^3 + 5*x + 8 over Finite Field of size 17]
sage: E = EllipticCurve(GF(13^4, 'a'),[2,8])
sage: E.isogenies_prime_degree(2)
[Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 2*x + 8 over Finite Field in a of size 13^4 to Elliptic Curve defined by y^2 = x^3 + 7*x + 4 over Finite Field in a of size 13^4, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 2*x + 8 over Finite Field in a of size 13^4 to Elliptic Curve defined by y^2 = x^3 + (8*a^3+2*a^2+7*a+5)*x + (12*a^3+3*a^2+4*a+4) over Finite Field in a of size 13^4, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 2*x + 8 over Finite Field in a of size 13^4 to Elliptic Curve defined by y^2 = x^3 + (5*a^3+11*a^2+6*a+11)*x + (a^3+10*a^2+9*a) over Finite Field in a of size 13^4]
sage: E.isogenies_prime_degree(3)
[Isogeny of degree 3 from Elliptic Curve defined by y^2 = x^3 + 2*x + 8 over Finite Field in a of size 13^4 to Elliptic Curve defined by y^2 = x^3 + 9*x + 11 over Finite Field in a of size 13^4]
sage: E.isogenies_prime_degree([2, 3, 5, 7, 13])
Traceback (most recent call last):
...
NotImplementedError: 2, 3, 5, 7 and 13-isogenies are not yet implemented in characteristic 2 and 3, and when the characteristic is the same as the degree of the isogeny.
sage: E.isogenies_prime_degree([2, 3, 5, 7])
[Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 2*x + 8 over Finite Field in a of size 13^4 to Elliptic Curve defined by y^2 = x^3 + 7*x + 4 over Finite Field in a of size 13^4, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 2*x + 8 over Finite Field in a of size 13^4 to Elliptic Curve defined by y^2 = x^3 + (8*a^3+2*a^2+7*a+5)*x + (12*a^3+3*a^2+4*a+4) over Finite Field in a of size 13^4, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 2*x + 8 over Finite Field in a of size 13^4 to Elliptic Curve defined by y^2 = x^3 + (5*a^3+11*a^2+6*a+11)*x + (a^3+10*a^2+9*a) over Finite Field in a of size 13^4, Isogeny of degree 3 from Elliptic Curve defined by y^2 = x^3 + 2*x + 8 over Finite Field in a of size 13^4 to Elliptic Curve defined by y^2 = x^3 + 9*x + 11 over Finite Field in a of size 13^4, Isogeny of degree 7 from Elliptic Curve defined by y^2 = x^3 + 2*x + 8 over Finite Field in a of size 13^4 to Elliptic Curve defined by y^2 = x^3 + 3*x + 11 over Finite Field in a of size 13^4, Isogeny of degree 7 from Elliptic Curve defined by y^2 = x^3 + 2*x + 8 over Finite Field in a of size 13^4 to Elliptic Curve defined by y^2 = x^3 + 9*x + 2 over Finite Field in a of size 13^4]
Examples over number fields (other than QQ)::
sage: QQroot2.<e> = NumberField(x^2-2)
sage: E = EllipticCurve(QQroot2,[1,1])
sage: E.isogenies_prime_degree(11)
Traceback (most recent call last):
...
NotImplementedError: Over general fields, only isogenies of degree 2, 3, 5, 7 or 13 have been implemented.
sage: E.isogenies_prime_degree(5)
[]
sage: E = EllipticCurve(QQroot2, [1,0,1,4, -6]); E
Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x + (-6) over Number Field in e with defining polynomial x^2 - 2
sage: E.isogenies_prime_degree(2)
[Isogeny of degree 2 from Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x + (-6) over Number Field in e with defining polynomial x^2 - 2 to Elliptic Curve defined by y^2 + x*y + y = x^3 + (-36)*x + (-70) over Number Field in e with defining polynomial x^2 - 2]
sage: E.isogenies_prime_degree(3)
[Isogeny of degree 3 from Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x + (-6) over Number Field in e with defining polynomial x^2 - 2 to Elliptic Curve defined by y^2 + x*y + y = x^3 + (-128/3)*x + 5662/27 over Number Field in e with defining polynomial x^2 - 2, Isogeny of degree 3 from Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x + (-6) over Number Field in e with defining polynomial x^2 - 2 to Elliptic Curve defined by y^2 + x*y + y = x^3 + (-171)*x + (-874) over Number Field in e with defining polynomial x^2 - 2]
"""
F = self.base_ring()
if rings.is_RealField(F):
raise NotImplementedError, "This code could be implemented for general real fields, but has not been yet."
if rings.is_ComplexField(F):
raise NotImplementedError, "This code could be implemented for general complex fields, but has not been yet."
if F == rings.QQbar:
raise NotImplementedError, "This code could be implemented for QQbar, but has not been yet."
from ell_curve_isogeny import isogenies_prime_degree_genus_0
if l is None:
l = [2, 3, 5, 7, 13]
if l in [2, 3, 5, 7, 13]:
return isogenies_prime_degree_genus_0(self, l)
if type(l) != list:
if l.is_prime():
raise NotImplementedError, "Over general fields, only isogenies of degree 2, 3, 5, 7 or 13 have been implemented."
else:
raise ValueError, "%s is not prime."%l
isogs = []
i = 0
while i<len(l):
isogenies = [f for f in self.isogenies_prime_degree(l[i]) if not f in isogs]
isogs.extend(isogenies)
i = i+1
return isogs
def has_rational_point(self,
point=False,
algorithm='default',
read_cache=True):
r"""
Returns True if and only if the conic ``self``
has a point over its base field `B`.
If ``point`` is True, then returns a second output, which is
a rational point if one exists.
Points are cached whenever they are found. Cached information
is used if and only if ``read_cache`` is True.
ALGORITHM:
The parameter ``algorithm`` specifies the algorithm
to be used:
- ``'default'`` -- If the base field is real or complex,
use an elementary native Sage implementation.
- ``'magma'`` (requires Magma to be installed) --
delegates the task to the Magma computer algebra
system.
EXAMPLES:
sage: Conic(RR, [1, 1, 1]).has_rational_point()
False
sage: Conic(CC, [1, 1, 1]).has_rational_point()
True
sage: Conic(RR, [1, 2, -3]).has_rational_point(point = True)
(True, (1.73205080756888 : 0.000000000000000 : 1.00000000000000))
Conics over polynomial rings can not be solved yet without Magma::
sage: R.<t> = QQ[]
sage: C = Conic([-2,t^2+1,t^2-1])
sage: C.has_rational_point()
Traceback (most recent call last):
...
NotImplementedError: has_rational_point not implemented for conics over base field Fraction Field of Univariate Polynomial Ring in t over Rational Field
But they can be solved with Magma::
sage: C.has_rational_point(algorithm='magma') # optional - magma
True
sage: C.has_rational_point(algorithm='magma', point=True) # optional - magma
(True, (t : 1 : 1))
sage: D = Conic([t,1,t^2])
sage: D.has_rational_point(algorithm='magma') # optional - magma
False
TESTS:
One of the following fields comes with an embedding into the complex
numbers, one does not. Check that they are both handled correctly by
the Magma interface.::
sage: K.<i> = QuadraticField(-1)
sage: K.coerce_embedding()
Generic morphism:
From: Number Field in i with defining polynomial x^2 + 1
To: Complex Lazy Field
Defn: i -> 1*I
sage: Conic(K, [1,1,1]).rational_point(algorithm='magma') # optional - magma
(-i : 1 : 0)
sage: x = QQ['x'].gen()
sage: L.<i> = NumberField(x^2+1, embedding=None)
sage: Conic(L, [1,1,1]).rational_point(algorithm='magma') # optional - magma
(-i : 1 : 0)
sage: L == K
False
"""
if read_cache:
if self._rational_point is not None:
if point:
return True, self._rational_point
else:
return True
B = self.base_ring()
if algorithm == 'magma':
from sage.interfaces.magma import magma
M = magma(self)
b = M.HasRationalPoint().sage()
if not point:
return b
if not b:
return False, None
M_pt = M.HasRationalPoint(nvals=2)[1]
# Various attempts will be made to convert `pt` to
# a Sage object. The end result will always be checked
# by self.point().
pt = [M_pt[1], M_pt[2], M_pt[3]]
# The first attempt is to use sequences. This is efficient and
# succeeds in cases where the Magma interface fails to convert
# number field elements, because embeddings between number fields
# may be lost on conversion to and from Magma.
# This should deal with all absolute number fields.
try:
return True, self.point([B(c.Eltseq().sage()) for c in pt])
except TypeError:
pass
# The second attempt tries to split Magma elements into
# numerators and denominators first. This is neccessary
# for the field of rational functions, because (at the moment of
# writing) fraction field elements are not converted automatically
# from Magma to Sage.
try:
return True, self.point( \
[B(c.Numerator().sage()/c.Denominator().sage()) for c in pt])
except (TypeError, NameError):
pass
# Finally, let the Magma interface handle conversion.
try:
return True, self.point([B(c.sage()) for c in pt])
except (TypeError, NameError):
pass
raise NotImplementedError, "No correct conversion implemented for converting the Magma point %s on %s to a correct Sage point on self (=%s)" % (
M_pt, M, self)
if algorithm != 'default':
raise ValueError, "Unknown algorithm: %s" % algorithm
if is_ComplexField(B):
if point:
[_, _, _, d, e, f] = self._coefficients
if d == 0:
return True, self.point([0, 1, 0])
return True, self.point(
[0, ((e**2 - 4 * d * f).sqrt() - e) / (2 * d), 1],
check=False)
return True
if is_RealField(B):
D, T = self.diagonal_matrix()
[a, b, c] = [D[0, 0], D[1, 1], D[2, 2]]
if a == 0:
ret = True, self.point(T * vector([1, 0, 0]), check=False)
elif a * c <= 0:
ret = True, self.point(T * vector([(-c / a).sqrt(), 0, 1]),
check=False)
elif b == 0:
ret = True, self.point(T * vector([0, 1, 0]), check=False)
elif b * c <= 0:
ret = True, self.point(T * vector([0, (-c / b).sqrt(), 0, 1]),
check=False)
else:
ret = False, None
if point:
return ret
return ret[0]
raise NotImplementedError, "has_rational_point not implemented for " \
"conics over base field %s" % B
def isogenies_prime_degree(self, l=None, max_l=31):
"""
Generic code, valid for all fields, for arbitrary prime `l` not equal to the characteristic.
INPUT:
- ``l`` -- either None, a prime or a list of primes.
- ``max_l`` -- a bound on the primes to be tested (ignored unless `l` is None).
OUTPUT:
(list) All `l`-isogenies for the given `l` with domain self.
METHOD:
Calls the generic function
``isogenies_prime_degree()``. This requires that
certain operations have been implemented over the base field,
such as root-finding for univariate polynomials.
EXAMPLES::
sage: F = QQbar
sage: E = EllipticCurve(F, [1,18]); E
Elliptic Curve defined by y^2 = x^3 + x + 18 over Algebraic Field
sage: E.isogenies_prime_degree()
Traceback (most recent call last):
...
NotImplementedError: This code could be implemented for QQbar, but has not been yet.
sage: F = CC
sage: E = EllipticCurve(F, [1,18]); E
Elliptic Curve defined by y^2 = x^3 + 1.00000000000000*x + 18.0000000000000 over Complex Field with 53 bits of precision
sage: E.isogenies_prime_degree(11)
Traceback (most recent call last):
...
NotImplementedError: This code could be implemented for general complex fields, but has not been yet.
Examples over finite fields::
sage: E = EllipticCurve(GF(next_prime(1000000)), [7,8])
sage: E.isogenies_prime_degree()
[Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 970389*x + 794257 over Finite Field of size 1000003, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 29783*x + 206196 over Finite Field of size 1000003, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 999960*x + 78 over Finite Field of size 1000003, Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 878063*x + 845666 over Finite Field of size 1000003, Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 375648*x + 342776 over Finite Field of size 1000003, Isogeny of degree 17 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 347438*x + 594729 over Finite Field of size 1000003, Isogeny of degree 17 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 674846*x + 7392 over Finite Field of size 1000003, Isogeny of degree 23 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 390065*x + 605596 over Finite Field of size 1000003]
sage: E.isogenies_prime_degree(2)
[Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 970389*x + 794257 over Finite Field of size 1000003, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 29783*x + 206196 over Finite Field of size 1000003, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 999960*x + 78 over Finite Field of size 1000003]
sage: E.isogenies_prime_degree(3)
[]
sage: E.isogenies_prime_degree(5)
[]
sage: E.isogenies_prime_degree(7)
[]
sage: E.isogenies_prime_degree(13)
[Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 878063*x + 845666 over Finite Field of size 1000003,
Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 375648*x + 342776 over Finite Field of size 1000003]
sage: E.isogenies_prime_degree([2, 3, 5, 7, 13])
[Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 970389*x + 794257 over Finite Field of size 1000003, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 29783*x + 206196 over Finite Field of size 1000003, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 999960*x + 78 over Finite Field of size 1000003, Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 878063*x + 845666 over Finite Field of size 1000003, Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 375648*x + 342776 over Finite Field of size 1000003]
sage: E.isogenies_prime_degree([2, 4])
Traceback (most recent call last):
...
ValueError: 4 is not prime.
sage: E.isogenies_prime_degree(4)
Traceback (most recent call last):
...
ValueError: 4 is not prime.
sage: E.isogenies_prime_degree(11)
[]
sage: E = EllipticCurve(GF(17),[2,0])
sage: E.isogenies_prime_degree(3)
[]
sage: E.isogenies_prime_degree(2)
[Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 2*x over Finite Field of size 17 to Elliptic Curve defined by y^2 = x^3 + 9*x over Finite Field of size 17, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 2*x over Finite Field of size 17 to Elliptic Curve defined by y^2 = x^3 + 5*x + 9 over Finite Field of size 17, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 2*x over Finite Field of size 17 to Elliptic Curve defined by y^2 = x^3 + 5*x + 8 over Finite Field of size 17]
sage: E = EllipticCurve(GF(13^4, 'a'),[2,8])
sage: E.isogenies_prime_degree(2)
[Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 2*x + 8 over Finite Field in a of size 13^4 to Elliptic Curve defined by y^2 = x^3 + 7*x + 4 over Finite Field in a of size 13^4, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 2*x + 8 over Finite Field in a of size 13^4 to Elliptic Curve defined by y^2 = x^3 + (8*a^3+2*a^2+7*a+5)*x + (12*a^3+3*a^2+4*a+4) over Finite Field in a of size 13^4, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 2*x + 8 over Finite Field in a of size 13^4 to Elliptic Curve defined by y^2 = x^3 + (5*a^3+11*a^2+6*a+11)*x + (a^3+10*a^2+9*a) over Finite Field in a of size 13^4]
sage: E.isogenies_prime_degree(3)
[Isogeny of degree 3 from Elliptic Curve defined by y^2 = x^3 + 2*x + 8 over Finite Field in a of size 13^4 to Elliptic Curve defined by y^2 = x^3 + 9*x + 11 over Finite Field in a of size 13^4]
Example to show that separable isogenies of degree equal to the characteristic are now implemented::
sage: E.isogenies_prime_degree(13)
[Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 2*x + 8 over Finite Field in a of size 13^4 to Elliptic Curve defined by y^2 = x^3 + 6*x + 5 over Finite Field in a of size 13^4]
Examples over number fields (other than QQ)::
sage: QQroot2.<e> = NumberField(x^2-2)
sage: E = EllipticCurve(QQroot2, j=8000)
sage: E.isogenies_prime_degree()
[Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + (-150528000)*x + (-629407744000) over Number Field in e with defining polynomial x^2 - 2 to Elliptic Curve defined by y^2 = x^3 + (-602112000)*x + 5035261952000 over Number Field in e with defining polynomial x^2 - 2,
Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + (-150528000)*x + (-629407744000) over Number Field in e with defining polynomial x^2 - 2 to Elliptic Curve defined by y^2 = x^3 + (903168000*e-1053696000)*x + (14161674240000*e-23288086528000) over Number Field in e with defining polynomial x^2 - 2,
Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + (-150528000)*x + (-629407744000) over Number Field in e with defining polynomial x^2 - 2 to Elliptic Curve defined by y^2 = x^3 + (-903168000*e-1053696000)*x + (-14161674240000*e-23288086528000) over Number Field in e with defining polynomial x^2 - 2]
sage: E = EllipticCurve(QQroot2, [1,0,1,4, -6]); E
Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x + (-6) over Number Field in e with defining polynomial x^2 - 2
sage: E.isogenies_prime_degree(2)
[Isogeny of degree 2 from Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x + (-6) over Number Field in e with defining polynomial x^2 - 2 to Elliptic Curve defined by y^2 + x*y + y = x^3 + (-36)*x + (-70) over Number Field in e with defining polynomial x^2 - 2]
sage: E.isogenies_prime_degree(3)
[Isogeny of degree 3 from Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x + (-6) over Number Field in e with defining polynomial x^2 - 2 to Elliptic Curve defined by y^2 + x*y + y = x^3 + (-128/3)*x + 5662/27 over Number Field in e with defining polynomial x^2 - 2, Isogeny of degree 3 from Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x + (-6) over Number Field in e with defining polynomial x^2 - 2 to Elliptic Curve defined by y^2 + x*y + y = x^3 + (-171)*x + (-874) over Number Field in e with defining polynomial x^2 - 2]
"""
F = self.base_ring()
if rings.is_RealField(F):
raise NotImplementedError, "This code could be implemented for general real fields, but has not been yet."
if rings.is_ComplexField(F):
raise NotImplementedError, "This code could be implemented for general complex fields, but has not been yet."
if F == rings.QQbar:
raise NotImplementedError, "This code could be implemented for QQbar, but has not been yet."
from isogeny_small_degree import isogenies_prime_degree
if l is None:
from sage.rings.all import prime_range
l = prime_range(max_l+1)
if type(l) != list:
try:
l = rings.ZZ(l)
except TypeError:
raise ValueError, "%s is not prime."%l
if l.is_prime():
return isogenies_prime_degree(self, l)
else:
raise ValueError, "%s is not prime."%l
L = list(set(l))
try:
L = [rings.ZZ(l) for l in L]
except TypeError:
raise ValueError, "%s is not a list of primes."%l
L.sort()
return sum([isogenies_prime_degree(self,l) for l in L],[])
def isogenies_prime_degree(self, l=None):
"""
Generic code, valid for all fields, for those l for which the
modular curve has genus 0.
INPUT:
- ``l`` -- either None, a prime or a list of primes, from [2,3,5,7,13].
OUTPUT:
(list) All `l`-isogenies for the given `l` with domain self.
METHOD:
Calls the generic function
``isogenies_prime_degree_genus_0()``. This requires that
certain operations have been implemented over the base field,
such as root-finding for univariate polynomials.
EXAMPLES::
sage: F = QQbar
sage: E = EllipticCurve(F, [1,18]); E
Elliptic Curve defined by y^2 = x^3 + x + 18 over Algebraic Field
sage: F = CC
sage: E = EllipticCurve(F, [1,18]); E
Elliptic Curve defined by y^2 = x^3 + 1.00000000000000*x + 18.0000000000000 over Complex Field with 53 bits of precision
sage: E.isogenies_prime_degree(11)
Traceback (most recent call last):
...
NotImplementedError: This code could be implemented for general complex fields, but has not been yet.
Examples over finite fields::
sage: E = EllipticCurve(GF(next_prime(1000000)), [7,8])
sage: E.isogenies_prime_degree()
[Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 970389*x + 794257 over Finite Field of size 1000003, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 29783*x + 206196 over Finite Field of size 1000003, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 999960*x + 78 over Finite Field of size 1000003, Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 878063*x + 845666 over Finite Field of size 1000003, Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 375648*x + 342776 over Finite Field of size 1000003]
sage: E.isogenies_prime_degree(2)
[Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 970389*x + 794257 over Finite Field of size 1000003, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 29783*x + 206196 over Finite Field of size 1000003, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 999960*x + 78 over Finite Field of size 1000003]
sage: E.isogenies_prime_degree(3)
[]
sage: E.isogenies_prime_degree(5)
[]
sage: E.isogenies_prime_degree(7)
[]
sage: E.isogenies_prime_degree(13)
[Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 878063*x + 845666 over Finite Field of size 1000003,
Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 375648*x + 342776 over Finite Field of size 1000003]
sage: E.isogenies_prime_degree([2, 3, 5, 7, 13])
[Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 970389*x + 794257 over Finite Field of size 1000003, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 29783*x + 206196 over Finite Field of size 1000003, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 999960*x + 78 over Finite Field of size 1000003, Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 878063*x + 845666 over Finite Field of size 1000003, Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 375648*x + 342776 over Finite Field of size 1000003]
sage: E.isogenies_prime_degree([2, 4])
Traceback (most recent call last):
...
ValueError: 4 is not prime.
sage: E.isogenies_prime_degree([2, 29])
Traceback (most recent call last):
...
NotImplementedError: Over general fields, only isogenies of degree 2, 3, 5, 7 or 13 have been implemented.
sage: E.isogenies_prime_degree(4)
Traceback (most recent call last):
...
ValueError: 4 is not prime.
sage: E.isogenies_prime_degree(11)
Traceback (most recent call last):
...
NotImplementedError: Over general fields, only isogenies of degree 2, 3, 5, 7 or 13 have been implemented.
sage: E = EllipticCurve(GF(17),[2,0])
sage: E.isogenies_prime_degree(3)
[]
sage: E.isogenies_prime_degree(2)
[Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 2*x over Finite Field of size 17 to Elliptic Curve defined by y^2 = x^3 + 9*x over Finite Field of size 17, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 2*x over Finite Field of size 17 to Elliptic Curve defined by y^2 = x^3 + 5*x + 9 over Finite Field of size 17, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 2*x over Finite Field of size 17 to Elliptic Curve defined by y^2 = x^3 + 5*x + 8 over Finite Field of size 17]
sage: E = EllipticCurve(GF(13^4, 'a'),[2,8])
sage: E.isogenies_prime_degree(2)
[Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 2*x + 8 over Finite Field in a of size 13^4 to Elliptic Curve defined by y^2 = x^3 + 7*x + 4 over Finite Field in a of size 13^4, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 2*x + 8 over Finite Field in a of size 13^4 to Elliptic Curve defined by y^2 = x^3 + (8*a^3+2*a^2+7*a+5)*x + (12*a^3+3*a^2+4*a+4) over Finite Field in a of size 13^4, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 2*x + 8 over Finite Field in a of size 13^4 to Elliptic Curve defined by y^2 = x^3 + (5*a^3+11*a^2+6*a+11)*x + (a^3+10*a^2+9*a) over Finite Field in a of size 13^4]
sage: E.isogenies_prime_degree(3)
[Isogeny of degree 3 from Elliptic Curve defined by y^2 = x^3 + 2*x + 8 over Finite Field in a of size 13^4 to Elliptic Curve defined by y^2 = x^3 + 9*x + 11 over Finite Field in a of size 13^4]
sage: E.isogenies_prime_degree([2, 3, 5, 7, 13])
Traceback (most recent call last):
...
NotImplementedError: 2, 3, 5, 7 and 13-isogenies are not yet implemented in characteristic 2 and 3, and when the characteristic is the same as the degree of the isogeny.
sage: E.isogenies_prime_degree([2, 3, 5, 7])
[Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 2*x + 8 over Finite Field in a of size 13^4 to Elliptic Curve defined by y^2 = x^3 + 7*x + 4 over Finite Field in a of size 13^4, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 2*x + 8 over Finite Field in a of size 13^4 to Elliptic Curve defined by y^2 = x^3 + (8*a^3+2*a^2+7*a+5)*x + (12*a^3+3*a^2+4*a+4) over Finite Field in a of size 13^4, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 2*x + 8 over Finite Field in a of size 13^4 to Elliptic Curve defined by y^2 = x^3 + (5*a^3+11*a^2+6*a+11)*x + (a^3+10*a^2+9*a) over Finite Field in a of size 13^4, Isogeny of degree 3 from Elliptic Curve defined by y^2 = x^3 + 2*x + 8 over Finite Field in a of size 13^4 to Elliptic Curve defined by y^2 = x^3 + 9*x + 11 over Finite Field in a of size 13^4, Isogeny of degree 7 from Elliptic Curve defined by y^2 = x^3 + 2*x + 8 over Finite Field in a of size 13^4 to Elliptic Curve defined by y^2 = x^3 + 3*x + 11 over Finite Field in a of size 13^4, Isogeny of degree 7 from Elliptic Curve defined by y^2 = x^3 + 2*x + 8 over Finite Field in a of size 13^4 to Elliptic Curve defined by y^2 = x^3 + 9*x + 2 over Finite Field in a of size 13^4]
Examples over number fields (other than QQ)::
sage: QQroot2.<e> = NumberField(x^2-2)
sage: E = EllipticCurve(QQroot2,[1,1])
sage: E.isogenies_prime_degree(11)
Traceback (most recent call last):
...
NotImplementedError: Over general fields, only isogenies of degree 2, 3, 5, 7 or 13 have been implemented.
sage: E.isogenies_prime_degree(5)
[]
sage: E = EllipticCurve(QQroot2, [1,0,1,4, -6]); E
Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x + (-6) over Number Field in e with defining polynomial x^2 - 2
sage: E.isogenies_prime_degree(2)
[Isogeny of degree 2 from Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x + (-6) over Number Field in e with defining polynomial x^2 - 2 to Elliptic Curve defined by y^2 + x*y + y = x^3 + (-36)*x + (-70) over Number Field in e with defining polynomial x^2 - 2]
sage: E.isogenies_prime_degree(3)
[Isogeny of degree 3 from Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x + (-6) over Number Field in e with defining polynomial x^2 - 2 to Elliptic Curve defined by y^2 + x*y + y = x^3 + (-171)*x + (-874) over Number Field in e with defining polynomial x^2 - 2, Isogeny of degree 3 from Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x + (-6) over Number Field in e with defining polynomial x^2 - 2 to Elliptic Curve defined by y^2 + x*y + y = x^3 + (-128/3)*x + 5662/27 over Number Field in e with defining polynomial x^2 - 2]
"""
F = self.base_ring()
if rings.is_RealField(F):
raise NotImplementedError, "This code could be implemented for general real fields, but has not been yet."
if rings.is_ComplexField(F):
raise NotImplementedError, "This code could be implemented for general complex fields, but has not been yet."
if F == rings.QQbar:
raise NotImplementedError, "This code could be implemented for QQbar, but has not been yet."
from ell_curve_isogeny import isogenies_prime_degree_genus_0
if l is None:
l = [2, 3, 5, 7, 13]
if l in [2, 3, 5, 7, 13]:
return isogenies_prime_degree_genus_0(self, l)
if type(l) != list:
if l.is_prime():
raise NotImplementedError, "Over general fields, only isogenies of degree 2, 3, 5, 7 or 13 have been implemented."
else:
raise ValueError, "%s is not prime." % l
isogs = []
i = 0
while i < len(l):
isogenies = [
f for f in self.isogenies_prime_degree(l[i]) if not f in isogs
]
isogs.extend(isogenies)
i = i + 1
return isogs