def base_extend(self, R):
r"""
Return the natural extension of ``self`` over `R`
INPUT:
- ``R`` -- a field. The new base field.
OUTPUT:
The Jacobian over the ring `R`.
EXAMPLES::
sage: R.<x> = QQ['x']
sage: H = HyperellipticCurve(x^3-10*x+9)
sage: Jac = H.jacobian(); Jac
Jacobian of Hyperelliptic Curve over Rational Field defined by y^2 = x^3 - 10*x + 9
sage: F.<a> = QQ.extension(x^2+1)
sage: Jac.base_extend(F)
Jacobian of Hyperelliptic Curve over Number Field in a with defining
polynomial x^2 + 1 defined by y^2 = x^3 - 10*x + 9
"""
if not is_Field(R):
raise ValueError('Not a field: '+str(R))
if self.base_ring() is R:
return self
if not R.has_coerce_map_from(self.base_ring()):
raise ValueError('no natural map from the base ring (=%s) to R (=%s)!'
% (self.base_ring(), R))
return self.change_ring(R)
def __init__(self, abvar, field_of_definition=QQ):
"""
A finite subgroup of a modular abelian variety.
INPUT:
- ``abvar`` - a modular abelian variety
- ``field_of_definition`` - a field over which this
group is defined.
EXAMPLES: This is an abstract base class, so there are no instances
of this class itself.
::
sage: A = J0(37)
sage: G = A.torsion_subgroup(3); G
Finite subgroup with invariants [3, 3, 3, 3] over QQ of Abelian variety J0(37) of dimension 2
sage: type(G)
<class 'sage.modular.abvar.finite_subgroup.FiniteSubgroup_lattice'>
sage: from sage.modular.abvar.finite_subgroup import FiniteSubgroup
sage: isinstance(G, FiniteSubgroup)
True
"""
if not is_Field(field_of_definition):
raise TypeError, "field_of_definition must be a field"
if not abelian_variety.is_ModularAbelianVariety(abvar):
raise TypeError, "abvar must be a modular abelian variety"
Module_old.__init__(self, ZZ)
self.__abvar = abvar
self.__field_of_definition = field_of_definition
def __init__(self, abvar, field_of_definition=QQ):
"""
A finite subgroup of a modular abelian variety.
INPUT:
- ``abvar`` - a modular abelian variety
- ``field_of_definition`` - a field over which this
group is defined.
EXAMPLES: This is an abstract base class, so there are no instances
of this class itself.
::
sage: A = J0(37)
sage: G = A.torsion_subgroup(3); G
Finite subgroup with invariants [3, 3, 3, 3] over QQ of Abelian variety J0(37) of dimension 2
sage: type(G)
<class 'sage.modular.abvar.finite_subgroup.FiniteSubgroup_lattice'>
sage: from sage.modular.abvar.finite_subgroup import FiniteSubgroup
sage: isinstance(G, FiniteSubgroup)
True
"""
if not is_Field(field_of_definition):
raise TypeError, "field_of_definition must be a field"
if not abelian_variety.is_ModularAbelianVariety(abvar):
raise TypeError, "abvar must be a modular abelian variety"
Module_old.__init__(self, ZZ)
self.__abvar = abvar
self.__field_of_definition = field_of_definition
def base_extend(self, R):
r"""
Return the natural extension of ``self`` over `R`
INPUT:
- ``R`` -- a field. The new base field.
OUTPUT:
The Jacobian over the ring `R`.
EXAMPLES::
sage: R.<x> = QQ['x']
sage: H = HyperellipticCurve(x^3-10*x+9)
sage: Jac = H.jacobian(); Jac
Jacobian of Hyperelliptic Curve over Rational Field defined by y^2 = x^3 - 10*x + 9
sage: F.<a> = QQ.extension(x^2+1)
sage: Jac.base_extend(F)
Jacobian of Hyperelliptic Curve over Number Field in a with defining
polynomial x^2 + 1 defined by y^2 = x^3 - 10*x + 9
"""
if not is_Field(R):
raise ValueError('Not a field: ' + str(R))
if self.base_ring() is R:
return self
if not R.has_coerce_map_from(self.base_ring()):
raise ValueError(
'no natural map from the base ring (=%s) to R (=%s)!' %
(self.base_ring(), R))
return self.change_ring(R)
def base_extend(self, S):
r"""
Returns the conic over ``S`` given by the same equation as ``self``.
EXAMPLES::
sage: c = Conic([1, 1, 1]); c
Projective Conic Curve over Rational Field defined by x^2 + y^2 + z^2
sage: c.has_rational_point()
False
sage: d = c.base_extend(QuadraticField(-1, 'i')); d
Projective Conic Curve over Number Field in i with defining polynomial x^2 + 1 defined by x^2 + y^2 + z^2
sage: d.rational_point(algorithm = 'rnfisnorm')
(i : 1 : 0)
"""
if is_Field(S):
B = self.base_ring()
if B == S:
return self
if not S.has_coerce_map_from(B):
raise ValueError, "No natural map from the base ring of self " \
"(= %s) to S (= %s)" % (self, S)
from constructor import Conic
con = Conic([S(c) for c in self.coefficients()], \
self.variable_names())
if self._rational_point != None:
pt = [S(c) for c in Sequence(self._rational_point)]
if not pt == [0,0,0]:
# The following line stores the point in the cache
# if (and only if) there is no point in the cache.
pt = con.point(pt)
return con
return ProjectiveCurve_generic.base_extend(self, S)
def EllipticCurve_from_c4c6(c4, c6):
"""
Return an elliptic curve with given `c_4` and
`c_6` invariants.
EXAMPLES::
sage: E = EllipticCurve_from_c4c6(17, -2005)
sage: E
Elliptic Curve defined by y^2 = x^3 - 17/48*x + 2005/864 over Rational Field
sage: E.c_invariants()
(17, -2005)
"""
try:
K = c4.parent()
except AttributeError:
K = rings.RationalField()
if not rings.is_Field(K):
K = K.fraction_field()
return EllipticCurve([-K(c4) / K(48), -K(c6) / K(864)])
def EllipticCurve_from_c4c6(c4, c6):
"""
Return an elliptic curve with given `c_4` and
`c_6` invariants.
EXAMPLES::
sage: E = EllipticCurve_from_c4c6(17, -2005)
sage: E
Elliptic Curve defined by y^2 = x^3 - 17/48*x + 2005/864 over Rational Field
sage: E.c_invariants()
(17, -2005)
"""
try:
K = c4.parent()
except AttributeError:
K = rings.RationalField()
if not rings.is_Field(K):
K = K.fraction_field()
return EllipticCurve([-K(c4)/K(48), -K(c6)/K(864)])
def __init__(self, C):
"""
TESTS::
sage: from sage.schemes.jacobians.abstract_jacobian import Jacobian_generic
sage: P2.<x, y, z> = ProjectiveSpace(QQ, 2)
sage: C = Curve(x^3 + y^3 + z^3)
sage: J = Jacobian_generic(C); J
Jacobian of Projective Curve over Rational Field defined by x^3 + y^3 + z^3
sage: type(J)
<class 'sage.schemes.jacobians.abstract_jacobian.Jacobian_generic_with_category'>
Note: this is an abstract parent, so we skip element tests::
sage: TestSuite(J).run(skip =["_test_an_element", "_test_elements", "_test_elements_eq", "_test_some_elements"])
::
sage: Jacobian_generic(ZZ)
Traceback (most recent call last):
...
TypeError: Argument (=Integer Ring) must be a scheme.
sage: Jacobian_generic(P2)
Traceback (most recent call last):
...
ValueError: C (=Projective Space of dimension 2 over Rational Field) must have dimension 1.
sage: P2.<x, y, z> = ProjectiveSpace(Zmod(6), 2)
sage: C = Curve(x + y + z)
sage: Jacobian_generic(C)
Traceback (most recent call last):
...
TypeError: C (=Projective Curve over Ring of integers modulo 6 defined by x + y + z) must be defined over a field.
"""
if not is_Scheme(C):
raise TypeError, "Argument (=%s) must be a scheme."%C
if not is_Field(C.base_ring()):
raise TypeError, "C (=%s) must be defined over a field."%C
if C.dimension() != 1:
raise ValueError, "C (=%s) must have dimension 1."%C
self.__curve = C
Scheme.__init__(self, C.base_scheme())
def __init__(self, C):
"""
TESTS::
sage: from sage.schemes.jacobians.abstract_jacobian import Jacobian_generic
sage: P2.<x, y, z> = ProjectiveSpace(QQ, 2)
sage: C = Curve(x^3 + y^3 + z^3)
sage: J = Jacobian_generic(C); J
Jacobian of Projective Curve over Rational Field defined by x^3 + y^3 + z^3
sage: type(J)
<class 'sage.schemes.jacobians.abstract_jacobian.Jacobian_generic_with_category'>
Note: this is an abstract parent, so we skip element tests::
sage: TestSuite(J).run(skip =["_test_an_element", "_test_elements", "_test_elements_eq", "_test_some_elements"])
::
sage: Jacobian_generic(ZZ)
Traceback (most recent call last):
...
TypeError: Argument (=Integer Ring) must be a scheme.
sage: Jacobian_generic(P2)
Traceback (most recent call last):
...
ValueError: C (=Projective Space of dimension 2 over Rational Field) must have dimension 1.
sage: P2.<x, y, z> = ProjectiveSpace(Zmod(6), 2)
sage: C = Curve(x + y + z)
sage: Jacobian_generic(C)
Traceback (most recent call last):
...
TypeError: C (=Projective Curve over Ring of integers modulo 6 defined by x + y + z) must be defined over a field.
"""
if not is_Scheme(C):
raise TypeError, "Argument (=%s) must be a scheme." % C
if not is_Field(C.base_ring()):
raise TypeError, "C (=%s) must be defined over a field." % C
if C.dimension() != 1:
raise ValueError, "C (=%s) must have dimension 1." % C
self.__curve = C
Scheme.__init__(self, C.base_scheme())
def ProjectiveSpace(n, R=None, names='x'):
r"""
Return projective space of dimension `n` over the ring `R`.
EXAMPLES: The dimension and ring can be given in either order.
::
sage: ProjectiveSpace(3, QQ)
Projective Space of dimension 3 over Rational Field
sage: ProjectiveSpace(5, QQ)
Projective Space of dimension 5 over Rational Field
sage: P = ProjectiveSpace(2, QQ, names='XYZ'); P
Projective Space of dimension 2 over Rational Field
sage: P.coordinate_ring()
Multivariate Polynomial Ring in X, Y, Z over Rational Field
The divide operator does base extension.
::
sage: ProjectiveSpace(5)/GF(17)
Projective Space of dimension 5 over Finite Field of size 17
The default base ring is `\ZZ`.
::
sage: ProjectiveSpace(5)
Projective Space of dimension 5 over Integer Ring
There is also an projective space associated each polynomial ring.
::
sage: R = GF(7)['x,y,z']
sage: P = ProjectiveSpace(R); P
Projective Space of dimension 2 over Finite Field of size 7
sage: P.coordinate_ring()
Multivariate Polynomial Ring in x, y, z over Finite Field of size 7
sage: P.coordinate_ring() is R
True
Projective spaces are not cached, i.e., there can be several with
the same base ring and dimension (to facilitate gluing
constructions).
"""
if is_MPolynomialRing(n) and R is None:
A = ProjectiveSpace(n.ngens() - 1, n.base_ring())
A._coordinate_ring = n
return A
if isinstance(R, (int, long, Integer)):
n, R = R, n
if R is None:
R = ZZ # default is the integers
if is_Field(R):
if is_FiniteField(R):
return ProjectiveSpace_finite_field(n, R, names)
if is_RationalField(R):
return ProjectiveSpace_rational_field(n, R, names)
else:
return ProjectiveSpace_field(n, R, names)
elif is_CommutativeRing(R):
return ProjectiveSpace_ring(n, R, names)
else:
raise TypeError, "R (=%s) must be a commutative ring" % R
def ProjectiveSpace(n, R=None, names="x"):
r"""
Return projective space of dimension `n` over the ring `R`.
EXAMPLES: The dimension and ring can be given in either order.
::
sage: ProjectiveSpace(3, QQ)
Projective Space of dimension 3 over Rational Field
sage: ProjectiveSpace(5, QQ)
Projective Space of dimension 5 over Rational Field
sage: P = ProjectiveSpace(2, QQ, names='XYZ'); P
Projective Space of dimension 2 over Rational Field
sage: P.coordinate_ring()
Multivariate Polynomial Ring in X, Y, Z over Rational Field
The divide operator does base extension.
::
sage: ProjectiveSpace(5)/GF(17)
Projective Space of dimension 5 over Finite Field of size 17
The default base ring is `\ZZ`.
::
sage: ProjectiveSpace(5)
Projective Space of dimension 5 over Integer Ring
There is also an projective space associated each polynomial ring.
::
sage: R = GF(7)['x,y,z']
sage: P = ProjectiveSpace(R); P
Projective Space of dimension 2 over Finite Field of size 7
sage: P.coordinate_ring()
Multivariate Polynomial Ring in x, y, z over Finite Field of size 7
sage: P.coordinate_ring() is R
True
Projective spaces are not cached, i.e., there can be several with
the same base ring and dimension (to facilitate gluing
constructions).
"""
if is_MPolynomialRing(n) and R is None:
A = ProjectiveSpace(n.ngens() - 1, n.base_ring())
A._coordinate_ring = n
return A
if isinstance(R, (int, long, Integer)):
n, R = R, n
if R is None:
R = ZZ # default is the integers
if is_Field(R):
if is_FiniteField(R):
return ProjectiveSpace_finite_field(n, R, names)
if is_RationalField(R):
return ProjectiveSpace_rational_field(n, R, names)
else:
return ProjectiveSpace_field(n, R, names)
elif is_CommutativeRing(R):
return ProjectiveSpace_ring(n, R, names)
else:
raise TypeError, "R (=%s) must be a commutative ring" % R
def EllipticCurve_from_j(j):
"""
Return an elliptic curve with given `j`-invariant.
EXAMPLES::
sage: E = EllipticCurve_from_j(0); E; E.j_invariant(); E.label()
Elliptic Curve defined by y^2 + y = x^3 over Rational Field
0
'27a3'
sage: E = EllipticCurve_from_j(1728); E; E.j_invariant(); E.label()
Elliptic Curve defined by y^2 = x^3 - x over Rational Field
1728
'32a2'
sage: E = EllipticCurve_from_j(1); E; E.j_invariant()
Elliptic Curve defined by y^2 + x*y = x^3 + 36*x + 3455 over Rational Field
1
"""
try:
K = j.parent()
except AttributeError:
K = rings.RationalField()
if not rings.is_Field(K):
K = K.fraction_field()
char = K.characteristic()
if char == 2:
if j == 0:
return EllipticCurve(K, [0, 0, 1, 0, 0])
else:
return EllipticCurve(K, [1, 0, 0, 0, 1 / j])
if char == 3:
if j == 0:
return EllipticCurve(K, [0, 0, 0, 1, 0])
else:
return EllipticCurve(K, [0, j, 0, 0, -j ** 2])
if K is rings.RationalField():
# we construct the minimal twist, i.e. the curve with minimal
# conductor with this j_invariant:
if j == 0:
return EllipticCurve(K, [0, 0, 1, 0, 0]) # 27a3
if j == 1728:
return EllipticCurve(K, [0, 0, 0, -1, 0]) # 32a2
n = j.numerator()
m = n - 1728 * j.denominator()
a4 = -3 * n * m
a6 = -2 * n * m ** 2
# Now E=[0,0,0,a4,a6] has j-invariant j=n/d
from sage.sets.set import Set
for p in Set(n.prime_divisors() + m.prime_divisors()):
e = min(a4.valuation(p) // 2, a6.valuation(p) // 3)
if e > 0:
p = p ** e
a4 /= p ** 2
a6 /= p ** 3
# Now E=[0,0,0,a4,a6] is minimal at all p != 2,3
tw = [-1, 2, -2, 3, -3, 6, -6]
E1 = EllipticCurve([0, 0, 0, a4, a6])
Elist = [E1] + [E1.quadratic_twist(t) for t in tw]
crv_cmp = lambda E, F: cmp(E.conductor(), F.conductor())
Elist.sort(cmp=crv_cmp)
return Elist[0]
# defaults for all other fields:
if j == 0:
return EllipticCurve(K, [0, 0, 0, 0, 1])
if j == 1728:
return EllipticCurve(K, [0, 0, 0, 1, 0])
k = j - 1728
return EllipticCurve(K, [0, 0, 0, -3 * j * k, -2 * j * k ** 2])
def EllipticCurve(x=None, y=None, j=None):
r"""
There are several ways to construct an elliptic curve:
.. math::
y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6.
- EllipticCurve([a1,a2,a3,a4,a6]): Elliptic curve with given
a-invariants. The invariants are coerced into the parent of the
first element. If all are integers, they are coerced into the
rational numbers.
- EllipticCurve([a4,a6]): Same as above, but a1=a2=a3=0.
- EllipticCurve(label): Returns the elliptic curve over Q from the
Cremona database with the given label. The label is a string, such
as "11a" or "37b2". The letters in the label *must* be lower case
(Cremona's new labeling).
- EllipticCurve(R, [a1,a2,a3,a4,a6]): Create the elliptic curve
over R with given a-invariants. Here R can be an arbitrary ring.
Note that addition need not be defined.
- EllipticCurve(j): Return an elliptic curve with j-invariant
`j`. Warning: this is deprecated. Use ``EllipticCurve_from_j(j)``
or ``EllipticCurve(j=j)`` instead.
In each case above where the input is a list of length 2 or 5, one
can instead give a 2 or 5-tuple instead.
EXAMPLES: We illustrate creating elliptic curves.
::
sage: EllipticCurve([0,0,1,-1,0])
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
We create a curve from a Cremona label::
sage: EllipticCurve('37b2')
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 1873*x - 31833 over Rational Field
sage: EllipticCurve('5077a')
Elliptic Curve defined by y^2 + y = x^3 - 7*x + 6 over Rational Field
sage: EllipticCurve('389a')
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field
Unicode labels are allowed::
sage: EllipticCurve(u'389a')
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field
We create curves over a finite field as follows::
sage: EllipticCurve([GF(5)(0),0,1,-1,0])
Elliptic Curve defined by y^2 + y = x^3 + 4*x over Finite Field of size 5
sage: EllipticCurve(GF(5), [0, 0,1,-1,0])
Elliptic Curve defined by y^2 + y = x^3 + 4*x over Finite Field of size 5
Elliptic curves over `\ZZ/N\ZZ` with `N` prime are of type
"elliptic curve over a finite field"::
sage: F = Zmod(101)
sage: EllipticCurve(F, [2, 3])
Elliptic Curve defined by y^2 = x^3 + 2*x + 3 over Ring of integers modulo 101
sage: E = EllipticCurve([F(2), F(3)])
sage: type(E)
<class 'sage.schemes.elliptic_curves.ell_finite_field.EllipticCurve_finite_field_with_category'>
sage: E.category()
Category of schemes over Ring of integers modulo 101
In contrast, elliptic curves over `\ZZ/N\ZZ` with `N` composite
are of type "generic elliptic curve"::
sage: F = Zmod(95)
sage: EllipticCurve(F, [2, 3])
Elliptic Curve defined by y^2 = x^3 + 2*x + 3 over Ring of integers modulo 95
sage: E = EllipticCurve([F(2), F(3)])
sage: type(E)
<class 'sage.schemes.elliptic_curves.ell_generic.EllipticCurve_generic_with_category'>
sage: E.category()
Category of schemes over Ring of integers modulo 95
The following is a curve over the complex numbers::
sage: E = EllipticCurve(CC, [0,0,1,-1,0])
sage: E
Elliptic Curve defined by y^2 + 1.00000000000000*y = x^3 + (-1.00000000000000)*x over Complex Field with 53 bits of precision
sage: E.j_invariant()
2988.97297297297
We can also create elliptic curves by giving the Weierstrass equation::
sage: x, y = var('x,y')
sage: EllipticCurve(y^2 + y == x^3 + x - 9)
Elliptic Curve defined by y^2 + y = x^3 + x - 9 over Rational Field
sage: R.<x,y> = GF(5)[]
sage: EllipticCurve(x^3 + x^2 + 2 - y^2 - y*x)
Elliptic Curve defined by y^2 + x*y = x^3 + x^2 + 2 over Finite Field of size 5
We can explicitly specify the `j`-invariant::
sage: E = EllipticCurve(j=1728); E; E.j_invariant(); E.label()
Elliptic Curve defined by y^2 = x^3 - x over Rational Field
1728
'32a2'
sage: E = EllipticCurve(j=GF(5)(2)); E; E.j_invariant()
Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field of size 5
2
See trac #6657::
sage: EllipticCurve(GF(144169),j=1728)
Elliptic Curve defined by y^2 = x^3 + x over Finite Field of size 144169
TESTS::
sage: R = ZZ['u', 'v']
sage: EllipticCurve(R, [1,1])
Elliptic Curve defined by y^2 = x^3 + x + 1 over Multivariate Polynomial Ring in u, v
over Integer Ring
We create a curve and a point over QQbar (see #6879)::
sage: E = EllipticCurve(QQbar,[0,1])
sage: E(0)
(0 : 1 : 0)
sage: E.base_field()
Algebraic Field
sage: E = EllipticCurve(RR,[1,2]); E; E.base_field()
Elliptic Curve defined by y^2 = x^3 + 1.00000000000000*x + 2.00000000000000 over Real Field with 53 bits of precision
Real Field with 53 bits of precision
sage: EllipticCurve(CC,[3,4]); E; E.base_field()
Elliptic Curve defined by y^2 = x^3 + 3.00000000000000*x + 4.00000000000000 over Complex Field with 53 bits of precision
Elliptic Curve defined by y^2 = x^3 + 1.00000000000000*x + 2.00000000000000 over Real Field with 53 bits of precision
Real Field with 53 bits of precision
sage: E = EllipticCurve(QQbar,[5,6]); E; E.base_field()
Elliptic Curve defined by y^2 = x^3 + 5*x + 6 over Algebraic Field
Algebraic Field
See trac #6657::
sage: EllipticCurve(3,j=1728)
Traceback (most recent call last):
...
ValueError: First parameter (if present) must be a ring when j is specified
sage: EllipticCurve(GF(5),j=3/5)
Traceback (most recent call last):
...
ValueError: First parameter must be a ring containing 3/5
If the universe of the coefficients is a general field, the object
constructed has type EllipticCurve_field. Otherwise it is
EllipticCurve_generic. See trac #9816::
sage: E = EllipticCurve([QQbar(1),3]); E
Elliptic Curve defined by y^2 = x^3 + x + 3 over Algebraic Field
sage: type(E)
<class 'sage.schemes.elliptic_curves.ell_field.EllipticCurve_field_with_category'>
sage: E = EllipticCurve([RR(1),3]); E
Elliptic Curve defined by y^2 = x^3 + 1.00000000000000*x + 3.00000000000000 over Real Field with 53 bits of precision
sage: type(E)
<class 'sage.schemes.elliptic_curves.ell_field.EllipticCurve_field_with_category'>
sage: E = EllipticCurve([i,i]); E
Elliptic Curve defined by y^2 = x^3 + I*x + I over Symbolic Ring
sage: type(E)
<class 'sage.schemes.elliptic_curves.ell_field.EllipticCurve_field_with_category'>
sage: E.category()
Category of schemes over Symbolic Ring
sage: is_field(SR)
True
sage: F = FractionField(PolynomialRing(QQ,'t'))
sage: t = F.gen()
sage: E = EllipticCurve([t,0]); E
Elliptic Curve defined by y^2 = x^3 + t*x over Fraction Field of Univariate Polynomial Ring in t over Rational Field
sage: type(E)
<class 'sage.schemes.elliptic_curves.ell_field.EllipticCurve_field_with_category'>
sage: E.category()
Category of schemes over Fraction Field of Univariate Polynomial Ring in t over Rational Field
See :trac:`12517`::
sage: E = EllipticCurve([1..5])
sage: EllipticCurve(E.a_invariants())
Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational Field
"""
import ell_generic, ell_field, ell_finite_field, ell_number_field, ell_rational_field, ell_padic_field # here to avoid circular includes
if j is not None:
if not x is None:
if rings.is_Ring(x):
try:
j = x(j)
except (ZeroDivisionError, ValueError, TypeError):
raise ValueError, "First parameter must be a ring containing %s" % j
else:
raise ValueError, "First parameter (if present) must be a ring when j is specified"
return EllipticCurve_from_j(j)
assert x is not None
if is_SymbolicEquation(x):
x = x.lhs() - x.rhs()
if parent(x) is SR:
x = x._polynomial_(rings.QQ["x", "y"])
if rings.is_MPolynomial(x) and y is None:
f = x
if f.degree() != 3:
raise ValueError, "Elliptic curves must be defined by a cubic polynomial."
if f.degrees() == (3, 2):
x, y = f.parent().gens()
elif f.degree() == (2, 3):
y, x = f.parent().gens()
elif len(f.parent().gens()) == 2 or len(f.parent().gens()) == 3 and f.is_homogeneous():
# We'd need a point too...
raise NotImplementedError, "Construction of an elliptic curve from a generic cubic not yet implemented."
else:
raise ValueError, "Defining polynomial must be a cubic polynomial in two variables."
try:
if f.coefficient(x ** 3) < 0:
f = -f
# is there a nicer way to extract the coefficients?
a1 = a2 = a3 = a4 = a6 = 0
for coeff, mon in f:
if mon == x ** 3:
assert coeff == 1
elif mon == x ** 2:
a2 = coeff
elif mon == x:
a4 = coeff
elif mon == 1:
a6 = coeff
elif mon == y ** 2:
assert coeff == -1
elif mon == x * y:
a1 = -coeff
elif mon == y:
a3 = -coeff
else:
assert False
return EllipticCurve([a1, a2, a3, a4, a6])
except AssertionError:
raise NotImplementedError, "Construction of an elliptic curve from a generic cubic not yet implemented."
if rings.is_Ring(x):
if rings.is_RationalField(x):
return ell_rational_field.EllipticCurve_rational_field(x, y)
elif rings.is_FiniteField(x) or (rings.is_IntegerModRing(x) and x.characteristic().is_prime()):
return ell_finite_field.EllipticCurve_finite_field(x, y)
elif rings.is_pAdicField(x):
return ell_padic_field.EllipticCurve_padic_field(x, y)
elif rings.is_NumberField(x):
return ell_number_field.EllipticCurve_number_field(x, y)
elif rings.is_Field(x):
return ell_field.EllipticCurve_field(x, y)
return ell_generic.EllipticCurve_generic(x, y)
if isinstance(x, unicode):
x = str(x)
if isinstance(x, str):
return ell_rational_field.EllipticCurve_rational_field(x)
if rings.is_RingElement(x) and y is None:
from sage.misc.misc import deprecation
deprecation("'EllipticCurve(j)' is deprecated; use 'EllipticCurve_from_j(j)' or 'EllipticCurve(j=j)' instead.")
# Fixed for all characteristics and cases by John Cremona
j = x
F = j.parent().fraction_field()
char = F.characteristic()
if char == 2:
if j == 0:
return EllipticCurve(F, [0, 0, 1, 0, 0])
else:
return EllipticCurve(F, [1, 0, 0, 0, 1 / j])
if char == 3:
if j == 0:
return EllipticCurve(F, [0, 0, 0, 1, 0])
else:
return EllipticCurve(F, [0, j, 0, 0, -j ** 2])
if j == 0:
return EllipticCurve(F, [0, 0, 0, 0, 1])
if j == 1728:
return EllipticCurve(F, [0, 0, 0, 1, 0])
k = j - 1728
return EllipticCurve(F, [0, 0, 0, -3 * j * k, -2 * j * k ** 2])
if not isinstance(x, (list, tuple)):
raise TypeError, "invalid input to EllipticCurve constructor"
x = Sequence(x)
if not (len(x) in [2, 5]):
raise ValueError, "sequence of coefficients must have length 2 or 5"
R = x.universe()
if isinstance(x[0], (rings.Rational, rings.Integer, int, long)):
return ell_rational_field.EllipticCurve_rational_field(x, y)
elif rings.is_NumberField(R):
return ell_number_field.EllipticCurve_number_field(x, y)
elif rings.is_pAdicField(R):
return ell_padic_field.EllipticCurve_padic_field(x, y)
elif rings.is_FiniteField(R) or (rings.is_IntegerModRing(R) and R.characteristic().is_prime()):
return ell_finite_field.EllipticCurve_finite_field(x, y)
elif rings.is_Field(R):
return ell_field.EllipticCurve_field(x, y)
return ell_generic.EllipticCurve_generic(x, y)
def EllipticCurve_from_j(j):
"""
Return an elliptic curve with given `j`-invariant.
EXAMPLES::
sage: E = EllipticCurve_from_j(0); E; E.j_invariant(); E.label()
Elliptic Curve defined by y^2 + y = x^3 over Rational Field
0
'27a3'
sage: E = EllipticCurve_from_j(1728); E; E.j_invariant(); E.label()
Elliptic Curve defined by y^2 = x^3 - x over Rational Field
1728
'32a2'
sage: E = EllipticCurve_from_j(1); E; E.j_invariant()
Elliptic Curve defined by y^2 + x*y = x^3 + 36*x + 3455 over Rational Field
1
"""
try:
K = j.parent()
except AttributeError:
K = rings.RationalField()
if not rings.is_Field(K):
K = K.fraction_field()
char=K.characteristic()
if char==2:
if j == 0:
return EllipticCurve(K, [ 0, 0, 1, 0, 0 ])
else:
return EllipticCurve(K, [ 1, 0, 0, 0, 1/j ])
if char == 3:
if j==0:
return EllipticCurve(K, [ 0, 0, 0, 1, 0 ])
else:
return EllipticCurve(K, [ 0, j, 0, 0, -j**2 ])
if K is rings.RationalField():
# we construct the minimal twist, i.e. the curve with minimal
# conductor with this j_invariant:
if j == 0:
return EllipticCurve(K, [ 0, 0, 1, 0, 0 ]) # 27a3
if j == 1728:
return EllipticCurve(K, [ 0, 0, 0, -1, 0 ]) # 32a2
n = j.numerator()
m = n-1728*j.denominator()
a4 = -3*n*m
a6 = -2*n*m**2
# Now E=[0,0,0,a4,a6] has j-invariant j=n/d
from sage.sets.set import Set
for p in Set(n.prime_divisors()+m.prime_divisors()):
e = min(a4.valuation(p)//2,a6.valuation(p)//3)
if e>0:
p = p**e
a4 /= p**2
a6 /= p**3
# Now E=[0,0,0,a4,a6] is minimal at all p != 2,3
tw = [-1,2,-2,3,-3,6,-6]
E1 = EllipticCurve([0,0,0,a4,a6])
Elist = [E1] + [E1.quadratic_twist(t) for t in tw]
crv_cmp = lambda E,F: cmp(E.conductor(),F.conductor())
Elist.sort(cmp=crv_cmp)
return Elist[0]
# defaults for all other fields:
if j == 0:
return EllipticCurve(K, [ 0, 0, 0, 0, 1 ])
if j == 1728:
return EllipticCurve(K, [ 0, 0, 0, 1, 0 ])
k=j-1728
return EllipticCurve(K, [0,0,0,-3*j*k, -2*j*k**2])
def EllipticCurve(x=None, y=None, j=None):
r"""
There are several ways to construct an elliptic curve:
.. math::
y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6.
- EllipticCurve([a1,a2,a3,a4,a6]): Elliptic curve with given
a-invariants. The invariants are coerced into the parent of the
first element. If all are integers, they are coerced into the
rational numbers.
- EllipticCurve([a4,a6]): Same as above, but a1=a2=a3=0.
- EllipticCurve(label): Returns the elliptic curve over Q from the
Cremona database with the given label. The label is a string, such
as "11a" or "37b2". The letters in the label *must* be lower case
(Cremona's new labeling).
- EllipticCurve(R, [a1,a2,a3,a4,a6]): Create the elliptic curve
over R with given a-invariants. Here R can be an arbitrary ring.
Note that addition need not be defined.
- EllipticCurve(j): Return an elliptic curve with j-invariant
`j`. Warning: this is deprecated. Use ``EllipticCurve_from_j(j)``
or ``EllipticCurve(j=j)`` instead.
In each case above where the input is a list of length 2 or 5, one
can instead give a 2 or 5-tuple instead.
EXAMPLES: We illustrate creating elliptic curves.
::
sage: EllipticCurve([0,0,1,-1,0])
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
We create a curve from a Cremona label::
sage: EllipticCurve('37b2')
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 1873*x - 31833 over Rational Field
sage: EllipticCurve('5077a')
Elliptic Curve defined by y^2 + y = x^3 - 7*x + 6 over Rational Field
sage: EllipticCurve('389a')
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field
Unicode labels are allowed::
sage: EllipticCurve(u'389a')
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field
We create curves over a finite field as follows::
sage: EllipticCurve([GF(5)(0),0,1,-1,0])
Elliptic Curve defined by y^2 + y = x^3 + 4*x over Finite Field of size 5
sage: EllipticCurve(GF(5), [0, 0,1,-1,0])
Elliptic Curve defined by y^2 + y = x^3 + 4*x over Finite Field of size 5
Elliptic curves over `\ZZ/N\ZZ` with `N` prime are of type
"elliptic curve over a finite field"::
sage: F = Zmod(101)
sage: EllipticCurve(F, [2, 3])
Elliptic Curve defined by y^2 = x^3 + 2*x + 3 over Ring of integers modulo 101
sage: E = EllipticCurve([F(2), F(3)])
sage: type(E)
<class 'sage.schemes.elliptic_curves.ell_finite_field.EllipticCurve_finite_field_with_category'>
sage: E.category()
Category of schemes over Ring of integers modulo 101
In contrast, elliptic curves over `\ZZ/N\ZZ` with `N` composite
are of type "generic elliptic curve"::
sage: F = Zmod(95)
sage: EllipticCurve(F, [2, 3])
Elliptic Curve defined by y^2 = x^3 + 2*x + 3 over Ring of integers modulo 95
sage: E = EllipticCurve([F(2), F(3)])
sage: type(E)
<class 'sage.schemes.elliptic_curves.ell_generic.EllipticCurve_generic_with_category'>
sage: E.category()
Category of schemes over Ring of integers modulo 95
The following is a curve over the complex numbers::
sage: E = EllipticCurve(CC, [0,0,1,-1,0])
sage: E
Elliptic Curve defined by y^2 + 1.00000000000000*y = x^3 + (-1.00000000000000)*x over Complex Field with 53 bits of precision
sage: E.j_invariant()
2988.97297297297
We can also create elliptic curves by giving the Weierstrass equation::
sage: x, y = var('x,y')
sage: EllipticCurve(y^2 + y == x^3 + x - 9)
Elliptic Curve defined by y^2 + y = x^3 + x - 9 over Rational Field
sage: R.<x,y> = GF(5)[]
sage: EllipticCurve(x^3 + x^2 + 2 - y^2 - y*x)
Elliptic Curve defined by y^2 + x*y = x^3 + x^2 + 2 over Finite Field of size 5
We can explicitly specify the `j`-invariant::
sage: E = EllipticCurve(j=1728); E; E.j_invariant(); E.label()
Elliptic Curve defined by y^2 = x^3 - x over Rational Field
1728
'32a2'
sage: E = EllipticCurve(j=GF(5)(2)); E; E.j_invariant()
Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field of size 5
2
See trac #6657::
sage: EllipticCurve(GF(144169),j=1728)
Elliptic Curve defined by y^2 = x^3 + x over Finite Field of size 144169
TESTS::
sage: R = ZZ['u', 'v']
sage: EllipticCurve(R, [1,1])
Elliptic Curve defined by y^2 = x^3 + x + 1 over Multivariate Polynomial Ring in u, v
over Integer Ring
We create a curve and a point over QQbar (see #6879)::
sage: E = EllipticCurve(QQbar,[0,1])
sage: E(0)
(0 : 1 : 0)
sage: E.base_field()
Algebraic Field
sage: E = EllipticCurve(RR,[1,2]); E; E.base_field()
Elliptic Curve defined by y^2 = x^3 + 1.00000000000000*x + 2.00000000000000 over Real Field with 53 bits of precision
Real Field with 53 bits of precision
sage: EllipticCurve(CC,[3,4]); E; E.base_field()
Elliptic Curve defined by y^2 = x^3 + 3.00000000000000*x + 4.00000000000000 over Complex Field with 53 bits of precision
Elliptic Curve defined by y^2 = x^3 + 1.00000000000000*x + 2.00000000000000 over Real Field with 53 bits of precision
Real Field with 53 bits of precision
sage: E = EllipticCurve(QQbar,[5,6]); E; E.base_field()
Elliptic Curve defined by y^2 = x^3 + 5*x + 6 over Algebraic Field
Algebraic Field
See trac #6657::
sage: EllipticCurve(3,j=1728)
Traceback (most recent call last):
...
ValueError: First parameter (if present) must be a ring when j is specified
sage: EllipticCurve(GF(5),j=3/5)
Traceback (most recent call last):
...
ValueError: First parameter must be a ring containing 3/5
If the universe of the coefficients is a general field, the object
constructed has type EllipticCurve_field. Otherwise it is
EllipticCurve_generic. See trac #9816::
sage: E = EllipticCurve([QQbar(1),3]); E
Elliptic Curve defined by y^2 = x^3 + x + 3 over Algebraic Field
sage: type(E)
<class 'sage.schemes.elliptic_curves.ell_field.EllipticCurve_field_with_category'>
sage: E = EllipticCurve([RR(1),3]); E
Elliptic Curve defined by y^2 = x^3 + 1.00000000000000*x + 3.00000000000000 over Real Field with 53 bits of precision
sage: type(E)
<class 'sage.schemes.elliptic_curves.ell_field.EllipticCurve_field_with_category'>
sage: E = EllipticCurve([i,i]); E
Elliptic Curve defined by y^2 = x^3 + I*x + I over Symbolic Ring
sage: type(E)
<class 'sage.schemes.elliptic_curves.ell_field.EllipticCurve_field_with_category'>
sage: E.category()
Category of schemes over Symbolic Ring
sage: is_field(SR)
True
sage: F = FractionField(PolynomialRing(QQ,'t'))
sage: t = F.gen()
sage: E = EllipticCurve([t,0]); E
Elliptic Curve defined by y^2 = x^3 + t*x over Fraction Field of Univariate Polynomial Ring in t over Rational Field
sage: type(E)
<class 'sage.schemes.elliptic_curves.ell_field.EllipticCurve_field_with_category'>
sage: E.category()
Category of schemes over Fraction Field of Univariate Polynomial Ring in t over Rational Field
See :trac:`12517`::
sage: E = EllipticCurve([1..5])
sage: EllipticCurve(E.a_invariants())
Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational Field
"""
import ell_generic, ell_field, ell_finite_field, ell_number_field, ell_rational_field, ell_padic_field # here to avoid circular includes
if j is not None:
if not x is None:
if rings.is_Ring(x):
try:
j = x(j)
except (ZeroDivisionError, ValueError, TypeError):
raise ValueError, "First parameter must be a ring containing %s"%j
else:
raise ValueError, "First parameter (if present) must be a ring when j is specified"
return EllipticCurve_from_j(j)
assert x is not None
if is_SymbolicEquation(x):
x = x.lhs() - x.rhs()
if parent(x) is SR:
x = x._polynomial_(rings.QQ['x', 'y'])
if rings.is_MPolynomial(x) and y is None:
f = x
if f.degree() != 3:
raise ValueError, "Elliptic curves must be defined by a cubic polynomial."
if f.degrees() == (3,2):
x, y = f.parent().gens()
elif f.degree() == (2,3):
y, x = f.parent().gens()
elif len(f.parent().gens()) == 2 or len(f.parent().gens()) == 3 and f.is_homogeneous():
# We'd need a point too...
raise NotImplementedError, "Construction of an elliptic curve from a generic cubic not yet implemented."
else:
raise ValueError, "Defining polynomial must be a cubic polynomial in two variables."
try:
if f.coefficient(x**3) < 0:
f = -f
# is there a nicer way to extract the coefficients?
a1 = a2 = a3 = a4 = a6 = 0
for coeff, mon in f:
if mon == x**3:
assert coeff == 1
elif mon == x**2:
a2 = coeff
elif mon == x:
a4 = coeff
elif mon == 1:
a6 = coeff
elif mon == y**2:
assert coeff == -1
elif mon == x*y:
a1 = -coeff
elif mon == y:
a3 = -coeff
else:
assert False
return EllipticCurve([a1, a2, a3, a4, a6])
except AssertionError:
raise NotImplementedError, "Construction of an elliptic curve from a generic cubic not yet implemented."
if rings.is_Ring(x):
if rings.is_RationalField(x):
return ell_rational_field.EllipticCurve_rational_field(x, y)
elif rings.is_FiniteField(x) or (rings.is_IntegerModRing(x) and x.characteristic().is_prime()):
return ell_finite_field.EllipticCurve_finite_field(x, y)
elif rings.is_pAdicField(x):
return ell_padic_field.EllipticCurve_padic_field(x, y)
elif rings.is_NumberField(x):
return ell_number_field.EllipticCurve_number_field(x, y)
elif rings.is_Field(x):
return ell_field.EllipticCurve_field(x, y)
return ell_generic.EllipticCurve_generic(x, y)
if isinstance(x, unicode):
x = str(x)
if isinstance(x, str):
return ell_rational_field.EllipticCurve_rational_field(x)
if rings.is_RingElement(x) and y is None:
from sage.misc.misc import deprecation
deprecation("'EllipticCurve(j)' is deprecated; use 'EllipticCurve_from_j(j)' or 'EllipticCurve(j=j)' instead.")
# Fixed for all characteristics and cases by John Cremona
j=x
F=j.parent().fraction_field()
char=F.characteristic()
if char==2:
if j==0:
return EllipticCurve(F, [ 0, 0, 1, 0, 0 ])
else:
return EllipticCurve(F, [ 1, 0, 0, 0, 1/j ])
if char==3:
if j==0:
return EllipticCurve(F, [ 0, 0, 0, 1, 0 ])
else:
return EllipticCurve(F, [ 0, j, 0, 0, -j**2 ])
if j == 0:
return EllipticCurve(F, [ 0, 0, 0, 0, 1 ])
if j == 1728:
return EllipticCurve(F, [ 0, 0, 0, 1, 0 ])
k=j-1728
return EllipticCurve(F, [0,0,0,-3*j*k, -2*j*k**2])
if not isinstance(x, (list, tuple)):
raise TypeError, "invalid input to EllipticCurve constructor"
x = Sequence(x)
if not (len(x) in [2,5]):
raise ValueError, "sequence of coefficients must have length 2 or 5"
R = x.universe()
if isinstance(x[0], (rings.Rational, rings.Integer, int, long)):
return ell_rational_field.EllipticCurve_rational_field(x, y)
elif rings.is_NumberField(R):
return ell_number_field.EllipticCurve_number_field(x, y)
elif rings.is_pAdicField(R):
return ell_padic_field.EllipticCurve_padic_field(x, y)
elif rings.is_FiniteField(R) or (rings.is_IntegerModRing(R) and R.characteristic().is_prime()):
return ell_finite_field.EllipticCurve_finite_field(x, y)
elif rings.is_Field(R):
return ell_field.EllipticCurve_field(x, y)
return ell_generic.EllipticCurve_generic(x, y)