def expand(self):
"""
EXAMPLES::
sage: X = SchubertPolynomialRing(ZZ)
sage: X([2,1,3]).expand()
x0
sage: map(lambda x: x.expand(), [X(p) for p in Permutations(3)])
[1, x0 + x1, x0, x0*x1, x0^2, x0^2*x1]
TESTS: Calling .expand() should always return an element of an
MPolynomialRing
::
sage: X = SchubertPolynomialRing(ZZ)
sage: f = X([1]); f
X[1]
sage: type(f.expand())
<type 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomial_libsingular'>
sage: f.expand()
1
sage: f = X([1,2])
sage: type(f.expand())
<type 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomial_libsingular'>
sage: f = X([1,3,2,4])
sage: type(f.expand())
<type 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomial_libsingular'>
"""
p = symmetrica.t_SCHUBERT_POLYNOM(self)
if not is_MPolynomial(p):
R = PolynomialRing(self.parent().base_ring(), 1, 'x')
p = R(p)
return p
def __init__(self, poly, ambient=None):
"""
Return the affine hypersurface in the space ambient
defined by the polynomial poly.
If ambient is not given, it will be constructed based on
poly.
EXAMPLES::
sage: A.<x, y, z> = AffineSpace(ZZ, 3)
sage: AffineHypersurface(x*y-z^3, A)
Affine hypersurface defined by -z^3 + x*y in Affine Space of dimension 3 over Integer Ring
::
sage: A.<x, y, z> = QQ[]
sage: AffineHypersurface(x*y-z^3)
Affine hypersurface defined by -z^3 + x*y in Affine Space of dimension 3 over Rational Field
TESTS::
sage: H = AffineHypersurface(x*y-z^3)
sage: H == loads(dumps(H))
True
"""
if not is_MPolynomial(poly):
raise TypeError, "Defining polynomial (= %s) must be a multivariate polynomial" % poly
if ambient == None:
R = poly.parent()
from affine_space import AffineSpace
ambient = AffineSpace(R.base_ring(), R.ngens())
ambient._coordinate_ring = R
AlgebraicScheme_subscheme_affine.__init__(self, ambient, [poly])
def _element_constructor_(self, x):
"""
Coerce x into self.
EXAMPLES::
sage: X = SchubertPolynomialRing(QQ)
sage: X._element_constructor_([2,1,3])
X[2, 1]
sage: X._element_constructor_(Permutation([2,1,3]))
X[2, 1]
::
sage: R.<x1, x2, x3> = QQ[]
sage: X(x1^2*x2)
X[3, 2, 1]
"""
if isinstance(x, list):
perm = permutation.Permutation_class(x).remove_extra_fixed_points()
return self._from_dict({ perm: self.base_ring()(1) })
elif isinstance(x, permutation.Permutation_class):
perm = x.remove_extra_fixed_points()
return self._from_dict({ perm: self.base_ring()(1) })
elif is_MPolynomial(x):
return symmetrica.t_POLYNOM_SCHUBERT(x)
else:
raise TypeError
def expand(self):
"""
EXAMPLES::
sage: X = SchubertPolynomialRing(ZZ)
sage: X([2,1,3]).expand()
x0
sage: map(lambda x: x.expand(), [X(p) for p in Permutations(3)])
[1, x0 + x1, x0, x0*x1, x0^2, x0^2*x1]
TESTS: Calling .expand() should always return an element of an
MPolynomialRing
::
sage: X = SchubertPolynomialRing(ZZ)
sage: f = X([1]); f
X[1]
sage: type(f.expand())
<type 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomial_libsingular'>
sage: f.expand()
1
sage: f = X([1,2])
sage: type(f.expand())
<type 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomial_libsingular'>
sage: f = X([1,3,2,4])
sage: type(f.expand())
<type 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomial_libsingular'>
"""
p = symmetrica.t_SCHUBERT_POLYNOM(self)
if not is_MPolynomial(p):
R = PolynomialRing(self.parent().base_ring(), 1, 'x')
p = R(p)
return p
def _element_constructor_(self, x):
"""
Coerce x into self.
EXAMPLES::
sage: X = SchubertPolynomialRing(QQ)
sage: X._element_constructor_([2,1,3])
X[2, 1]
sage: X._element_constructor_(Permutation([2,1,3]))
X[2, 1]
::
sage: R.<x1, x2, x3> = QQ[]
sage: X(x1^2*x2)
X[3, 2, 1]
"""
if isinstance(x, list):
perm = permutation.Permutation_class(x).remove_extra_fixed_points()
return self._from_dict({perm: self.base_ring()(1)})
elif isinstance(x, permutation.Permutation_class):
perm = x.remove_extra_fixed_points()
return self._from_dict({perm: self.base_ring()(1)})
elif is_MPolynomial(x):
return symmetrica.t_POLYNOM_SCHUBERT(x)
else:
raise TypeError
def __init__(self, poly, ambient=None):
"""
Return the affine hypersurface in the space ambient
defined by the polynomial poly.
If ambient is not given, it will be constructed based on
poly.
EXAMPLES::
sage: A.<x, y, z> = AffineSpace(ZZ, 3)
sage: AffineHypersurface(x*y-z^3, A)
Affine hypersurface defined by -z^3 + x*y in Affine Space of dimension 3 over Integer Ring
::
sage: A.<x, y, z> = QQ[]
sage: AffineHypersurface(x*y-z^3)
Affine hypersurface defined by -z^3 + x*y in Affine Space of dimension 3 over Rational Field
TESTS::
sage: H = AffineHypersurface(x*y-z^3)
sage: H == loads(dumps(H))
True
"""
if not is_MPolynomial(poly):
raise TypeError, "Defining polynomial (= %s) must be a multivariate polynomial"%poly
if ambient == None:
R = poly.parent()
from affine_space import AffineSpace
ambient = AffineSpace(R.base_ring(), R.ngens())
ambient._coordinate_ring = R
AlgebraicScheme_subscheme_affine.__init__(self, ambient, [poly])
def Jacobian(X, **kwds):
"""
Return the Jacobian.
INPUT:
- ``X`` -- polynomial, algebraic variety, or anything else that
has a Jacobian elliptic curve.
- ``kwds`` -- optional keyword arguments.
The input ``X`` can be one of the following:
* A polynomial, see :func:`Jacobian_of_equation` for details.
* A curve, see :func:`Jacobian_of_curve` for details.
EXAMPLES::
sage: R.<u,v,w> = QQ[]
sage: Jacobian(u^3+v^3+w^3)
Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field
sage: C = Curve(u^3+v^3+w^3)
sage: Jacobian(C)
Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field
sage: P2.<u,v,w> = ProjectiveSpace(2, QQ)
sage: C = P2.subscheme(u^3+v^3+w^3)
sage: Jacobian(C)
Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field
sage: Jacobian(C, morphism=True)
Scheme morphism:
From: Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
u^3 + v^3 + w^3
To: Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field
Defn: Defined on coordinates by sending (u : v : w) to
(u*v^7*w + u*v^4*w^4 + u*v*w^7 :
v^9 + 3/2*v^6*w^3 - 3/2*v^3*w^6 - w^9 :
-v^6*w^3 - v^3*w^6)
"""
try:
return X.jacobian(**kwds)
except AttributeError:
pass
morphism = kwds.pop('morphism', False)
from sage.rings.all import is_MPolynomial
if is_MPolynomial(X):
if morphism:
from sage.schemes.plane_curves.constructor import Curve
return Jacobian_of_equation(X, curve=Curve(X), **kwds)
else:
return Jacobian_of_equation(X, **kwds)
from sage.schemes.all import is_Scheme
if is_Scheme(X) and X.dimension() == 1:
return Jacobian_of_curve(X, morphism=morphism, **kwds)
def QuarticCurve(F, PP=None, check=False):
"""
Returns the quartic curve defined by the polynomial F.
INPUT:
- F -- a polynomial in three variables, homogeneous of degree 4
- PP -- a projective plane (default:None)
- check -- whether to check for smoothness or not (default:False)
EXAMPLES::
sage: x,y,z=PolynomialRing(QQ,['x','y','z']).gens()
sage: QuarticCurve(x**4+y**4+z**4)
Quartic Curve over Rational Field defined by x^4 + y^4 + z^4
TESTS::
sage: QuarticCurve(x**3+y**3)
Traceback (most recent call last):
...
ValueError: Argument F (=x^3 + y^3) must be a homogeneous polynomial of degree 4
sage: QuarticCurve(x**4+y**4+z**3)
Traceback (most recent call last):
...
ValueError: Argument F (=x^4 + y^4 + z^3) must be a homogeneous polynomial of degree 4
sage: x,y=PolynomialRing(QQ,['x','y']).gens()
sage: QuarticCurve(x**4+y**4)
Traceback (most recent call last):
...
ValueError: Argument F (=x^4 + y^4) must be a polynomial in 3 variables
"""
if not is_MPolynomial(F):
raise ValueError("Argument F (=%s) must be a multivariate polynomial"%F)
P = F.parent()
if not P.ngens() == 3:
raise ValueError("Argument F (=%s) must be a polynomial in 3 variables"%F)
if not(F.is_homogeneous() and F.degree()==4):
raise ValueError("Argument F (=%s) must be a homogeneous polynomial of degree 4"%F)
if not PP is None:
if not is_ProjectiveSpace(PP) and PP.dimension == 2:
raise ValueError("Argument PP (=%s) must be a projective plane"%PP)
else:
PP = ProjectiveSpace(P)
if check:
raise NotImplementedError("Argument checking (for nonsingularity) is not implemented.")
return QuarticCurve_generic(PP, F)
def _element_constructor_(self, x):
"""
Coerce x into self.
EXAMPLES::
sage: X = SchubertPolynomialRing(QQ)
sage: X._element_constructor_([2,1,3])
X[2, 1]
sage: X._element_constructor_(Permutation([2,1,3]))
X[2, 1]
sage: R.<x1, x2, x3> = QQ[]
sage: X(x1^2*x2)
X[3, 2, 1]
TESTS:
We check that :trac:`12924` is fixed::
sage: X = SchubertPolynomialRing(QQ)
sage: X._element_constructor_([1,2,1])
Traceback (most recent call last):
...
ValueError: The input [1, 2, 1] is not a valid permutation
"""
if isinstance(x, list):
#checking the input to avoid symmetrica crashing Sage, see trac 12924
if not x in Permutations():
raise ValueError, "The input %s is not a valid permutation" % (
x)
perm = permutation.Permutation_class(x).remove_extra_fixed_points()
return self._from_dict({perm: self.base_ring()(1)})
elif isinstance(x, permutation.Permutation_class):
if not list(x) in Permutations():
raise ValueError, "The input %s is not a valid permutation" % (
x)
perm = x.remove_extra_fixed_points()
return self._from_dict({perm: self.base_ring()(1)})
elif is_MPolynomial(x):
return symmetrica.t_POLYNOM_SCHUBERT(x)
else:
raise TypeError
def QuarticCurve(F, PP=None, check=False):
"""
Returns the quartic curve defined by F.
"""
if not PP is None:
if not is_ProjectiveSpace(PP) and PP.dimension == 2:
raise TypeError, "Argument PP (=%s) must be a projective plane" % PP
elif not is_MPolynomial(F):
raise TypeError, \
"Argument F (=%s) must be a homogeneous multivariate polynomial"%F
else:
if not F.is_homogeneous():
"Argument F (=%s) must be a homogeneous polynomial" % F
P = F.parent()
if P.ngens() != 3:
"Argument F (=%s) must be a homogeneous multivariate polynomial in 3 variables" % F
PP = ProjectiveSpace(P)
if check:
raise TypeError, "Argument checking (for nonsingularity) is not implemented."
return QuarticCurve_generic(PP, F)
def QuarticCurve(F,PP=None,check=False):
"""
Returns the quartic curve defined by F.
"""
if not PP is None:
if not is_ProjectiveSpace(PP) and PP.dimension == 2:
raise TypeError, "Argument PP (=%s) must be a projective plane"%PP
elif not is_MPolynomial(F):
raise TypeError, \
"Argument F (=%s) must be a homogeneous multivariate polynomial"%F
else:
if not F.is_homogeneous():
"Argument F (=%s) must be a homogeneous polynomial"%F
P = F.parent()
if P.ngens() != 3:
"Argument F (=%s) must be a homogeneous multivariate polynomial in 3 variables"%F
PP = ProjectiveSpace(P)
if check:
raise TypeError, "Argument checking (for nonsingularity) is not implemented."
return QuarticCurve_generic(PP, F)
def _element_constructor_(self, x):
"""
Coerce x into self.
EXAMPLES::
sage: X = SchubertPolynomialRing(QQ)
sage: X._element_constructor_([2,1,3])
X[2, 1]
sage: X._element_constructor_(Permutation([2,1,3]))
X[2, 1]
sage: R.<x1, x2, x3> = QQ[]
sage: X(x1^2*x2)
X[3, 2, 1]
TESTS:
We check that :trac:`12924` is fixed::
sage: X = SchubertPolynomialRing(QQ)
sage: X._element_constructor_([1,2,1])
Traceback (most recent call last):
...
ValueError: The input [1, 2, 1] is not a valid permutation
"""
if isinstance(x, list):
#checking the input to avoid symmetrica crashing Sage, see trac 12924
if not x in Permutations():
raise ValueError, "The input %s is not a valid permutation"%(x)
perm = permutation.Permutation(x).remove_extra_fixed_points()
return self._from_dict({ perm: self.base_ring()(1) })
elif isinstance(x, permutation.Permutation):
if not list(x) in Permutations():
raise ValueError, "The input %s is not a valid permutation"%(x)
perm = x.remove_extra_fixed_points()
return self._from_dict({ perm: self.base_ring()(1) })
elif is_MPolynomial(x):
return symmetrica.t_POLYNOM_SCHUBERT(x)
else:
raise TypeError
def __init__(self, poly, ambient=None):
"""
Return the projective hypersurface in the space ambient
defined by the polynomial poly.
If ambient is not given, it will be constructed based on
poly.
EXAMPLES::
sage: P.<x, y, z> = ProjectiveSpace(ZZ, 2)
sage: ProjectiveHypersurface(x-y, P)
Projective hypersurface defined by x - y in Projective Space of dimension 2 over Integer Ring
::
sage: R.<x, y, z> = QQ[]
sage: ProjectiveHypersurface(x-y)
Projective hypersurface defined by x - y in Projective Space of dimension 2 over Rational Field
TESTS::
sage: H = ProjectiveHypersurface(x-y)
sage: H == loads(dumps(H))
True
"""
if not is_MPolynomial(poly):
raise TypeError, \
"Defining polynomial (=%s) must be a multivariate polynomial."%poly
if not poly.is_homogeneous():
raise TypeError, "Defining polynomial (=%s) must be homogeneous."%poly
if ambient == None:
R = poly.parent()
from projective_space import ProjectiveSpace
ambient = ProjectiveSpace(R.base_ring(), R.ngens()-1)
ambient._coordinate_ring = R
AlgebraicScheme_subscheme_projective.__init__(self, ambient, [poly])
def __init__(self, poly, ambient=None):
"""
Return the projective hypersurface in the space ambient
defined by the polynomial poly.
If ambient is not given, it will be constructed based on
poly.
EXAMPLES::
sage: P.<x, y, z> = ProjectiveSpace(ZZ, 2)
sage: ProjectiveHypersurface(x-y, P)
Projective hypersurface defined by x - y in Projective Space of dimension 2 over Integer Ring
::
sage: R.<x, y, z> = QQ[]
sage: ProjectiveHypersurface(x-y)
Projective hypersurface defined by x - y in Projective Space of dimension 2 over Rational Field
TESTS::
sage: H = ProjectiveHypersurface(x-y)
sage: H == loads(dumps(H))
True
"""
if not is_MPolynomial(poly):
raise TypeError, \
"Defining polynomial (=%s) must be a multivariate polynomial."%poly
if not poly.is_homogeneous():
raise TypeError, "Defining polynomial (=%s) must be homogeneous." % poly
if ambient == None:
R = poly.parent()
from projective_space import ProjectiveSpace
ambient = ProjectiveSpace(R.base_ring(), R.ngens() - 1)
ambient._coordinate_ring = R
AlgebraicScheme_subscheme_projective.__init__(self, ambient, [poly])
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 Conic(base_field, F=None, names=None, unique=True):
r"""
Return the plane projective conic curve defined by ``F``
over ``base_field``.
The input form ``Conic(F, names=None)`` is also accepted,
in which case the fraction field of the base ring of ``F``
is used as base field.
INPUT:
- ``base_field`` -- The base field of the conic.
- ``names`` -- a list, tuple, or comma separated string
of three variable names specifying the names
of the coordinate functions of the ambient
space `\Bold{P}^3`. If not specified or read
off from ``F``, then this defaults to ``'x,y,z'``.
- ``F`` -- a polynomial, list, matrix, ternary quadratic form,
or list or tuple of 5 points in the plane.
If ``F`` is a polynomial or quadratic form,
then the output is the curve in the projective plane
defined by ``F = 0``.
If ``F`` is a polynomial, then it must be a polynomial
of degree at most 2 in 2 variables, or a homogeneous
polynomial in of degree 2 in 3 variables.
If ``F`` is a matrix, then the output is the zero locus
of `(x,y,z) F (x,y,z)^t`.
If ``F`` is a list of coefficients, then it has
length 3 or 6 and gives the coefficients of
the monomials `x^2, y^2, z^2` or all 6 monomials
`x^2, xy, xz, y^2, yz, z^2` in lexicographic order.
If ``F`` is a list of 5 points in the plane, then the output
is a conic through those points.
- ``unique`` -- Used only if ``F`` is a list of points in the plane.
If the conic through the points is not unique, then
raise ``ValueError`` if and only if ``unique`` is True
OUTPUT:
A plane projective conic curve defined by ``F`` over a field.
EXAMPLES:
Conic curves given by polynomials ::
sage: X,Y,Z = QQ['X,Y,Z'].gens()
sage: Conic(X^2 - X*Y + Y^2 - Z^2)
Projective Conic Curve over Rational Field defined by X^2 - X*Y + Y^2 - Z^2
sage: x,y = GF(7)['x,y'].gens()
sage: Conic(x^2 - x + 2*y^2 - 3, 'U,V,W')
Projective Conic Curve over Finite Field of size 7 defined by U^2 + 2*V^2 - U*W - 3*W^2
Conic curves given by matrices ::
sage: Conic(matrix(QQ, [[1, 2, 0], [4, 0, 0], [7, 0, 9]]), 'x,y,z')
Projective Conic Curve over Rational Field defined by x^2 + 6*x*y + 7*x*z + 9*z^2
sage: x,y,z = GF(11)['x,y,z'].gens()
sage: C = Conic(x^2+y^2-2*z^2); C
Projective Conic Curve over Finite Field of size 11 defined by x^2 + y^2 - 2*z^2
sage: Conic(C.symmetric_matrix(), 'x,y,z')
Projective Conic Curve over Finite Field of size 11 defined by x^2 + y^2 - 2*z^2
Conics given by coefficients ::
sage: Conic(QQ, [1,2,3])
Projective Conic Curve over Rational Field defined by x^2 + 2*y^2 + 3*z^2
sage: Conic(GF(7), [1,2,3,4,5,6], 'X')
Projective Conic Curve over Finite Field of size 7 defined by X0^2 + 2*X0*X1 - 3*X1^2 + 3*X0*X2 - 2*X1*X2 - X2^2
The conic through a set of points ::
sage: C = Conic(QQ, [[10,2],[3,4],[-7,6],[7,8],[9,10]]); C
Projective Conic Curve over Rational Field defined by x^2 + 13/4*x*y - 17/4*y^2 - 35/2*x*z + 91/4*y*z - 37/2*z^2
sage: C.rational_point()
(10 : 2 : 1)
sage: C.point([3,4])
(3 : 4 : 1)
sage: a=AffineSpace(GF(13),2)
sage: Conic([a([x,x^2]) for x in range(5)])
Projective Conic Curve over Finite Field of size 13 defined by x^2 - y*z
"""
if not (is_IntegralDomain(base_field) or base_field == None):
if names is None:
names = F
F = base_field
base_field = None
if isinstance(F, (list,tuple)):
if len(F) == 1:
return Conic(base_field, F[0], names)
if names == None:
names = 'x,y,z'
if len(F) == 5:
L=[]
for f in F:
if isinstance(f, SchemeMorphism_point_affine):
C = Sequence(f, universe = base_field)
if len(C) != 2:
raise TypeError, "points in F (=%s) must be planar"%F
C.append(1)
elif isinstance(f, SchemeMorphism_point_projective_field):
C = Sequence(f, universe = base_field)
elif isinstance(f, (list, tuple)):
C = Sequence(f, universe = base_field)
if len(C) == 2:
C.append(1)
else:
raise TypeError, "F (=%s) must be a sequence of planar " \
"points" % F
if len(C) != 3:
raise TypeError, "points in F (=%s) must be planar" % F
P = C.universe()
if not is_IntegralDomain(P):
raise TypeError, "coordinates of points in F (=%s) must " \
"be in an integral domain" % F
L.append(Sequence([C[0]**2, C[0]*C[1], C[0]*C[2], C[1]**2,
C[1]*C[2], C[2]**2], P.fraction_field()))
M=Matrix(L)
if unique and M.rank() != 5:
raise ValueError, "points in F (=%s) do not define a unique " \
"conic" % F
con = Conic(base_field, Sequence(M.right_kernel().gen()), names)
con.point(F[0])
return con
F = Sequence(F, universe = base_field)
base_field = F.universe().fraction_field()
temp_ring = PolynomialRing(base_field, 3, names)
(x,y,z) = temp_ring.gens()
if len(F) == 3:
return Conic(F[0]*x**2 + F[1]*y**2 + F[2]*z**2)
if len(F) == 6:
return Conic(F[0]*x**2 + F[1]*x*y + F[2]*x*z + F[3]*y**2 + \
F[4]*y*z + F[5]*z**2)
raise TypeError, "F (=%s) must be a sequence of 3 or 6" \
"coefficients" % F
if is_QuadraticForm(F):
F = F.matrix()
if is_Matrix(F) and F.is_square() and F.ncols() == 3:
if names == None:
names = 'x,y,z'
temp_ring = PolynomialRing(F.base_ring(), 3, names)
F = vector(temp_ring.gens()) * F * vector(temp_ring.gens())
if not is_MPolynomial(F):
raise TypeError, "F (=%s) must be a three-variable polynomial or " \
"a sequence of points or coefficients" % F
if F.total_degree() != 2:
raise TypeError, "F (=%s) must have degree 2" % F
if base_field == None:
base_field = F.base_ring()
if not is_IntegralDomain(base_field):
raise ValueError, "Base field (=%s) must be a field" % base_field
base_field = base_field.fraction_field()
if names == None:
names = F.parent().variable_names()
pol_ring = PolynomialRing(base_field, 3, names)
if F.parent().ngens() == 2:
(x,y,z) = pol_ring.gens()
F = pol_ring(F(x/z,y/z)*z**2)
if F == 0:
raise ValueError, "F must be nonzero over base field %s" % base_field
if F.total_degree() != 2:
raise TypeError, "F (=%s) must have degree 2 over base field %s" % \
(F, base_field)
if F.parent().ngens() == 3:
P2 = ProjectiveSpace(2, base_field, names)
if is_PrimeFiniteField(base_field):
return ProjectiveConic_prime_finite_field(P2, F)
if is_FiniteField(base_field):
return ProjectiveConic_finite_field(P2, F)
if is_RationalField(base_field):
return ProjectiveConic_rational_field(P2, F)
if is_NumberField(base_field):
return ProjectiveConic_number_field(P2, F)
return ProjectiveConic_field(P2, F)
raise TypeError, "Number of variables of F (=%s) must be 2 or 3" % F
def Curve(F):
"""
Return the plane or space curve defined by `F`, where
`F` can be either a multivariate polynomial, a list or
tuple of polynomials, or an algebraic scheme.
If `F` is in two variables the curve is affine, and if it
is homogenous in `3` variables, then the curve is
projective.
EXAMPLE: A projective plane curve
::
sage: x,y,z = QQ['x,y,z'].gens()
sage: C = Curve(x^3 + y^3 + z^3); C
Projective Curve over Rational Field defined by x^3 + y^3 + z^3
sage: C.genus()
1
EXAMPLE: Affine plane curves
::
sage: x,y = GF(7)['x,y'].gens()
sage: C = Curve(y^2 + x^3 + x^10); C
Affine Curve over Finite Field of size 7 defined by x^10 + x^3 + y^2
sage: C.genus()
0
sage: x, y = QQ['x,y'].gens()
sage: Curve(x^3 + y^3 + 1)
Affine Curve over Rational Field defined by x^3 + y^3 + 1
EXAMPLE: A projective space curve
::
sage: x,y,z,w = QQ['x,y,z,w'].gens()
sage: C = Curve([x^3 + y^3 - z^3 - w^3, x^5 - y*z^4]); C
Projective Space Curve over Rational Field defined by x^3 + y^3 - z^3 - w^3, x^5 - y*z^4
sage: C.genus()
13
EXAMPLE: An affine space curve
::
sage: x,y,z = QQ['x,y,z'].gens()
sage: C = Curve([y^2 + x^3 + x^10 + z^7, x^2 + y^2]); C
Affine Space Curve over Rational Field defined by x^10 + z^7 + x^3 + y^2, x^2 + y^2
sage: C.genus()
47
EXAMPLE: We can also make non-reduced non-irreducible curves.
::
sage: x,y,z = QQ['x,y,z'].gens()
sage: Curve((x-y)*(x+y))
Projective Conic Curve over Rational Field defined by x^2 - y^2
sage: Curve((x-y)^2*(x+y)^2)
Projective Curve over Rational Field defined by x^4 - 2*x^2*y^2 + y^4
EXAMPLE: A union of curves is a curve.
::
sage: x,y,z = QQ['x,y,z'].gens()
sage: C = Curve(x^3 + y^3 + z^3)
sage: D = Curve(x^4 + y^4 + z^4)
sage: C.union(D)
Projective Curve over Rational Field defined by
x^7 + x^4*y^3 + x^3*y^4 + y^7 + x^4*z^3 + y^4*z^3 + x^3*z^4 + y^3*z^4 + z^7
The intersection is not a curve, though it is a scheme.
::
sage: X = C.intersection(D); X
Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
x^3 + y^3 + z^3,
x^4 + y^4 + z^4
Note that the intersection has dimension `0`.
::
sage: X.dimension()
0
sage: I = X.defining_ideal(); I
Ideal (x^3 + y^3 + z^3, x^4 + y^4 + z^4) of Multivariate Polynomial Ring in x, y, z over Rational Field
EXAMPLE: In three variables, the defining equation must be
homogeneous.
If the parent polynomial ring is in three variables, then the
defining ideal must be homogeneous.
::
sage: x,y,z = QQ['x,y,z'].gens()
sage: Curve(x^2+y^2)
Projective Conic Curve over Rational Field defined by x^2 + y^2
sage: Curve(x^2+y^2+z)
Traceback (most recent call last):
...
TypeError: x^2 + y^2 + z is not a homogeneous polynomial!
The defining polynomial must always be nonzero::
sage: P1.<x,y> = ProjectiveSpace(1,GF(5))
sage: Curve(0*x)
Traceback (most recent call last):
...
ValueError: defining polynomial of curve must be nonzero
"""
if is_AlgebraicScheme(F):
return Curve(F.defining_polynomials())
if isinstance(F, (list, tuple)):
if len(F) == 1:
return Curve(F[0])
F = Sequence(F)
P = F.universe()
if not is_MPolynomialRing(P):
raise TypeError, "universe of F must be a multivariate polynomial ring"
for f in F:
if not f.is_homogeneous():
A = AffineSpace(P.ngens(), P.base_ring())
A._coordinate_ring = P
return AffineSpaceCurve_generic(A, F)
A = ProjectiveSpace(P.ngens() - 1, P.base_ring())
A._coordinate_ring = P
return ProjectiveSpaceCurve_generic(A, F)
if not is_MPolynomial(F):
raise TypeError, "F (=%s) must be a multivariate polynomial" % F
P = F.parent()
k = F.base_ring()
if F.parent().ngens() == 2:
if F == 0:
raise ValueError, "defining polynomial of curve must be nonzero"
A2 = AffineSpace(2, P.base_ring())
A2._coordinate_ring = P
if is_FiniteField(k):
if k.is_prime_field():
return AffineCurve_prime_finite_field(A2, F)
else:
return AffineCurve_finite_field(A2, F)
else:
return AffineCurve_generic(A2, F)
elif F.parent().ngens() == 3:
if F == 0:
raise ValueError, "defining polynomial of curve must be nonzero"
P2 = ProjectiveSpace(2, P.base_ring())
P2._coordinate_ring = P
if F.total_degree() == 2 and k.is_field():
return Conic(F)
if is_FiniteField(k):
if k.is_prime_field():
return ProjectiveCurve_prime_finite_field(P2, F)
else:
return ProjectiveCurve_finite_field(P2, F)
else:
return ProjectiveCurve_generic(P2, F)
else:
raise TypeError, "Number of variables of F (=%s) must be 2 or 3" % F
def Curve(F):
"""
Return the plane or space curve defined by `F`, where
`F` can be either a multivariate polynomial, a list or
tuple of polynomials, or an algebraic scheme.
If `F` is in two variables the curve is affine, and if it
is homogenous in `3` variables, then the curve is
projective.
EXAMPLE: A projective plane curve
::
sage: x,y,z = QQ['x,y,z'].gens()
sage: C = Curve(x^3 + y^3 + z^3); C
Projective Curve over Rational Field defined by x^3 + y^3 + z^3
sage: C.genus()
1
EXAMPLE: Affine plane curves
::
sage: x,y = GF(7)['x,y'].gens()
sage: C = Curve(y^2 + x^3 + x^10); C
Affine Curve over Finite Field of size 7 defined by x^10 + x^3 + y^2
sage: C.genus()
0
sage: x, y = QQ['x,y'].gens()
sage: Curve(x^3 + y^3 + 1)
Affine Curve over Rational Field defined by x^3 + y^3 + 1
EXAMPLE: A projective space curve
::
sage: x,y,z,w = QQ['x,y,z,w'].gens()
sage: C = Curve([x^3 + y^3 - z^3 - w^3, x^5 - y*z^4]); C
Projective Space Curve over Rational Field defined by x^3 + y^3 - z^3 - w^3, x^5 - y*z^4
sage: C.genus()
13
EXAMPLE: An affine space curve
::
sage: x,y,z = QQ['x,y,z'].gens()
sage: C = Curve([y^2 + x^3 + x^10 + z^7, x^2 + y^2]); C
Affine Space Curve over Rational Field defined by x^10 + z^7 + x^3 + y^2, x^2 + y^2
sage: C.genus()
47
EXAMPLE: We can also make non-reduced non-irreducible curves.
::
sage: x,y,z = QQ['x,y,z'].gens()
sage: Curve((x-y)*(x+y))
Projective Conic Curve over Rational Field defined by x^2 - y^2
sage: Curve((x-y)^2*(x+y)^2)
Projective Curve over Rational Field defined by x^4 - 2*x^2*y^2 + y^4
EXAMPLE: A union of curves is a curve.
::
sage: x,y,z = QQ['x,y,z'].gens()
sage: C = Curve(x^3 + y^3 + z^3)
sage: D = Curve(x^4 + y^4 + z^4)
sage: C.union(D)
Projective Curve over Rational Field defined by
x^7 + x^4*y^3 + x^3*y^4 + y^7 + x^4*z^3 + y^4*z^3 + x^3*z^4 + y^3*z^4 + z^7
The intersection is not a curve, though it is a scheme.
::
sage: X = C.intersection(D); X
Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
x^3 + y^3 + z^3,
x^4 + y^4 + z^4
Note that the intersection has dimension `0`.
::
sage: X.dimension()
0
sage: I = X.defining_ideal(); I
Ideal (x^3 + y^3 + z^3, x^4 + y^4 + z^4) of Multivariate Polynomial Ring in x, y, z over Rational Field
EXAMPLE: In three variables, the defining equation must be
homogeneous.
If the parent polynomial ring is in three variables, then the
defining ideal must be homogeneous.
::
sage: x,y,z = QQ['x,y,z'].gens()
sage: Curve(x^2+y^2)
Projective Conic Curve over Rational Field defined by x^2 + y^2
sage: Curve(x^2+y^2+z)
Traceback (most recent call last):
...
TypeError: x^2 + y^2 + z is not a homogeneous polynomial!
The defining polynomial must always be nonzero::
sage: P1.<x,y> = ProjectiveSpace(1,GF(5))
sage: Curve(0*x)
Traceback (most recent call last):
...
ValueError: defining polynomial of curve must be nonzero
"""
if is_AlgebraicScheme(F):
return Curve(F.defining_polynomials())
if isinstance(F, (list, tuple)):
if len(F) == 1:
return Curve(F[0])
F = Sequence(F)
P = F.universe()
if not is_MPolynomialRing(P):
raise TypeError, "universe of F must be a multivariate polynomial ring"
for f in F:
if not f.is_homogeneous():
A = AffineSpace(P.ngens(), P.base_ring())
A._coordinate_ring = P
return AffineSpaceCurve_generic(A, F)
A = ProjectiveSpace(P.ngens()-1, P.base_ring())
A._coordinate_ring = P
return ProjectiveSpaceCurve_generic(A, F)
if not is_MPolynomial(F):
raise TypeError, "F (=%s) must be a multivariate polynomial"%F
P = F.parent()
k = F.base_ring()
if F.parent().ngens() == 2:
if F == 0:
raise ValueError, "defining polynomial of curve must be nonzero"
A2 = AffineSpace(2, P.base_ring())
A2._coordinate_ring = P
if is_FiniteField(k):
if k.is_prime_field():
return AffineCurve_prime_finite_field(A2, F)
else:
return AffineCurve_finite_field(A2, F)
else:
return AffineCurve_generic(A2, F)
elif F.parent().ngens() == 3:
if F == 0:
raise ValueError, "defining polynomial of curve must be nonzero"
P2 = ProjectiveSpace(2, P.base_ring())
P2._coordinate_ring = P
if F.total_degree() == 2 and k.is_field():
return Conic(F)
if is_FiniteField(k):
if k.is_prime_field():
return ProjectiveCurve_prime_finite_field(P2, F)
else:
return ProjectiveCurve_finite_field(P2, F)
else:
return ProjectiveCurve_generic(P2, F)
else:
raise TypeError, "Number of variables of F (=%s) must be 2 or 3"%F
def EllipticCurve(x=None, y=None, j=None, minimal_twist=True):
r"""
Construct an elliptic curve.
In Sage, an elliptic curve is always specified by its a-invariants
.. math::
y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6.
INPUT:
There are several ways to construct an elliptic curve:
- ``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 `a_1=a_2=a_3=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=j0)`` or ``EllipticCurve_from_j(j0)``: Return
an elliptic curve with j-invariant ``j0``.
- ``EllipticCurve(polynomial)``: Read off the a-invariants from
the polynomial coefficients, see
:func:`EllipticCurve_from_Weierstrass_polynomial`.
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
Old Cremona labels are allowed::
sage: EllipticCurve('2400FF')
Elliptic Curve defined by y^2 = x^3 + x^2 + 2*x + 8 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
By default, when a rational value of `j` is given, the constructed
curve is a minimal twist (minimal conductor for curves with that
`j`-invariant). This can be changed by setting the optional
parameter ``minimal_twist``, which is True by default, to False::
sage: EllipticCurve(j=100)
Elliptic Curve defined by y^2 = x^3 + x^2 + 3392*x + 307888 over Rational Field
sage: E =EllipticCurve(j=100); E
Elliptic Curve defined by y^2 = x^3 + x^2 + 3392*x + 307888 over Rational Field
sage: E.conductor()
33129800
sage: E.j_invariant()
100
sage: E =EllipticCurve(j=100, minimal_twist=False); E
Elliptic Curve defined by y^2 = x^3 + 488400*x - 530076800 over Rational Field
sage: E.conductor()
298168200
sage: E.j_invariant()
100
Without this option, constructing the curve could take a long time
since both `j` and `j-1728` have to be factored to compute the
minimal twist (see :trac:`13100`)::
sage: E = EllipticCurve_from_j(2^256+1,minimal_twist=False)
sage: E.j_invariant() == 2^256+1
True
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: SR in Fields()
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
See :trac:`11773`::
sage: E = EllipticCurve()
Traceback (most recent call last):
...
TypeError: invalid input to EllipticCurve constructor
"""
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, minimal_twist)
if x is None:
raise TypeError, "invalid input to EllipticCurve constructor"
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):
if y is None:
return EllipticCurve_from_Weierstrass_polynomial(x)
else:
return EllipticCurve_from_cubic(x, y, morphism=False)
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 x in _Fields:
return ell_field.EllipticCurve_field(x, y)
return ell_generic.EllipticCurve_generic(x, y)
if isinstance(x, unicode):
x = str(x)
if isinstance(x, basestring):
return ell_rational_field.EllipticCurve_rational_field(x)
if rings.is_RingElement(x) and y is None:
raise TypeError, "invalid input to EllipticCurve constructor"
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 R in _Fields:
return ell_field.EllipticCurve_field(x, y)
return ell_generic.EllipticCurve_generic(x, y)
def EllipticCurve(x=None, y=None, j=None, minimal_twist=True):
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=j0) or EllipticCurve_from_j(j0): Return an
elliptic curve with j-invariant `j0`.
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
Old Cremona labels are allowed::
sage: EllipticCurve('2400FF')
Elliptic Curve defined by y^2 = x^3 + x^2 + 2*x + 8 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
By default, when a rational value of `j` is given, the constructed
curve is a minimal twist (minimal conductor for curves with that
`j`-invariant). This can be changed by setting the optional
parameter ``minimal_twist``, which is True by default, to False::
sage: EllipticCurve(j=100)
Elliptic Curve defined by y^2 = x^3 + x^2 + 3392*x + 307888 over Rational Field
sage: E =EllipticCurve(j=100); E
Elliptic Curve defined by y^2 = x^3 + x^2 + 3392*x + 307888 over Rational Field
sage: E.conductor()
33129800
sage: E.j_invariant()
100
sage: E =EllipticCurve(j=100, minimal_twist=False); E
Elliptic Curve defined by y^2 = x^3 + 488400*x - 530076800 over Rational Field
sage: E.conductor()
298168200
sage: E.j_invariant()
100
Without this option, constructing the curve could take a long time
since both `j` and `j-1728` have to be factored to compute the
minimal twist (see :trac:`13100`)::
sage: E = EllipticCurve_from_j(2^256+1,minimal_twist=False)
sage: E.j_invariant() == 2^256+1
True
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: SR in Fields()
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
See :trac:`11773`::
sage: E = EllipticCurve()
Traceback (most recent call last):
...
TypeError: invalid input to EllipticCurve constructor
"""
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, minimal_twist)
if x is None:
raise TypeError, "invalid input to EllipticCurve constructor"
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 x in _Fields:
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:
raise TypeError, "invalid input to EllipticCurve constructor"
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 R in _Fields:
return ell_field.EllipticCurve_field(x, y)
return ell_generic.EllipticCurve_generic(x, y)
def Conic(base_field, F=None, names=None, unique=True):
r"""
Return the plane projective conic curve defined by ``F``
over ``base_field``.
The input form ``Conic(F, names=None)`` is also accepted,
in which case the fraction field of the base ring of ``F``
is used as base field.
INPUT:
- ``base_field`` -- The base field of the conic.
- ``names`` -- a list, tuple, or comma separated string
of three variable names specifying the names
of the coordinate functions of the ambient
space `\Bold{P}^3`. If not specified or read
off from ``F``, then this defaults to ``'x,y,z'``.
- ``F`` -- a polynomial, list, matrix, ternary quadratic form,
or list or tuple of 5 points in the plane.
If ``F`` is a polynomial or quadratic form,
then the output is the curve in the projective plane
defined by ``F = 0``.
If ``F`` is a polynomial, then it must be a polynomial
of degree at most 2 in 2 variables, or a homogeneous
polynomial in of degree 2 in 3 variables.
If ``F`` is a matrix, then the output is the zero locus
of `(x,y,z) F (x,y,z)^t`.
If ``F`` is a list of coefficients, then it has
length 3 or 6 and gives the coefficients of
the monomials `x^2, y^2, z^2` or all 6 monomials
`x^2, xy, xz, y^2, yz, z^2` in lexicographic order.
If ``F`` is a list of 5 points in the plane, then the output
is a conic through those points.
- ``unique`` -- Used only if ``F`` is a list of points in the plane.
If the conic through the points is not unique, then
raise ``ValueError`` if and only if ``unique`` is True
OUTPUT:
A plane projective conic curve defined by ``F`` over a field.
EXAMPLES:
Conic curves given by polynomials ::
sage: X,Y,Z = QQ['X,Y,Z'].gens()
sage: Conic(X^2 - X*Y + Y^2 - Z^2)
Projective Conic Curve over Rational Field defined by X^2 - X*Y + Y^2 - Z^2
sage: x,y = GF(7)['x,y'].gens()
sage: Conic(x^2 - x + 2*y^2 - 3, 'U,V,W')
Projective Conic Curve over Finite Field of size 7 defined by U^2 + 2*V^2 - U*W - 3*W^2
Conic curves given by matrices ::
sage: Conic(matrix(QQ, [[1, 2, 0], [4, 0, 0], [7, 0, 9]]), 'x,y,z')
Projective Conic Curve over Rational Field defined by x^2 + 6*x*y + 7*x*z + 9*z^2
sage: x,y,z = GF(11)['x,y,z'].gens()
sage: C = Conic(x^2+y^2-2*z^2); C
Projective Conic Curve over Finite Field of size 11 defined by x^2 + y^2 - 2*z^2
sage: Conic(C.symmetric_matrix(), 'x,y,z')
Projective Conic Curve over Finite Field of size 11 defined by x^2 + y^2 - 2*z^2
Conics given by coefficients ::
sage: Conic(QQ, [1,2,3])
Projective Conic Curve over Rational Field defined by x^2 + 2*y^2 + 3*z^2
sage: Conic(GF(7), [1,2,3,4,5,6], 'X')
Projective Conic Curve over Finite Field of size 7 defined by X0^2 + 2*X0*X1 - 3*X1^2 + 3*X0*X2 - 2*X1*X2 - X2^2
The conic through a set of points ::
sage: C = Conic(QQ, [[10,2],[3,4],[-7,6],[7,8],[9,10]]); C
Projective Conic Curve over Rational Field defined by x^2 + 13/4*x*y - 17/4*y^2 - 35/2*x*z + 91/4*y*z - 37/2*z^2
sage: C.rational_point()
(10 : 2 : 1)
sage: C.point([3,4])
(3 : 4 : 1)
sage: a=AffineSpace(GF(13),2)
sage: Conic([a([x,x^2]) for x in range(5)])
Projective Conic Curve over Finite Field of size 13 defined by x^2 - y*z
"""
if not (is_IntegralDomain(base_field) or base_field == None):
if names is None:
names = F
F = base_field
base_field = None
if isinstance(F, (list,tuple)):
if len(F) == 1:
return Conic(base_field, F[0], names)
if names == None:
names = 'x,y,z'
if len(F) == 5:
L=[]
for f in F:
if isinstance(f, SchemeMorphism_point_affine):
C = Sequence(f, universe = base_field)
if len(C) != 2:
raise TypeError, "points in F (=%s) must be planar"%F
C.append(1)
elif isinstance(f, SchemeMorphism_point_projective_field):
C = Sequence(f, universe = base_field)
elif isinstance(f, (list, tuple)):
C = Sequence(f, universe = base_field)
if len(C) == 2:
C.append(1)
else:
raise TypeError, "F (=%s) must be a sequence of planar " \
"points" % F
if len(C) != 3:
raise TypeError, "points in F (=%s) must be planar" % F
P = C.universe()
if not is_IntegralDomain(P):
raise TypeError, "coordinates of points in F (=%s) must " \
"be in an integral domain" % F
L.append(Sequence([C[0]**2, C[0]*C[1], C[0]*C[2], C[1]**2,
C[1]*C[2], C[2]**2], P.fraction_field()))
M=Matrix(L)
if unique and M.rank() != 5:
raise ValueError, "points in F (=%s) do not define a unique " \
"conic" % F
con = Conic(base_field, Sequence(M.right_kernel().gen()), names)
con.point(F[0])
return con
F = Sequence(F, universe = base_field)
base_field = F.universe().fraction_field()
temp_ring = PolynomialRing(base_field, 3, names)
(x,y,z) = temp_ring.gens()
if len(F) == 3:
return Conic(F[0]*x**2 + F[1]*y**2 + F[2]*z**2)
if len(F) == 6:
return Conic(F[0]*x**2 + F[1]*x*y + F[2]*x*z + F[3]*y**2 + \
F[4]*y*z + F[5]*z**2)
raise TypeError, "F (=%s) must be a sequence of 3 or 6" \
"coefficients" % F
if is_QuadraticForm(F):
F = F.matrix()
if is_Matrix(F) and F.is_square() and F.ncols() == 3:
if names == None:
names = 'x,y,z'
temp_ring = PolynomialRing(F.base_ring(), 3, names)
F = vector(temp_ring.gens()) * F * vector(temp_ring.gens())
if not is_MPolynomial(F):
raise TypeError, "F (=%s) must be a three-variable polynomial or " \
"a sequence of points or coefficients" % F
if F.total_degree() != 2:
raise TypeError, "F (=%s) must have degree 2" % F
if base_field == None:
base_field = F.base_ring()
if not is_IntegralDomain(base_field):
raise ValueError, "Base field (=%s) must be a field" % base_field
base_field = base_field.fraction_field()
if names == None:
names = F.parent().variable_names()
pol_ring = PolynomialRing(base_field, 3, names)
if F.parent().ngens() == 2:
(x,y,z) = pol_ring.gens()
F = pol_ring(F(x/z,y/z)*z**2)
if F == 0:
raise ValueError, "F must be nonzero over base field %s" % base_field
if F.total_degree() != 2:
raise TypeError, "F (=%s) must have degree 2 over base field %s" % \
(F, base_field)
if F.parent().ngens() == 3:
P2 = ProjectiveSpace(2, base_field, names)
if is_PrimeFiniteField(base_field):
return ProjectiveConic_prime_finite_field(P2, F)
if is_FiniteField(base_field):
return ProjectiveConic_finite_field(P2, F)
if is_RationalField(base_field):
return ProjectiveConic_rational_field(P2, F)
if is_NumberField(base_field):
return ProjectiveConic_number_field(P2, F)
return ProjectiveConic_field(P2, F)
raise TypeError, "Number of variables of F (=%s) must be 2 or 3" % F
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.
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'>
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'>
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'>
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'>
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'>
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'>
"""
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):
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)