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 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 lseries_dokchitser(E, prec=53):
"""
Return the Dokchitser L-series object associated to the elliptic
curve E, which may be defined over the rational numbers or a
number field. Also prec is the number of bits of precision to
which evaluation of the L-series occurs.
INPUT:
- E -- elliptic curve over a number field (or QQ)
- prec -- integer (default: 53) precision in *bits*
OUTPUT:
- Dokchitser L-function object
EXAMPLES::
A curve over Q(sqrt(5)), for which we have an optimized implementation::
sage: from psage.ellcurve.lseries.lseries_nf import lseries_dokchitser
sage: K.<a> = NumberField(x^2-x-1); E = EllipticCurve([0,-a,a,0,0])
sage: L = lseries_dokchitser(E); L
Dokchitser L-function of Elliptic Curve defined by y^2 + a*y = x^3 + (-a)*x^2 over Number Field in a with defining polynomial x^2 - x - 1
sage: L(1)
0.422214159001667
sage: L.taylor_series(1,5)
0.422214159001667 + 0.575883864741340*z - 0.102163426876721*z^2 - 0.158119743123727*z^3 + 0.120350687595265*z^4 + O(z^5)
Higher precision::
sage: L = lseries_dokchitser(E, 200)
sage: L(1)
0.42221415900166715092967967717023093014455137669360598558872
A curve over Q(i)::
sage: K.<i> = NumberField(x^2 + 1)
sage: E = EllipticCurve(K, [1,0])
sage: E.conductor().norm()
256
sage: L = lseries_dokchitser(E, 10)
sage: L.taylor_series(1,5)
0.86 + 0.58*z - 0.62*z^2 + 0.19*z^3 + 0.18*z^4 + O(z^5)
More examples::
sage: lseries_dokchitser(EllipticCurve([0,-1,1,0,0]), 10)(1)
0.25
sage: K.<i> = NumberField(x^2+1)
sage: lseries_dokchitser(EllipticCurve(K, [0,-1,1,0,0]), 10)(1)
0.37
sage: K.<a> = NumberField(x^2-x-1)
sage: lseries_dokchitser(EllipticCurve(K, [0,-1,1,0,0]), 10)(1)
0.72
sage: E = EllipticCurve([0,-1,1,0,0])
sage: E.quadratic_twist(2).rank()
1
sage: K.<d> = NumberField(x^2-2)
sage: L = lseries_dokchitser(EllipticCurve(K, [0,-1,1,0,0]), 10)
sage: L(1)
0
sage: L.taylor_series(1, 5)
0.58*z + 0.20*z^2 - 0.50*z^3 + 0.28*z^4 + O(z^5)
You can use this function as an algorithm to compute the sign of the functional
equation (global root number)::
sage: E = EllipticCurve([1..5])
sage: E.root_number()
-1
sage: L = lseries_dokchitser(E,32); L
Dokchitser L-function of Elliptic Curve defined by y^2 + x*y = x^3 - x^2 + 4*x + 3 over Rational Field
sage: L.eps
-1
Over QQ, this isn't so useful (since Sage has a root_number
method), but over number fields it is very useful::
sage: K.<a> = NumberField(x^2 - x - 1)
sage: E1=EllipticCurve([0,-a-1,1,a,0]); E0 = EllipticCurve([0,-a,a,0,0])
sage: lseries_dokchitser(E1, 16).eps
-1
sage: E1.rank()
1
sage: lseries_dokchitser(E0, 16).eps
1
sage: E0.rank()
0
"""
# The code assumes in various places that we have a global minimal model,
# for example, in anlist_sqrt5 above.
E = E.global_minimal_model()
# Check that we're over a number field.
K = E.base_field()
if not is_NumberField(K):
raise TypeError("base field must be a number field")
# Compute norm of the conductor -- awkward because QQ elements have no norm method (they should).
N = E.conductor()
if K != QQ:
N = N.norm()
# We guess the sign epsilon in the functional equation to be +1
# first. If our guess is wrong then we just choose the other
# possibility.
epsilon = 1
# Define the Dokchitser L-function object with all parameters set:
L = Dokchitser(conductor=N * K.discriminant()**2,
gammaV=[0] * K.degree() + [1] * K.degree(),
weight=2,
eps=epsilon,
poles=[],
prec=prec)
# Find out how many coefficients of the Dirichlet series are needed
# to compute to the requested precision.
n = L.num_coeffs()
# print "num coeffs = %s"%n
# Compute the Dirichlet series coefficients
coeffs = anlist(E, n)[1:]
# Define a string that when evaluated in PARI defines a function
# a(k), which returns the Dirichlet coefficient a_k.
s = 'v=%s; a(k)=v[k];' % coeffs
# Actually tell the L-series / PARI about the coefficients.
L.init_coeffs('a(k)', pari_precode=s)
# Test that the functional equation is satisfied. This will very,
# very, very likely if we chose the sign of the functional
# equation incorrectly, or made any mistake in computing the
# Dirichlet series coefficients.
tiny = max(1e-8, old_div(1.0, 2**(prec - 1)))
if abs(L.check_functional_equation()) > tiny:
# The test failed, so we try the other choice of functional equation.
epsilon *= -1
L.eps = epsilon
# It is not necessary to recreate L -- just tell PARI the different sign.
L._gp_eval('sgn = %s' % epsilon)
# Once again, verify that the functional equation is
# satisfied. If it is, then we've got it. If it isn't, then
# there is definitely some other subtle bug, probably in computed
# the Dirichlet series coefficients.
if abs(L.check_functional_equation()) > tiny:
raise RuntimeError(
"Functional equation not numerically satisfied for either choice of sign"
)
L.rename('Dokchitser L-function of %s' % E)
return L
def check_prime(K, P):
r"""
Function to check that `P` determines a prime of `K`, and return that ideal.
INPUT:
- ``K`` -- a number field (including `\QQ`).
- ``P`` -- an element of ``K`` or a (fractional) ideal of ``K``.
OUTPUT:
- If ``K`` is `\QQ`: the prime integer equal to or which generates `P`.
- If ``K`` is not `\QQ`: the prime ideal equal to or generated by `P`.
.. note::
If `P` is not a prime and does not generate a prime, a TypeError is raised.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.ell_local_data import check_prime
sage: check_prime(QQ,3)
3
sage: check_prime(QQ,ZZ.ideal(31))
31
sage: K.<a>=NumberField(x^2-5)
sage: check_prime(K,a)
Fractional ideal (a)
sage: check_prime(K,a+1)
Fractional ideal (a + 1)
sage: [check_prime(K,P) for P in K.primes_above(31)]
[Fractional ideal (5/2*a + 1/2), Fractional ideal (5/2*a - 1/2)]
"""
if K is QQ:
if isinstance(P, (int, long, Integer)):
P = Integer(P)
if P.is_prime():
return P
else:
raise TypeError, "%s is not prime" % P
else:
if is_Ideal(P) and P.base_ring() is ZZ and P.is_prime():
return P.gen()
raise TypeError, "%s is not a prime ideal of %s" % (P, ZZ)
if not is_NumberField(K):
raise TypeError, "%s is not a number field" % K
if is_NumberFieldFractionalIdeal(P):
if P.is_prime():
return P
else:
raise TypeError, "%s is not a prime ideal of %s" % (P, K)
if is_NumberFieldElement(P):
if P in K:
P = K.ideal(P)
else:
raise TypeError, "%s is not an element of %s" % (P, K)
if P.is_prime():
return P
else:
raise TypeError, "%s is not a prime ideal of %s" % (P, K)
raise TypeError, "%s is not a valid prime of %s" % (P, K)
def check_prime(K, P):
r"""
Function to check that `P` determines a prime of `K`, and return that ideal.
INPUT:
- ``K`` -- a number field (including `\QQ`).
- ``P`` -- an element of ``K`` or a (fractional) ideal of ``K``.
OUTPUT:
- If ``K`` is `\QQ`: the prime integer equal to or which generates `P`.
- If ``K`` is not `\QQ`: the prime ideal equal to or generated by `P`.
.. note::
If `P` is not a prime and does not generate a prime, a TypeError is raised.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.ell_local_data import check_prime
sage: check_prime(QQ,3)
3
sage: check_prime(QQ,ZZ.ideal(31))
31
sage: K.<a>=NumberField(x^2-5)
sage: check_prime(K,a)
Fractional ideal (a)
sage: check_prime(K,a+1)
Fractional ideal (a + 1)
sage: [check_prime(K,P) for P in K.primes_above(31)]
[Fractional ideal (5/2*a + 1/2), Fractional ideal (5/2*a - 1/2)]
"""
if K is QQ:
if isinstance(P, (int, long, Integer)):
P = Integer(P)
if P.is_prime():
return P
else:
raise TypeError, "%s is not prime" % P
else:
if is_Ideal(P) and P.base_ring() is ZZ and P.is_prime():
return P.gen()
raise TypeError, "%s is not a prime ideal of %s" % (P, ZZ)
if not is_NumberField(K):
raise TypeError, "%s is not a number field" % K
if is_NumberFieldFractionalIdeal(P):
if P.is_prime():
return P
else:
raise TypeError, "%s is not a prime ideal of %s" % (P, K)
if is_NumberFieldElement(P):
if P in K:
P = K.ideal(P)
else:
raise TypeError, "%s is not an element of %s" % (P, K)
if P.is_prime():
return P
else:
raise TypeError, "%s is not a prime ideal of %s" % (P, K)
raise TypeError, "%s is not a valid prime of %s" % (P, K)
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 lseries_dokchitser(E, prec=53):
"""
Return the Dokchitser L-series object associated to the elliptic
curve E, which may be defined over the rational numbers or a
number field. Also prec is the number of bits of precision to
which evaluation of the L-series occurs.
INPUT:
- E -- elliptic curve over a number field (or QQ)
- prec -- integer (default: 53) precision in *bits*
OUTPUT:
- Dokchitser L-function object
EXAMPLES::
A curve over Q(sqrt(5)), for which we have an optimized implementation::
sage: from psage.ellcurve.lseries.lseries_nf import lseries_dokchitser
sage: K.<a> = NumberField(x^2-x-1); E = EllipticCurve([0,-a,a,0,0])
sage: L = lseries_dokchitser(E); L
Dokchitser L-function of Elliptic Curve defined by y^2 + a*y = x^3 + (-a)*x^2 over Number Field in a with defining polynomial x^2 - x - 1
sage: L(1)
0.422214159001667
sage: L.taylor_series(1,5)
0.422214159001667 + 0.575883864741340*z - 0.102163426876721*z^2 - 0.158119743123727*z^3 + 0.120350687595265*z^4 + O(z^5)
Higher precision::
sage: L = lseries_dokchitser(E, 200)
sage: L(1)
0.42221415900166715092967967717023093014455137669360598558872
A curve over Q(i)::
sage: K.<i> = NumberField(x^2 + 1)
sage: E = EllipticCurve(K, [1,0])
sage: E.conductor().norm()
256
sage: L = lseries_dokchitser(E, 10)
sage: L.taylor_series(1,5)
0.86 + 0.58*z - 0.62*z^2 + 0.19*z^3 + 0.18*z^4 + O(z^5)
More examples::
sage: lseries_dokchitser(EllipticCurve([0,-1,1,0,0]), 10)(1)
0.25
sage: K.<i> = NumberField(x^2+1)
sage: lseries_dokchitser(EllipticCurve(K, [0,-1,1,0,0]), 10)(1)
0.37
sage: K.<a> = NumberField(x^2-x-1)
sage: lseries_dokchitser(EllipticCurve(K, [0,-1,1,0,0]), 10)(1)
0.72
sage: E = EllipticCurve([0,-1,1,0,0])
sage: E.quadratic_twist(2).rank()
1
sage: K.<d> = NumberField(x^2-2)
sage: L = lseries_dokchitser(EllipticCurve(K, [0,-1,1,0,0]), 10)
sage: L(1)
0
sage: L.taylor_series(1, 5)
0.58*z + 0.20*z^2 - 0.50*z^3 + 0.28*z^4 + O(z^5)
You can use this function as an algorithm to compute the sign of the functional
equation (global root number)::
sage: E = EllipticCurve([1..5])
sage: E.root_number()
-1
sage: L = lseries_dokchitser(E,32); L
Dokchitser L-function of Elliptic Curve defined by y^2 + x*y = x^3 - x^2 + 4*x + 3 over Rational Field
sage: L.eps
-1
Over QQ, this isn't so useful (since Sage has a root_number
method), but over number fields it is very useful::
sage: K.<a> = NumberField(x^2 - x - 1)
sage: E1=EllipticCurve([0,-a-1,1,a,0]); E0 = EllipticCurve([0,-a,a,0,0])
sage: lseries_dokchitser(E1, 16).eps
-1
sage: E1.rank()
1
sage: lseries_dokchitser(E0, 16).eps
1
sage: E0.rank()
0
"""
# The code asssumes in various places that we have a global minimal model,
# for example, in anlist_sqrt5 above.
E = E.global_minimal_model()
# Check that we're over a number field.
K = E.base_field()
if not is_NumberField(K):
raise TypeError, "base field must be a number field"
# Compute norm of the conductor -- awkward because QQ elements have no norm method (they should).
N = E.conductor()
if K != QQ:
N = N.norm()
# We guess the sign epsilon in the functional equation to be +1
# first. If our guess is wrong then we just choose the other
# possibility.
epsilon = 1
# Define the Dokchitser L-function object with all parameters set:
L = Dokchitser(conductor = N * K.discriminant()**2,
gammaV = [0]*K.degree() + [1]*K.degree(),
weight = 2, eps = epsilon, poles = [], prec = prec)
# Find out how many coefficients of the Dirichlet series are needed
# to compute to the requested precision.
n = L.num_coeffs()
# print "num coeffs = %s"%n
# Compute the Dirichlet series coefficients
coeffs = anlist(E, n)[1:]
# Define a string that when evaluated in PARI defines a function
# a(k), which returns the Dirichlet coefficient a_k.
s = 'v=%s; a(k)=v[k];'%coeffs
# Actually tell the L-series / PARI about the coefficients.
L.init_coeffs('a(k)', pari_precode = s)
# Test that the functional equation is satisfied. This will very,
# very, very likely if we chose the sign of the functional
# equation incorrectly, or made any mistake in computing the
# Dirichlet series coefficients.
tiny = max(1e-8, 1.0/2**(prec-1))
if abs(L.check_functional_equation()) > tiny:
# The test failed, so we try the other choice of functional equation.
epsilon *= -1
L.eps = epsilon
# It is not necessary to recreate L -- just tell PARI the different sign.
L._gp_eval('sgn = %s'%epsilon)
# Once again, verify that the functional equation is
# satisfied. If it is, then we've got it. If it isn't, then
# there is definitely some other subtle bug, probably in computed
# the Dirichlet series coefficients.
if abs(L.check_functional_equation()) > tiny:
raise RuntimeError, "Functional equation not numerically satisfied for either choice of sign"
L.rename('Dokchitser L-function of %s'%E)
return L
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)