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 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)