Base class for Jacobians of curves¶

sage.schemes.jacobians.abstract_jacobian.
Jacobian
(C)¶ EXAMPLES:
sage: from sage.schemes.jacobians.abstract_jacobian import Jacobian sage: P2.<x, y, z> = ProjectiveSpace(QQ, 2) sage: C = Curve(x^3 + y^3 + z^3) sage: Jacobian(C) Jacobian of Projective Plane Curve over Rational Field defined by x^3 + y^3 + z^3

class
sage.schemes.jacobians.abstract_jacobian.
Jacobian_generic
(C)¶ Bases:
sage.schemes.generic.scheme.Scheme
Base class for Jacobians of projective curves.
The input must be a projective curve over a field.
EXAMPLES:
sage: from sage.schemes.jacobians.abstract_jacobian import Jacobian sage: P2.<x, y, z> = ProjectiveSpace(QQ, 2) sage: C = Curve(x^3 + y^3 + z^3) sage: J = Jacobian(C); J Jacobian of Projective Plane Curve over Rational Field defined by x^3 + y^3 + z^3

base_extend
(R)¶ Return the natural extension of
self
over \(R\)INPUT:
R
– a field. The new base field.
OUTPUT:
The Jacobian over the ring \(R\).
EXAMPLES:
sage: R.<x> = QQ['x'] sage: H = HyperellipticCurve(x^310*x+9) sage: Jac = H.jacobian(); Jac Jacobian of Hyperelliptic Curve over Rational Field defined by y^2 = x^3  10*x + 9 sage: F.<a> = QQ.extension(x^2+1) sage: Jac.base_extend(F) Jacobian of Hyperelliptic Curve over Number Field in a with defining polynomial x^2 + 1 defined by y^2 = x^3  10*x + 9

change_ring
(R)¶ Return the Jacobian over the ring \(R\).
INPUT:
R
– a field. The new base ring.
OUTPUT:
The Jacobian over the ring \(R\).
EXAMPLES:
sage: R.<x> = QQ['x'] sage: H = HyperellipticCurve(x^310*x+9) sage: Jac = H.jacobian(); Jac Jacobian of Hyperelliptic Curve over Rational Field defined by y^2 = x^3  10*x + 9 sage: Jac.change_ring(RDF) Jacobian of Hyperelliptic Curve over Real Double Field defined by y^2 = x^3  10.0*x + 9.0

curve
()¶ Return the curve of which self is the Jacobian.
EXAMPLES:
sage: from sage.schemes.jacobians.abstract_jacobian import Jacobian sage: P2.<x, y, z> = ProjectiveSpace(QQ, 2) sage: J = Jacobian(Curve(x^3 + y^3 + z^3)) sage: J.curve() Projective Plane Curve over Rational Field defined by x^3 + y^3 + z^3


sage.schemes.jacobians.abstract_jacobian.
is_Jacobian
(J)¶ Return True if \(J\) is of type Jacobian_generic.
EXAMPLES:
sage: from sage.schemes.jacobians.abstract_jacobian import Jacobian, is_Jacobian sage: P2.<x, y, z> = ProjectiveSpace(QQ, 2) sage: C = Curve(x^3 + y^3 + z^3) sage: J = Jacobian(C) sage: is_Jacobian(J) True
sage: E = EllipticCurve('37a1') sage: is_Jacobian(E) False