Jacobians of function fields¶

This module provides base classes for Jacobians of function fields.

Jacobian¶

The Jacobian of a function field is created by default in the Hess model, with a base divisor of degree \(g\) the genus of the function field. The base divisor is automatically chosen if not given.

Sage

sage: P2.<x,y,z> = ProjectiveSpace(GF(29), 2)
sage: C = Curve(x^3 + 5*z^3 - y^2*z, P2)
sage: F = C.function_field()
sage: J = F.jacobian()
sage: J
Jacobian of Function field in z defined by z^3 + 23*y^2*z + 6 (Hess model)
sage: J.base_divisor().degree()
1

Python

>>> from sage.all import *
>>> P2 = ProjectiveSpace(GF(Integer(29)), Integer(2), names=('x', 'y', 'z',)); (x, y, z,) = P2._first_ngens(3)
>>> C = Curve(x**Integer(3) + Integer(5)*z**Integer(3) - y**Integer(2)*z, P2)
>>> F = C.function_field()
>>> J = F.jacobian()
>>> J
Jacobian of Function field in z defined by z^3 + 23*y^2*z + 6 (Hess model)
>>> J.base_divisor().degree()
1

Sage Live

P2.<x,y,z> = ProjectiveSpace(GF(29), 2)
C = Curve(x^3 + 5*z^3 - y^2*z, P2)
F = C.function_field()
J = F.jacobian()
J
J.base_divisor().degree()

Explicitly specify a model if you want Jacobians in different models.

Sage

sage: J_km = F.jacobian(model='km_large')
sage: J_km
Jacobian of Function field in z defined by z^3 + 23*y^2*z + 6 (Khuri-Makdisi large model)

Python

>>> from sage.all import *
>>> J_km = F.jacobian(model='km_large')
>>> J_km
Jacobian of Function field in z defined by z^3 + 23*y^2*z + 6 (Khuri-Makdisi large model)

Sage Live

J_km = F.jacobian(model='km_large')
J_km

Group of rational points¶

The group of rational points of a Jacobian is created from the Jacobian. A point of the Jacobian group is determined by a divisor of degree zero. To represent the point, a divisor of the form \(D-B\) is selected where \(D\) is an effective divisor of the same degree with the base divisor \(B\). Hence the point is simply represented by the divisor \(D\).

Sage

sage: G = J.group()
sage: G.order()
30
sage: pl1 = C([1,8,1]).place()
sage: pl2 = C([2,10,1]).place()
sage: p1 = G.point(pl1 - pl2)
sage: p1
[Place (y + 1, z + 6)]
sage: p2 = G.point(pl2 - pl1)
sage: p2
[Place (y + 28, z + 6)]
sage: p1 + p2 == G.zero()
True
sage: p1.order()
5

Python

>>> from sage.all import *
>>> G = J.group()
>>> G.order()
30
>>> pl1 = C([Integer(1),Integer(8),Integer(1)]).place()
>>> pl2 = C([Integer(2),Integer(10),Integer(1)]).place()
>>> p1 = G.point(pl1 - pl2)
>>> p1
[Place (y + 1, z + 6)]
>>> p2 = G.point(pl2 - pl1)
>>> p2
[Place (y + 28, z + 6)]
>>> p1 + p2 == G.zero()
True
>>> p1.order()
5

Sage Live

G = J.group()
G.order()
pl1 = C([1,8,1]).place()
pl2 = C([2,10,1]).place()
p1 = G.point(pl1 - pl2)
p1
p2 = G.point(pl2 - pl1)
p2
p1 + p2 == G.zero()
p1.order()

We can get the corresponding point in the Jacobian in a different model.

Sage

sage: p1km = J_km(p1)
sage: p1km.order()
5
sage: p1km
Point of Jacobian determined by
[ 1  0  0  0  0  0 11  0  0]
[ 0  1  0  0  0  0 18  0  0]
[ 0  0  1  0  0  0 11  0  0]
[ 0  0  0  1  0  0 18  1  0]
[ 0  0  0  0  1  0 25  0 19]
[ 0  0  0  0  0  1  8  8  0]

Python

>>> from sage.all import *
>>> p1km = J_km(p1)
>>> p1km.order()
5
>>> p1km
Point of Jacobian determined by
[ 1  0  0  0  0  0 11  0  0]
[ 0  1  0  0  0  0 18  0  0]
[ 0  0  1  0  0  0 11  0  0]
[ 0  0  0  1  0  0 18  1  0]
[ 0  0  0  0  1  0 25  0 19]
[ 0  0  0  0  0  1  8  8  0]

Sage Live

p1km = J_km(p1)
p1km.order()
p1km

AUTHORS:

Kwankyu Lee (2022-01-24): initial version

class sage.rings.function_field.jacobian_base.JacobianGroupFunctor(base_field, field)[source]¶

Bases: ConstructionFunctor

A construction functor for Jacobian groups.

EXAMPLES:

Sage

sage: k = GF(7)
sage: P2.<x,y,z> = ProjectiveSpace(k, 2)
sage: C = Curve(x^3 + 5*z^3 - y^2*z, P2)
sage: J = C.jacobian(model='hess')
sage: G = J.group()
sage: F, obj = G.construction()
sage: F
JacobianGroupFunctor

Python

>>> from sage.all import *
>>> k = GF(Integer(7))
>>> P2 = ProjectiveSpace(k, Integer(2), names=('x', 'y', 'z',)); (x, y, z,) = P2._first_ngens(3)
>>> C = Curve(x**Integer(3) + Integer(5)*z**Integer(3) - y**Integer(2)*z, P2)
>>> J = C.jacobian(model='hess')
>>> G = J.group()
>>> F, obj = G.construction()
>>> F
JacobianGroupFunctor

Sage Live

k = GF(7)
P2.<x,y,z> = ProjectiveSpace(k, 2)
C = Curve(x^3 + 5*z^3 - y^2*z, P2)
J = C.jacobian(model='hess')
G = J.group()
F, obj = G.construction()
F

merge(other)[source]¶

Return the functor merging self and other.

INPUT:

other – a functor

EXAMPLES:

Sage

sage: k = GF(7)
sage: P2.<x,y,z> = ProjectiveSpace(k, 2)
sage: C = Curve(x^3 + 5*z^3 - y^2*z, P2)
sage: J = C.jacobian(model='hess')
sage: K2 = k.extension(2)
sage: G2 = J.group(K2)
sage: K3 = k.extension(3)
sage: G3 = J.group(K3)
sage: sage.categories.pushout.pushout(G2, G3)  # indirect doctest
Group of rational points of Jacobian over Finite Field in z6 of size 7^6 (Hess model)

Python

>>> from sage.all import *
>>> k = GF(Integer(7))
>>> P2 = ProjectiveSpace(k, Integer(2), names=('x', 'y', 'z',)); (x, y, z,) = P2._first_ngens(3)
>>> C = Curve(x**Integer(3) + Integer(5)*z**Integer(3) - y**Integer(2)*z, P2)
>>> J = C.jacobian(model='hess')
>>> K2 = k.extension(Integer(2))
>>> G2 = J.group(K2)
>>> K3 = k.extension(Integer(3))
>>> G3 = J.group(K3)
>>> sage.categories.pushout.pushout(G2, G3)  # indirect doctest
Group of rational points of Jacobian over Finite Field in z6 of size 7^6 (Hess model)

Sage Live

k = GF(7)
P2.<x,y,z> = ProjectiveSpace(k, 2)
C = Curve(x^3 + 5*z^3 - y^2*z, P2)
J = C.jacobian(model='hess')
K2 = k.extension(2)
G2 = J.group(K2)
K3 = k.extension(3)
G3 = J.group(K3)
sage.categories.pushout.pushout(G2, G3)  # indirect doctest

rank = 20¶

class sage.rings.function_field.jacobian_base.JacobianGroup_base(parent, function_field, base_div)[source]¶

Bases: Parent

Groups of rational points of Jacobians.

INPUT:

parent – a Jacobian
function_field – a function field
base_div – an effective divisor of the function field

EXAMPLES:

Sage

sage: P2.<x,y,z> = ProjectiveSpace(GF(7), 2)
sage: C = Curve(x^3 + 5*z^3 - y^2*z, P2)
sage: J = C.jacobian(model='hess')
sage: J.group()
Group of rational points of Jacobian over Finite Field of size 7 (Hess model)

Python

>>> from sage.all import *
>>> P2 = ProjectiveSpace(GF(Integer(7)), Integer(2), names=('x', 'y', 'z',)); (x, y, z,) = P2._first_ngens(3)
>>> C = Curve(x**Integer(3) + Integer(5)*z**Integer(3) - y**Integer(2)*z, P2)
>>> J = C.jacobian(model='hess')
>>> J.group()
Group of rational points of Jacobian over Finite Field of size 7 (Hess model)

Sage Live

P2.<x,y,z> = ProjectiveSpace(GF(7), 2)
C = Curve(x^3 + 5*z^3 - y^2*z, P2)
J = C.jacobian(model='hess')
J.group()

base_divisor()[source]¶

Return the base divisor that is used to represent points of this group.

EXAMPLES:

Sage

sage: P2.<x,y,z> = ProjectiveSpace(GF(7), 2)
sage: C = Curve(x^3 + 5*z^3 - y^2*z, P2)
sage: b = C([0,1,0]).place()
sage: J = C.jacobian(model='hess', base_div=b)
sage: G = J.group()
sage: G.base_divisor()
Place (1/y, 1/y*z)
sage: _ == 1*b
True

Python

>>> from sage.all import *
>>> P2 = ProjectiveSpace(GF(Integer(7)), Integer(2), names=('x', 'y', 'z',)); (x, y, z,) = P2._first_ngens(3)
>>> C = Curve(x**Integer(3) + Integer(5)*z**Integer(3) - y**Integer(2)*z, P2)
>>> b = C([Integer(0),Integer(1),Integer(0)]).place()
>>> J = C.jacobian(model='hess', base_div=b)
>>> G = J.group()
>>> G.base_divisor()
Place (1/y, 1/y*z)
>>> _ == Integer(1)*b
True

Sage Live

P2.<x,y,z> = ProjectiveSpace(GF(7), 2)
C = Curve(x^3 + 5*z^3 - y^2*z, P2)
b = C([0,1,0]).place()
J = C.jacobian(model='hess', base_div=b)
G = J.group()
G.base_divisor()
_ == 1*b

The base divisor is the denominator (negative part) of the divisor of degree zero that represents a point.

Sage

sage: p = C([-1,2,1]).place()
sage: G.point(p - b).divisor()
- Place (1/y, 1/y*z)
 + Place (y + 2, z + 1)

Python

>>> from sage.all import *
>>> p = C([-Integer(1),Integer(2),Integer(1)]).place()
>>> G.point(p - b).divisor()
- Place (1/y, 1/y*z)
 + Place (y + 2, z + 1)

Sage Live

p = C([-1,2,1]).place()
G.point(p - b).divisor()

construction()[source]¶

Return the data for a functorial construction of this Jacobian group.

EXAMPLES:

Sage

sage: k = GF(7)
sage: P2.<x,y,z> = ProjectiveSpace(k, 2)
sage: C = Curve(x^3 + 5*z^3 - y^2*z, P2)
sage: J = C.jacobian(model='hess')
sage: K2 = k.extension(2)
sage: G2 = J.group(K2)
sage: K3= k.extension(3)
sage: G3 = J.group(K3)
sage: p1, p2 = G2.get_points(2)
sage: q1, q2 = G3.get_points(2)
sage: (p1 + q1).parent() is (p2 + q2).parent()
True

Python

>>> from sage.all import *
>>> k = GF(Integer(7))
>>> P2 = ProjectiveSpace(k, Integer(2), names=('x', 'y', 'z',)); (x, y, z,) = P2._first_ngens(3)
>>> C = Curve(x**Integer(3) + Integer(5)*z**Integer(3) - y**Integer(2)*z, P2)
>>> J = C.jacobian(model='hess')
>>> K2 = k.extension(Integer(2))
>>> G2 = J.group(K2)
>>> K3= k.extension(Integer(3))
>>> G3 = J.group(K3)
>>> p1, p2 = G2.get_points(Integer(2))
>>> q1, q2 = G3.get_points(Integer(2))
>>> (p1 + q1).parent() is (p2 + q2).parent()
True

Sage Live

k = GF(7)
P2.<x,y,z> = ProjectiveSpace(k, 2)
C = Curve(x^3 + 5*z^3 - y^2*z, P2)
J = C.jacobian(model='hess')
K2 = k.extension(2)
G2 = J.group(K2)
K3= k.extension(3)
G3 = J.group(K3)
p1, p2 = G2.get_points(2)
q1, q2 = G3.get_points(2)
(p1 + q1).parent() is (p2 + q2).parent()

function_field()[source]¶

Return the function field to which this Jacobian group attached.

EXAMPLES:

Sage

sage: P2.<x,y,z> = ProjectiveSpace(GF(7), 2)
sage: C = Curve(x^3 + 5*z^3 - y^2*z, P2)
sage: J = C.jacobian(model='hess')
sage: G = J.group()
sage: G.function_field()
Function field in z defined by z^3 + 4*y^2*z + 3

Python

>>> from sage.all import *
>>> P2 = ProjectiveSpace(GF(Integer(7)), Integer(2), names=('x', 'y', 'z',)); (x, y, z,) = P2._first_ngens(3)
>>> C = Curve(x**Integer(3) + Integer(5)*z**Integer(3) - y**Integer(2)*z, P2)
>>> J = C.jacobian(model='hess')
>>> G = J.group()
>>> G.function_field()
Function field in z defined by z^3 + 4*y^2*z + 3

Sage Live

P2.<x,y,z> = ProjectiveSpace(GF(7), 2)
C = Curve(x^3 + 5*z^3 - y^2*z, P2)
J = C.jacobian(model='hess')
G = J.group()
G.function_field()

parent()[source]¶

Return the Jacobian to which this Jacobian group belongs.

EXAMPLES:

Sage

sage: P2.<x,y,z> = ProjectiveSpace(GF(7), 2)
sage: C = Curve(x^3 + 5*z^3 - y^2*z, P2)
sage: J = C.jacobian(model='hess')
sage: G = J.group()
sage: G.parent()
Jacobian of Projective Plane Curve over Finite Field of size 7
 defined by x^3 - y^2*z - 2*z^3 (Hess model)

Python

>>> from sage.all import *
>>> P2 = ProjectiveSpace(GF(Integer(7)), Integer(2), names=('x', 'y', 'z',)); (x, y, z,) = P2._first_ngens(3)
>>> C = Curve(x**Integer(3) + Integer(5)*z**Integer(3) - y**Integer(2)*z, P2)
>>> J = C.jacobian(model='hess')
>>> G = J.group()
>>> G.parent()
Jacobian of Projective Plane Curve over Finite Field of size 7
 defined by x^3 - y^2*z - 2*z^3 (Hess model)

Sage Live

P2.<x,y,z> = ProjectiveSpace(GF(7), 2)
C = Curve(x^3 + 5*z^3 - y^2*z, P2)
J = C.jacobian(model='hess')
G = J.group()
G.parent()

class sage.rings.function_field.jacobian_base.JacobianGroup_finite_field_base(parent, function_field, base_div)[source]¶

Bases: JacobianGroup_base

Jacobian groups of function fields over finite fields.

EXAMPLES:

Sage

sage: P2.<x,y,z> = ProjectiveSpace(GF(7), 2)
sage: C = Curve(x^3 + 5*z^3 - y^2*z, P2)
sage: b = C([0,1,0]).place()
sage: J = C.jacobian(model='hess', base_div=b)
sage: J.group()
Group of rational points of Jacobian over Finite Field of size 7 (Hess model)

Python

>>> from sage.all import *
>>> P2 = ProjectiveSpace(GF(Integer(7)), Integer(2), names=('x', 'y', 'z',)); (x, y, z,) = P2._first_ngens(3)
>>> C = Curve(x**Integer(3) + Integer(5)*z**Integer(3) - y**Integer(2)*z, P2)
>>> b = C([Integer(0),Integer(1),Integer(0)]).place()
>>> J = C.jacobian(model='hess', base_div=b)
>>> J.group()
Group of rational points of Jacobian over Finite Field of size 7 (Hess model)

Sage Live

P2.<x,y,z> = ProjectiveSpace(GF(7), 2)
C = Curve(x^3 + 5*z^3 - y^2*z, P2)
b = C([0,1,0]).place()
J = C.jacobian(model='hess', base_div=b)
J.group()

get_points(n)[source]¶

Return \(n\) points of the Jacobian group.

If \(n\) is greater than the order of the group, then returns all points of the group.

INPUT:

n – integer

EXAMPLES:

Sage

sage: k = GF(7)
sage: A.<x,y> = AffineSpace(k,2)
sage: C = Curve(y^2 + x^3 + 2*x + 1).projective_closure()
sage: J = C.jacobian(model='hess')
sage: G = J.group()
sage: pts = G.get_points(G.order())
sage: len(pts)
11

Python

>>> from sage.all import *
>>> k = GF(Integer(7))
>>> A = AffineSpace(k,Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
>>> C = Curve(y**Integer(2) + x**Integer(3) + Integer(2)*x + Integer(1)).projective_closure()
>>> J = C.jacobian(model='hess')
>>> G = J.group()
>>> pts = G.get_points(G.order())
>>> len(pts)
11

Sage Live

k = GF(7)
A.<x,y> = AffineSpace(k,2)
C = Curve(y^2 + x^3 + 2*x + 1).projective_closure()
J = C.jacobian(model='hess')
G = J.group()
pts = G.get_points(G.order())
len(pts)

order(algorithm='numeric')[source]¶

Return the order of the Jacobian group.

INPUT:

algorithm – 'numeric' (default) or 'algebraic'

EXAMPLES:

Sage

sage: P2.<x,y,z> = ProjectiveSpace(GF(7), 2)
sage: C = Curve(x^3 + 5*z^3 - y^2*z, P2)
sage: b = C([0,1,0]).place()
sage: J = C.jacobian(model='hess', base_div=b)
sage: G = J.group()
sage: G.order()
7

Python

>>> from sage.all import *
>>> P2 = ProjectiveSpace(GF(Integer(7)), Integer(2), names=('x', 'y', 'z',)); (x, y, z,) = P2._first_ngens(3)
>>> C = Curve(x**Integer(3) + Integer(5)*z**Integer(3) - y**Integer(2)*z, P2)
>>> b = C([Integer(0),Integer(1),Integer(0)]).place()
>>> J = C.jacobian(model='hess', base_div=b)
>>> G = J.group()
>>> G.order()
7

Sage Live

P2.<x,y,z> = ProjectiveSpace(GF(7), 2)
C = Curve(x^3 + 5*z^3 - y^2*z, P2)
b = C([0,1,0]).place()
J = C.jacobian(model='hess', base_div=b)
G = J.group()
G.order()

class sage.rings.function_field.jacobian_base.JacobianPoint_base[source]¶

Bases: ModuleElement

Abstract base class of points of Jacobian groups.

class sage.rings.function_field.jacobian_base.JacobianPoint_finite_field_base[source]¶

Bases: JacobianPoint_base

Points of Jacobians over finite fields.

frobenius()[source]¶

Return the image of the point acted by the Frobenius automorphism.

EXAMPLES:

Sage

sage: k = GF(7)
sage: A.<x,y> = AffineSpace(k,2)
sage: C = Curve(y^2 + x^3 + 2*x + 1).projective_closure()
sage: J = C.jacobian(model='hess')
sage: G1 = J.group()
sage: G1.order()
11
sage: K = k.extension(3)
sage: G3 = J.group(K)
sage: pts1 = G1.get_points(11)
sage: pts3 = G3.get_points(12)
sage: pt = next(pt for pt in pts3 if pt not in pts1)
sage: pt.frobenius() == pt
False
sage: pt.frobenius().frobenius().frobenius() == pt
True

Python

>>> from sage.all import *
>>> k = GF(Integer(7))
>>> A = AffineSpace(k,Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
>>> C = Curve(y**Integer(2) + x**Integer(3) + Integer(2)*x + Integer(1)).projective_closure()
>>> J = C.jacobian(model='hess')
>>> G1 = J.group()
>>> G1.order()
11
>>> K = k.extension(Integer(3))
>>> G3 = J.group(K)
>>> pts1 = G1.get_points(Integer(11))
>>> pts3 = G3.get_points(Integer(12))
>>> pt = next(pt for pt in pts3 if pt not in pts1)
>>> pt.frobenius() == pt
False
>>> pt.frobenius().frobenius().frobenius() == pt
True

Sage Live

k = GF(7)
A.<x,y> = AffineSpace(k,2)
C = Curve(y^2 + x^3 + 2*x + 1).projective_closure()
J = C.jacobian(model='hess')
G1 = J.group()
G1.order()
K = k.extension(3)
G3 = J.group(K)
pts1 = G1.get_points(11)
pts3 = G3.get_points(12)
pt = next(pt for pt in pts3 if pt not in pts1)
pt.frobenius() == pt
pt.frobenius().frobenius().frobenius() == pt

order()[source]¶

Return the order of this point.

EXAMPLES:

Sage

sage: P2.<x,y,z> = ProjectiveSpace(GF(29), 2)
sage: C = Curve(x^3 + 5*z^3 - y^2*z, P2)
sage: F = C.function_field()
sage: h = C.function(y/x).divisor_of_poles()
sage: J = C.jacobian(model='km_large', base_div=h)
sage: G = J.group()
sage: b = F.get_place(1)
sage: pl = C([-1,2,1]).place()
sage: p = G.point(pl - b)
sage: p.order()
15

Python

>>> from sage.all import *
>>> P2 = ProjectiveSpace(GF(Integer(29)), Integer(2), names=('x', 'y', 'z',)); (x, y, z,) = P2._first_ngens(3)
>>> C = Curve(x**Integer(3) + Integer(5)*z**Integer(3) - y**Integer(2)*z, P2)
>>> F = C.function_field()
>>> h = C.function(y/x).divisor_of_poles()
>>> J = C.jacobian(model='km_large', base_div=h)
>>> G = J.group()
>>> b = F.get_place(Integer(1))
>>> pl = C([-Integer(1),Integer(2),Integer(1)]).place()
>>> p = G.point(pl - b)
>>> p.order()
15

Sage Live

P2.<x,y,z> = ProjectiveSpace(GF(29), 2)
C = Curve(x^3 + 5*z^3 - y^2*z, P2)
F = C.function_field()
h = C.function(y/x).divisor_of_poles()
J = C.jacobian(model='km_large', base_div=h)
G = J.group()
b = F.get_place(1)
pl = C([-1,2,1]).place()
p = G.point(pl - b)
p.order()

ALGORITHM: Shanks’ Baby Step Giant Step

class sage.rings.function_field.jacobian_base.Jacobian_base(function_field, base_div, **kwds)[source]¶

Bases: Parent

Jacobians of function fields.

EXAMPLES:

Sage

sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: F.<y> = K.extension(Y^2 + Y + x + 1/x)
sage: F.jacobian()
Jacobian of Function field in y defined by y^2 + y + (x^2 + 1)/x (Hess model)

Python

>>> from sage.all import *
>>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1)
>>> F = K.extension(Y**Integer(2) + Y + x + Integer(1)/x, names=('y',)); (y,) = F._first_ngens(1)
>>> F.jacobian()
Jacobian of Function field in y defined by y^2 + y + (x^2 + 1)/x (Hess model)

Sage Live

K.<x> = FunctionField(GF(2)); _.<Y> = K[]
F.<y> = K.extension(Y^2 + Y + x + 1/x)
F.jacobian()

base_curve()[source]¶

Return the base curve or the function field that abstractly defines a curve.

EXAMPLES:

Sage

sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: F.<y> = K.extension(Y^2 + Y + x + 1/x)
sage: J = F.jacobian()
sage: J.base_curve()
Function field in y defined by y^2 + y + (x^2 + 1)/x

Python

>>> from sage.all import *
>>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1)
>>> F = K.extension(Y**Integer(2) + Y + x + Integer(1)/x, names=('y',)); (y,) = F._first_ngens(1)
>>> J = F.jacobian()
>>> J.base_curve()
Function field in y defined by y^2 + y + (x^2 + 1)/x

Sage Live

K.<x> = FunctionField(GF(2)); _.<Y> = K[]
F.<y> = K.extension(Y^2 + Y + x + 1/x)
J = F.jacobian()
J.base_curve()

base_divisor()[source]¶

Return the base divisor used to construct the Jacobian.

EXAMPLES:

Sage

sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: F.<y> = K.extension(Y^2 + Y + x + 1/x)
sage: b = F.get_place(1)
sage: J = F.jacobian(base_div=b)
sage: J.base_divisor() == b
True

Python

>>> from sage.all import *
>>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1)
>>> F = K.extension(Y**Integer(2) + Y + x + Integer(1)/x, names=('y',)); (y,) = F._first_ngens(1)
>>> b = F.get_place(Integer(1))
>>> J = F.jacobian(base_div=b)
>>> J.base_divisor() == b
True

Sage Live

K.<x> = FunctionField(GF(2)); _.<Y> = K[]
F.<y> = K.extension(Y^2 + Y + x + 1/x)
b = F.get_place(1)
J = F.jacobian(base_div=b)
J.base_divisor() == b

curve()[source]¶

Return the projective curve to which this Jacobian is attached.

If the Jacobian was constructed from a function field, then returns nothing.

EXAMPLES:

Sage

sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: F.<y> = K.extension(Y^2 + Y + x + 1/x)
sage: J = F.jacobian()
sage: J.curve()

Python

>>> from sage.all import *
>>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1)
>>> F = K.extension(Y**Integer(2) + Y + x + Integer(1)/x, names=('y',)); (y,) = F._first_ngens(1)
>>> J = F.jacobian()
>>> J.curve()

Sage Live

K.<x> = FunctionField(GF(2)); _.<Y> = K[]
F.<y> = K.extension(Y^2 + Y + x + 1/x)
J = F.jacobian()
J.curve()

facade_for()[source]¶

Return the system of groups that this Jacobian is a facade for.

The Jacobian can be seen as a facade for all groups of rational points over field extensions of the base constant field of the function field. This method returns only the internally constructed system of such groups.

EXAMPLES:

Sage

sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: F.<y> = K.extension(Y^2 + Y + x + 1/x)
sage: J = F.jacobian()
sage: J.facade_for()
[Group of rational points of Jacobian over Finite Field of size 2 (Hess model)]

Python

>>> from sage.all import *
>>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1)
>>> F = K.extension(Y**Integer(2) + Y + x + Integer(1)/x, names=('y',)); (y,) = F._first_ngens(1)
>>> J = F.jacobian()
>>> J.facade_for()
[Group of rational points of Jacobian over Finite Field of size 2 (Hess model)]

Sage Live

K.<x> = FunctionField(GF(2)); _.<Y> = K[]
F.<y> = K.extension(Y^2 + Y + x + 1/x)
J = F.jacobian()
J.facade_for()

group(k_ext=None)[source]¶

Return the group of rational points of the Jacobian.

EXAMPLES:

Sage

sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: F.<y> = K.extension(Y^2 + Y + x + 1/x)
sage: b = F.get_place(1)
sage: J = F.jacobian(base_div=b)
sage: J.group()
Group of rational points of Jacobian over Finite Field of size 2 (Hess model)

Python

>>> from sage.all import *
>>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1)
>>> F = K.extension(Y**Integer(2) + Y + x + Integer(1)/x, names=('y',)); (y,) = F._first_ngens(1)
>>> b = F.get_place(Integer(1))
>>> J = F.jacobian(base_div=b)
>>> J.group()
Group of rational points of Jacobian over Finite Field of size 2 (Hess model)

Sage Live

K.<x> = FunctionField(GF(2)); _.<Y> = K[]
F.<y> = K.extension(Y^2 + Y + x + 1/x)
b = F.get_place(1)
J = F.jacobian(base_div=b)
J.group()

set_base_place(place)[source]¶

Set place as the base place.

INPUT:

place – a rational place of the function field

The base place \(B\) is used to map a rational place \(P\) of the function field to the point of the Jacobian defined by the divisor \(P - B\).

EXAMPLES:

Sage

sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: F.<y> = K.extension(Y^2 + Y + x + 1/x)
sage: J = F.jacobian()
sage: B = F.get_place(1)
sage: J.set_base_place(B)
sage: Q = F.places()[-1]
sage: J(Q)
[Place (x + 1, x*y + 1)]
sage: J(Q).parent()
Group of rational points of Jacobian over Finite Field of size 2 (Hess model)
sage: J(B)
[Place (x, x*y)]
sage: J(B).is_zero()
True

Python

>>> from sage.all import *
>>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1)
>>> F = K.extension(Y**Integer(2) + Y + x + Integer(1)/x, names=('y',)); (y,) = F._first_ngens(1)
>>> J = F.jacobian()
>>> B = F.get_place(Integer(1))
>>> J.set_base_place(B)
>>> Q = F.places()[-Integer(1)]
>>> J(Q)
[Place (x + 1, x*y + 1)]
>>> J(Q).parent()
Group of rational points of Jacobian over Finite Field of size 2 (Hess model)
>>> J(B)
[Place (x, x*y)]
>>> J(B).is_zero()
True

Sage Live

K.<x> = FunctionField(GF(2)); _.<Y> = K[]
F.<y> = K.extension(Y^2 + Y + x + 1/x)
J = F.jacobian()
B = F.get_place(1)
J.set_base_place(B)
Q = F.places()[-1]
J(Q)
J(Q).parent()
J(B)
J(B).is_zero()