Jacobian of a general hyperelliptic curve¶

AUTHORS:

David Kohel (2006): initial version
Sabrina Kunzweiler, Gareth Ma, Giacomo Pope (2024): adapt to smooth model

class sage.schemes.hyperelliptic_curves.jacobian_generic.HyperellipticJacobian_generic(C, category=None)[source]¶

Bases: Jacobian_generic

This is the base class for Jacobians of hyperelliptic curves.

We represent elements of the Jacobian by tuples of the form \((u, v : n)\), where

\((u,v)\) is the Mumford representative of a divisor \(P_1 + ... + P_r\),
\(n\) is a non-negative integer

This tuple represents the equivalence class

\[[P_1 + ... + P_r + n \cdot \infty_+ + m\cdot \infty_- - D_\infty],\]

where \(m = g - \deg(u) - n\), and \(\infty_+\), \(\infty_-\) are the points at infinity of the hyperelliptic curve,

\[D_\infty = \lceil g/2 \rceil \infty_+ + \lfloor g/2 \rfloor \infty_-.\]

Here, \(\infty_- = \infty_+\), if the hyperelliptic curve is ramified.

Such a representation exists and is unique, unless the genus \(g\) is odd and the curve is inert.

If the hyperelliptic curve is ramified or inert, then \(n\) can be deduced from \(\deg(u)\) and \(g\). In these cases, \(n\) is omitted in the description.

abelian_group(*args, **kwds)[source]¶

Return the group of rational points on this Jacobian as an AdditiveAbelianGroupWrapper object.

EXAMPLES:

Sage

sage: x = polygen(GF(11^4))
sage: C = HyperellipticCurve(x^5 + x + 1)
sage: J = C.jacobian()
sage: J.abelian_group()
Additive abelian group isomorphic to Z/1677896 + Z/8 + Z/4 + Z/4
  embedded in Abelian group of points over Finite Field in z4 of size 11^4
    on Jacobian of Hyperelliptic Curve over Finite Field in z4 of size 11^4 defined by y^2 = x^5 + x + 1

Python

>>> from sage.all import *
>>> x = polygen(GF(Integer(11)**Integer(4)))
>>> C = HyperellipticCurve(x**Integer(5) + x + Integer(1))
>>> J = C.jacobian()
>>> J.abelian_group()
Additive abelian group isomorphic to Z/1677896 + Z/8 + Z/4 + Z/4
  embedded in Abelian group of points over Finite Field in z4 of size 11^4
    on Jacobian of Hyperelliptic Curve over Finite Field in z4 of size 11^4 defined by y^2 = x^5 + x + 1

See also

sage.schemes.hyperelliptic_curves.jacobian_homset_generic.abelian_group().

cardinality()[source]¶: alias of order().

count_points(*args, **kwds)[source]¶

Compute the number of points on this Jacobian, possibly also over extension fields.

EXAMPLES:

Sage

sage: R.<x> = GF(3663031)[]
sage: J = HyperellipticCurve(x^5 + 1758294*x^4 + 1908793*x^3 + 3033920*x^2 + 3445698*x + 3020661).jacobian()
sage: J.count_points() == J.order()
True
sage: J.count_points(3)
[13403849798842,
 180037321758127724648694244,
 2415703144684957707144219559418205805522]

Python

>>> from sage.all import *
>>> R = GF(Integer(3663031))['x']; (x,) = R._first_ngens(1)
>>> J = HyperellipticCurve(x**Integer(5) + Integer(1758294)*x**Integer(4) + Integer(1908793)*x**Integer(3) + Integer(3033920)*x**Integer(2) + Integer(3445698)*x + Integer(3020661)).jacobian()
>>> J.count_points() == J.order()
True
>>> J.count_points(Integer(3))
[13403849798842,
 180037321758127724648694244,
 2415703144684957707144219559418205805522]

See also

sage.schemes.hyperelliptic_curves.jacobian_homset_generic.count_points().

dimension()[source]¶

Return the dimension of this Jacobian.

EXAMPLES:

Sage

sage: R.<x> = QQ[]
sage: H = HyperellipticCurve(x^2, x^4+1); H
Hyperelliptic Curve over Rational Field defined by y^2 + (x^4 + 1)*y = x^2
sage: J = Jacobian(H)
sage: J.dimension()
3

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> H = HyperellipticCurve(x**Integer(2), x**Integer(4)+Integer(1)); H
Hyperelliptic Curve over Rational Field defined by y^2 + (x^4 + 1)*y = x^2
>>> J = Jacobian(H)
>>> J.dimension()
3

geometric_endomorphism_algebra_is_field(B=200, proof=False)[source]¶

Return whether the geometric endomorphism algebra is a field.

This implies that the Jacobian of the curve is geometrically simple. It is based on Algorithm 4.10 from [Lom2019]

INPUT:

B – (default: 200) the bound which appears in the statement of the algorithm from [Lom2019]
proof – boolean (default: False); whether or not to insist on a provably correct answer. This is related to the warning in the docstring of this module: if this function returns False, then strictly speaking this has not been proven to be False until one has exhibited a non-trivial endomorphism, which these methods are not designed to carry out. If one is convinced that this method should return True, but it is returning False, then this can be exhibited by increasing \(B\).

OUTPUT:

Boolean indicating whether or not the geometric endomorphism algebra is a field.

EXAMPLES:

This is LMFDB curve 262144.d.524288.2 which has QM. Although its Jacobian is geometrically simple, the geometric endomorphism algebra is not a field:

Sage

sage: R.<x> = QQ[]
sage: f = x^5 + x^4 + 4*x^3 + 8*x^2 + 5*x + 1
sage: C = HyperellipticCurve(f)
sage: J = C.jacobian()
sage: J.geometric_endomorphism_algebra_is_field()
False

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> f = x**Integer(5) + x**Integer(4) + Integer(4)*x**Integer(3) + Integer(8)*x**Integer(2) + Integer(5)*x + Integer(1)
>>> C = HyperellipticCurve(f)
>>> J = C.jacobian()
>>> J.geometric_endomorphism_algebra_is_field()
False

This is LMFDB curve 50000.a.200000.1:

Sage

sage: f = 8*x^5 + 1
sage: C = HyperellipticCurve(f)
sage: J = C.jacobian()
sage: J.geometric_endomorphism_algebra_is_field()
True

[Python]

>>> from sage.all import *
>>> f = Integer(8)*x**Integer(5) + Integer(1)
>>> C = HyperellipticCurve(f)
>>> J = C.jacobian()
>>> J.geometric_endomorphism_algebra_is_field()
True

geometric_endomorphism_ring_is_ZZ(B=200, proof=False)[source]¶

Return whether the geometric endomorphism ring of self is the integer ring \(\ZZ\).

INPUT:

B – (default: 200) the bound which appears in the statement of the algorithm from [Lom2019]
proof – boolean (default: False); whether or not to insist on a provably correct answer. This is related to the warning in the module docstring of \(jacobian_endomorphisms.py\): if this function returns False, then strictly speaking this has not been proven to be False until one has exhibited a non-trivial endomorphism, which the methods in that module are not designed to carry out. If one is convinced that this method should return True, but it is returning False, then this can be exhibited by increasing \(B\).

OUTPUT:

Boolean indicating whether or not the geometric endomorphism ring is isomorphic to the integer ring.

EXAMPLES:

This is LMFDB curve 603.a.603.2:

Sage

sage: R.<x> = QQ[]
sage: f = 4*x^5 + x^4 - 4*x^3 + 2*x^2 + 4*x + 1
sage: C = HyperellipticCurve(f)
sage: J = C.jacobian()
sage: J.geometric_endomorphism_ring_is_ZZ()
True

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> f = Integer(4)*x**Integer(5) + x**Integer(4) - Integer(4)*x**Integer(3) + Integer(2)*x**Integer(2) + Integer(4)*x + Integer(1)
>>> C = HyperellipticCurve(f)
>>> J = C.jacobian()
>>> J.geometric_endomorphism_ring_is_ZZ()
True

This is LMFDB curve 1152.a.147456.1 whose geometric endomorphism ring is isomorphic to the group of 2x2 matrices over \(\QQ\):

Sage

sage: f = x^6 - 2*x^4 + 2*x^2 - 1
sage: C = HyperellipticCurve(f)
sage: J = C.jacobian()
sage: J.geometric_endomorphism_ring_is_ZZ()
False

[Python]

>>> from sage.all import *
>>> f = x**Integer(6) - Integer(2)*x**Integer(4) + Integer(2)*x**Integer(2) - Integer(1)
>>> C = HyperellipticCurve(f)
>>> J = C.jacobian()
>>> J.geometric_endomorphism_ring_is_ZZ()
False

This is LMFDB curve 20736.k.373248.1 whose geometric endomorphism ring is isomorphic to the group of 2x2 matrices over a CM field:

Sage

sage: f = x^6 + 8
sage: C = HyperellipticCurve(f)
sage: J = C.jacobian()
sage: J.geometric_endomorphism_ring_is_ZZ()
False

Python

>>> from sage.all import *
>>> f = x**Integer(6) + Integer(8)
>>> C = HyperellipticCurve(f)
>>> J = C.jacobian()
>>> J.geometric_endomorphism_ring_is_ZZ()
False

This is LMFDB curve 708.a.181248.1:

Sage

sage: R.<x> = QQ[]
sage: f = -3*x^6 - 16*x^5 + 36*x^4 + 194*x^3 - 164*x^2 - 392*x - 143
sage: C = HyperellipticCurve(f)
sage: J = C.jacobian()
sage: J.geometric_endomorphism_ring_is_ZZ()
True

[Python]

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> f = -Integer(3)*x**Integer(6) - Integer(16)*x**Integer(5) + Integer(36)*x**Integer(4) + Integer(194)*x**Integer(3) - Integer(164)*x**Integer(2) - Integer(392)*x - Integer(143)
>>> C = HyperellipticCurve(f)
>>> J = C.jacobian()
>>> J.geometric_endomorphism_ring_is_ZZ()
True

This is LMFDB curve 10609.a.10609.1 whose geometric endomorphism ring is an order in a real quadratic field:

Sage

sage: f = x^6 + 2*x^4 + 2*x^3 + 5*x^2 + 6*x + 1
sage: C = HyperellipticCurve(f)
sage: J = C.jacobian()
sage: J.geometric_endomorphism_ring_is_ZZ()
False

Python

>>> from sage.all import *
>>> f = x**Integer(6) + Integer(2)*x**Integer(4) + Integer(2)*x**Integer(3) + Integer(5)*x**Integer(2) + Integer(6)*x + Integer(1)
>>> C = HyperellipticCurve(f)
>>> J = C.jacobian()
>>> J.geometric_endomorphism_ring_is_ZZ()
False

This is LMFDB curve 160000.c.800000.1 whose geometric endomorphism ring is an order in a CM field:

Sage

sage: f = x^5 - 1
sage: C = HyperellipticCurve(f)
sage: J = C.jacobian()
sage: J.geometric_endomorphism_ring_is_ZZ()
False

[Python]

>>> from sage.all import *
>>> f = x**Integer(5) - Integer(1)
>>> C = HyperellipticCurve(f)
>>> J = C.jacobian()
>>> J.geometric_endomorphism_ring_is_ZZ()
False

This is LMFDB curve 262144.d.524288.2 whose geometric endomorphism ring is an order in a quaternion algebra:

Sage

sage: f = x^5 + x^4 + 4*x^3 + 8*x^2 + 5*x + 1
sage: C = HyperellipticCurve(f)
sage: J = C.jacobian()
sage: J.geometric_endomorphism_ring_is_ZZ()
False

Python

>>> from sage.all import *
>>> f = x**Integer(5) + x**Integer(4) + Integer(4)*x**Integer(3) + Integer(8)*x**Integer(2) + Integer(5)*x + Integer(1)
>>> C = HyperellipticCurve(f)
>>> J = C.jacobian()
>>> J.geometric_endomorphism_ring_is_ZZ()
False

This is LMFDB curve 578.a.2312.1 whose geometric endomorphism ring is \(\QQ \times \QQ\):

Sage

sage: f = 4*x^5 - 7*x^4 + 10*x^3 - 7*x^2 + 4*x
sage: C = HyperellipticCurve(f)
sage: J = C.jacobian()
sage: J.geometric_endomorphism_ring_is_ZZ()
False

[Python]

>>> from sage.all import *
>>> f = Integer(4)*x**Integer(5) - Integer(7)*x**Integer(4) + Integer(10)*x**Integer(3) - Integer(7)*x**Integer(2) + Integer(4)*x
>>> C = HyperellipticCurve(f)
>>> J = C.jacobian()
>>> J.geometric_endomorphism_ring_is_ZZ()
False

is_parent_of(element)[source]¶: Return whether self is the parent of element.

lift_u(*args, **kwds)[source]¶: Return one or all points with given \(u\)-coordinate.

See also

sage.schemes.hyperelliptic_curves.jacobian_homset_generic.lift_u().

list(*args, **kwds)[source]¶: alias of points().

order()[source]¶

Compute the order of the Jacobian.

EXAMPLES:

Sage

sage: R.<x> = GF(3663031)[]
sage: HyperellipticCurve(x^5 + 1758294*x^4 + 1908793*x^3 + 3033920*x^2 + 3445698*x + 3020661).jacobian().cardinality()
13403849798842

Python

>>> from sage.all import *
>>> R = GF(Integer(3663031))['x']; (x,) = R._first_ngens(1)
>>> HyperellipticCurve(x**Integer(5) + Integer(1758294)*x**Integer(4) + Integer(1908793)*x**Integer(3) + Integer(3033920)*x**Integer(2) + Integer(3445698)*x + Integer(3020661)).jacobian().cardinality()
13403849798842

See also

sage.schemes.hyperelliptic_curves.jacobian_homset_generic.order().

point(check, *mumford, **kwargs)[source]¶

Return a point on the Jacobian, given:

No arguments or the integer \(0\); return \(0 \in J\);
A point \(P\) on \(J = Jac(C)\), return \(P\);
A point \(P\) on the curve \(H\) such that \(J = Jac(H)\); return \([P - P_0]\), where \(P_0\) is the distinguished point of \(H\). By default, \(P_0 = \infty\);
Two points \(P, Q\) on the curve \(H\) such that \(J = Jac(H)\); return \([P - Q]\);
Polynomials \((u, v)\) such that \(v^2 + hv - f \equiv 0 \pmod u\); return \([(u(x), y - v(x))]\).

EXAMPLES:

Sage

sage: x = polygen(GF(101))
sage: C = HyperellipticCurve(x^5 + x)
sage: J = C.jacobian()
sage: J.point(0)
(1, 0)
sage: J.point(C(-5, 1))
(x + 5, 1)
sage: J.point(C(-5, 1), C(0, 0))
(x^2 + 5*x, 20*x)
sage: J.point(C(-5, 1), C(0, 0)) - J.point(C(0, 0))
(x + 5, 1)
sage: J.point(x, 0)
(x, 0)

Python

>>> from sage.all import *
>>> x = polygen(GF(Integer(101)))
>>> C = HyperellipticCurve(x**Integer(5) + x)
>>> J = C.jacobian()
>>> J.point(Integer(0))
(1, 0)
>>> J.point(C(-Integer(5), Integer(1)))
(x + 5, 1)
>>> J.point(C(-Integer(5), Integer(1)), C(Integer(0), Integer(0)))
(x^2 + 5*x, 20*x)
>>> J.point(C(-Integer(5), Integer(1)), C(Integer(0), Integer(0))) - J.point(C(Integer(0), Integer(0)))
(x + 5, 1)
>>> J.point(x, Integer(0))
(x, 0)

points(*args, **kwds)[source]¶

Return all rational points on the Jacobian.

EXAMPLES:

Sage

sage: x = polygen(GF(11))
sage: C = HyperellipticCurve(x^5 + x + 1)
sage: J = C.jacobian()
sage: pts = J.points()
sage: sorted(pts)
[(1, 0), (x, 1), (x, 10), (x + 2, 0), (x + 8, 4),
 ..., (x^2 + 10*x + 6, 3*x), (x^2 + 10*x + 6, 8*x),
 (x^2 + 10*x + 7, 5*x + 6), (x^2 + 10*x + 7, 6*x + 5)]
sage: len(pts) == J.order()
True

Python

>>> from sage.all import *
>>> x = polygen(GF(Integer(11)))
>>> C = HyperellipticCurve(x**Integer(5) + x + Integer(1))
>>> J = C.jacobian()
>>> pts = J.points()
>>> sorted(pts)
[(1, 0), (x, 1), (x, 10), (x + 2, 0), (x + 8, 4),
 ..., (x^2 + 10*x + 6, 3*x), (x^2 + 10*x + 6, 8*x),
 (x^2 + 10*x + 7, 5*x + 6), (x^2 + 10*x + 7, 6*x + 5)]
>>> len(pts) == J.order()
True

See also

sage.schemes.hyperelliptic_curves.jacobian_homset_generic.points().

random_element(*args, **kwds)[source]¶

Return a random element of the Jacobian.

EXAMPLES:

Sage

sage: x = polygen(GF(11))
sage: J = HyperellipticCurve(x^5 + x + 1).jacobian()
sage: elt = J.random_element(); elt  # random
(x^2 + 3*x + 10, 1)
sage: elt in J
True

Python

>>> from sage.all import *
>>> x = polygen(GF(Integer(11)))
>>> J = HyperellipticCurve(x**Integer(5) + x + Integer(1)).jacobian()
>>> elt = J.random_element(); elt  # random
(x^2 + 3*x + 10, 1)
>>> elt in J
True

See also

sage.schemes.hyperelliptic_curves.jacobian_homset_generic.random_element().

rational_points(*args, **kwds)[source]¶: alias of points().

some_elements()[source]¶

Return some rational points on the Jacobian.

EXAMPLES:

Sage

sage: x = polygen(GF(11))
sage: J = HyperellipticCurve(x^5 + x + 1).jacobian()
sage: elts = J.some_elements(); elts  # random
[(1, 0), (x^2 + 7*x + 3, 10*x + 7), (x^2 + 10*x, 5*x + 1), (x^2 + 8*x + 10, 1)]
sage: all(elt in J for elt in elts)
True

Python

>>> from sage.all import *
>>> x = polygen(GF(Integer(11)))
>>> J = HyperellipticCurve(x**Integer(5) + x + Integer(1)).jacobian()
>>> elts = J.some_elements(); elts  # random
[(1, 0), (x^2 + 7*x + 3, 10*x + 7), (x^2 + 10*x, 5*x + 1), (x^2 + 8*x + 10, 1)]
>>> all(elt in J for elt in elts)
True