Rational point sets on a Jacobian of a hyperelliptic curve (inert case)

Note

In the inert case, the current implementation of hyperelliptic Jacobians only supports arithmetic on curves of even genus.

AUTHORS:

• Sabrina Kunzweiler, Gareth Ma, Giacomo Pope (2024): adapt to smooth model

class sage.schemes.hyperelliptic_curves.jacobian_homset_inert.HyperellipticJacobianHomsetInert(Y, X, **kwds)

Bases: HyperellipticJacobianHomset

Create the Jacobian Hom-set of a hyperelliptic curve without rational points at infinity.

Element

alias of MumfordDivisorClassFieldInert

zero(check=True)

Return the zero element of the Jacobian.

The Mumford presentation of the zero element is given by \((1, 0 : g/2)\), \(g\) is the genus of the hyperelliptic curve.

EXAMPLES:

Sage

sage: R.<x> = QQ[]
sage: H = HyperellipticCurve(2*x^6 + 1)
sage: H.is_inert()
True
sage: J = H.jacobian()
sage: J.zero()
(1, 0 : 1)

sage: H = HyperellipticCurve(3*x^10 + 1)
sage: J = H.jacobian()
sage: J.zero()
(1, 0 : 2)

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> H = HyperellipticCurve(Integer(2)*x**Integer(6) + Integer(1))
>>> H.is_inert()
True
>>> J = H.jacobian()
>>> J.zero()
(1, 0 : 1)

>>> H = HyperellipticCurve(Integer(3)*x**Integer(10) + Integer(1))
>>> J = H.jacobian()
>>> J.zero()
(1, 0 : 2)