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

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.

NOTE: We require that the genus is even if the hyperelliptic curve is inert.

EXAMPLES:

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)