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

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# 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)