Hyperelliptic curves of genus 2 over a general ring - Elliptic curves

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# Hyperelliptic curves of genus 2 over a general ring

class sage.schemes.hyperelliptic_curves.hyperelliptic_g2.HyperellipticCurve_g2(PP, f, h=None, names=None, genus=None)

  Bases: `HyperellipticCurve_generic`

  absolute_igusa_invariants_kohel()

    Return the three absolute Igusa invariants used by Kohel [KohECHIDNA].

    See also

    `sage.schemes.hyperelliptic_curves.invariants()`

    EXAMPLES:

    Sage

        sage: R.<x> = QQ[]
        sage: HyperellipticCurve(x^5 - 1).absolute_igusa_invariants_kohel()
        (0, 0, 0)
        sage: HyperellipticCurve(x^5 - x + 1, x^2).absolute_igusa_invariants_kohel()
        (-1030567/178769, 259686400/178769, 20806400/178769)
        sage: HyperellipticCurve((x^5 - x + 1)(3*x + 1), (x^2)(3*x + 1)).absolute_igusa_invariants_kohel()
        (-1030567/178769, 259686400/178769, 20806400/178769)

    Python

        >>> from sage.all import *
        >>> R = QQ['x']; (x,) = R._first_ngens(1)
        >>> HyperellipticCurve(x**Integer(5) - Integer(1)).absolute_igusa_invariants_kohel()
        (0, 0, 0)
        >>> HyperellipticCurve(x**Integer(5) - x + Integer(1), x**Integer(2)).absolute_igusa_invariants_kohel()
        (-1030567/178769, 259686400/178769, 20806400/178769)
        >>> HyperellipticCurve((x**Integer(5) - x + Integer(1))(Integer(3)*x + Integer(1)), (x**Integer(2))(Integer(3)*x + Integer(1))).absolute_igusa_invariants_kohel()
        (-1030567/178769, 259686400/178769, 20806400/178769)

  absolute_igusa_invariants_wamelen()

    Return the three absolute Igusa invariants used by van Wamelen [Wam1999].

    EXAMPLES:

    Sage

        sage: R.<x> = QQ[]
        sage: HyperellipticCurve(x^5 - 1).absolute_igusa_invariants_wamelen()
        (0, 0, 0)
        sage: HyperellipticCurve((x^5 - 1)(x - 2), (x^2)(x - 2)).absolute_igusa_invariants_wamelen()
        (0, 0, 0)

    Python

        >>> from sage.all import *
        >>> R = QQ['x']; (x,) = R._first_ngens(1)
        >>> HyperellipticCurve(x**Integer(5) - Integer(1)).absolute_igusa_invariants_wamelen()
        (0, 0, 0)
        >>> HyperellipticCurve((x**Integer(5) - Integer(1))(x - Integer(2)), (x**Integer(2))(x - Integer(2))).absolute_igusa_invariants_wamelen()
        (0, 0, 0)

  clebsch_invariants()

    Return the Clebsch invariants \((A, B, C, D)\) of Mestre, p 317, [Mes1991].

    See also

    `sage.schemes.hyperelliptic_curves.invariants()`

    EXAMPLES:

    Sage

        sage: R.<x> = QQ[]
        sage: f = x^5 - x^4 + 3
        sage: HyperellipticCurve(f).clebsch_invariants()
        (0, -2048/375, -4096/25, -4881645568/84375)
        sage: HyperellipticCurve(f(2*x)).clebsch_invariants()
        (0, -8388608/375, -1073741824/25, -5241627016305836032/84375)

        sage: HyperellipticCurve(f, x).clebsch_invariants()
        (-8/15, 17504/5625, -23162896/140625, -420832861216768/7119140625)
        sage: HyperellipticCurve(f(2*x), 2*x).clebsch_invariants()
        (-512/15, 71696384/5625, -6072014209024/140625, -451865844002031331704832/7119140625)

    Python

        >>> from sage.all import *
        >>> R = QQ['x']; (x,) = R._first_ngens(1)
        >>> f = x**Integer(5) - x**Integer(4) + Integer(3)
        >>> HyperellipticCurve(f).clebsch_invariants()
        (0, -2048/375, -4096/25, -4881645568/84375)
        >>> HyperellipticCurve(f(Integer(2)*x)).clebsch_invariants()
        (0, -8388608/375, -1073741824/25, -5241627016305836032/84375)

        >>> HyperellipticCurve(f, x).clebsch_invariants()
        (-8/15, 17504/5625, -23162896/140625, -420832861216768/7119140625)
        >>> HyperellipticCurve(f(Integer(2)*x), Integer(2)*x).clebsch_invariants()
        (-512/15, 71696384/5625, -6072014209024/140625, -451865844002031331704832/7119140625)

  igusa_clebsch_invariants()

    Return the Igusa-Clebsch invariants \(I_2, I_4, I_6, I_{10}\) of Igusa and Clebsch [IJ1960].

    See also

    `sage.schemes.hyperelliptic_curves.invariants()`

    EXAMPLES:

    Sage

        sage: R.<x> = QQ[]
        sage: f = x^5 - x + 2
        sage: HyperellipticCurve(f).igusa_clebsch_invariants()
        (-640, -20480, 1310720, 52160364544)
        sage: HyperellipticCurve(f(2*x)).igusa_clebsch_invariants()
        (-40960, -83886080, 343597383680, 56006764965979488256)

        sage: HyperellipticCurve(f, x).igusa_clebsch_invariants()
        (-640, 17920, -1966656, 52409511936)
        sage: HyperellipticCurve(f(2*x), 2*x).igusa_clebsch_invariants()
        (-40960, 73400320, -515547070464, 56274284941110411264)

    Python

        >>> from sage.all import *
        >>> R = QQ['x']; (x,) = R._first_ngens(1)
        >>> f = x**Integer(5) - x + Integer(2)
        >>> HyperellipticCurve(f).igusa_clebsch_invariants()
        (-640, -20480, 1310720, 52160364544)
        >>> HyperellipticCurve(f(Integer(2)*x)).igusa_clebsch_invariants()
        (-40960, -83886080, 343597383680, 56006764965979488256)

        >>> HyperellipticCurve(f, x).igusa_clebsch_invariants()
        (-640, 17920, -1966656, 52409511936)
        >>> HyperellipticCurve(f(Integer(2)*x), Integer(2)*x).igusa_clebsch_invariants()
        (-40960, 73400320, -515547070464, 56274284941110411264)

  is_odd_degree()

    Return `True` if the curve is an odd degree model.

    EXAMPLES:

    Sage

        sage: R.<x> = QQ[]
        sage: f = x^5 - x^4 + 3
        sage: HyperellipticCurve(f).is_odd_degree()
        True

    Python

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

  jacobian()

    Return the Jacobian of the hyperelliptic curve.

    EXAMPLES:

    Sage

        sage: R.<x> = QQ[]
        sage: f = x^5 - x^4 + 3
        sage: HyperellipticCurve(f).jacobian()
        Jacobian of Hyperelliptic Curve over Rational Field defined by y^2 = x^5 - x^4 + 3

    Python

        >>> from sage.all import *
        >>> R = QQ['x']; (x,) = R._first_ngens(1)
        >>> f = x**Integer(5) - x**Integer(4) + Integer(3)
        >>> HyperellipticCurve(f).jacobian()
        Jacobian of Hyperelliptic Curve over Rational Field defined by y^2 = x^5 - x^4 + 3

  kummer_morphism()

    Return the morphism of an odd degree hyperelliptic curve to the Kummer surface of its Jacobian.

    This could be extended to an even degree model if a prescribed embedding in its Jacobian is fixed.

    EXAMPLES:

    Sage

        sage: R.<x> = QQ[]
        sage: f = x^5 - x^4 + 3
        sage: HyperellipticCurve(f).kummer_morphism()  # not tested

    Python

        >>> from sage.all import *
        >>> R = QQ['x']; (x,) = R._first_ngens(1)
        >>> f = x**Integer(5) - x**Integer(4) + Integer(3)
        >>> HyperellipticCurve(f).kummer_morphism()  # not tested