CoCalc -- hyperelliptic_g2

GitHub Repository: sagemath/sagelib
Path: blob/master/sage/schemes/hyperelliptic_curves/hyperelliptic_g2_generic.py
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#*****************************************************************************
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#  Copyright (C) 2006 David Kohel <[email protected]>
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#  Distributed under the terms of the GNU General Public License (GPL)
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#                  http://www.gnu.org/licenses/
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#*****************************************************************************
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import hyperelliptic_generic
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import jacobian_g2
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import invariants
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class HyperellipticCurve_g2_generic(hyperelliptic_generic.HyperellipticCurve_generic):
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    def is_odd_degree(self):
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        """
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        Returns True if the curve is an odd degree model.
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        """
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        f, h = self.hyperelliptic_polynomials()
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        df = f.degree()
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        if h.degree() < 3: 
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            return df%2 == 1
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        elif df < 6:
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            return False
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        else:
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            a0 = f.leading_coefficient()
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            c0 = h.leading_coefficient()
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            return (c0**2 + 4*a0) == 0
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    def jacobian(self):
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        return jacobian_g2.HyperellipticJacobian_g2(self)
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    def kummer_morphism(self):
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        """
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        Returns the morphism of an odd degree hyperelliptic curve to the Kummer
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        surface of its Jacobian.  (This could be extended to an even degree model
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        if a prescribed embedding in its Jacobian is fixed.)
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        """
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        try:
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            return self._kummer_morphism
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        except AttributeError:
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            pass
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        if not self.is_odd_degree():
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            raise TypeError, \
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                  "Kummer embedding not determined for even degree model curves."
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        J = self.jacobian()
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        K = J.kummer_surface()
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        return self._kummer_morphism
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    def clebsch_invariants(self):
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        r"""
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        Return the Clebsch invariants `(A, B, C, D)` of Mestre, p 317, [M]_.
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        SEEALSO::
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        .. sage.schemes.hyperelliptic_curves.invariants
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        EXAMPLES::
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            sage: R.<x> = QQ[]
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            sage: f = x^5 - x^4 + 3
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            sage: HyperellipticCurve(f).clebsch_invariants()
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            (0, -2048/375, -4096/25, -4881645568/84375)
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            sage: HyperellipticCurve(f(2*x)).clebsch_invariants()
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            (0, -8388608/375, -1073741824/25, -5241627016305836032/84375)
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            sage: HyperellipticCurve(f, x).clebsch_invariants()
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            (-8/15, 17504/5625, -23162896/140625, -420832861216768/7119140625)
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            sage: HyperellipticCurve(f(2*x), 2*x).clebsch_invariants()
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            (-512/15, 71696384/5625, -6072014209024/140625, -451865844002031331704832/7119140625)
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        TESTS::
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            sage: magma(HyperellipticCurve(f)).ClebschInvariants() # optional - magma
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            [ 0, -2048/375, -4096/25, -4881645568/84375 ]
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            sage: magma(HyperellipticCurve(f(2*x))).ClebschInvariants() # optional - magma
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            [ 0, -8388608/375, -1073741824/25, -5241627016305836032/84375 ]
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            sage: magma(HyperellipticCurve(f, x)).ClebschInvariants() # optional - magma
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            [ -8/15, 17504/5625, -23162896/140625, -420832861216768/7119140625 ]
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            sage: magma(HyperellipticCurve(f(2*x), 2*x)).ClebschInvariants() # optional - magma
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            [ -512/15, 71696384/5625, -6072014209024/140625, -451865844002031331704832/7119140625 ]
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        """
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        f, h = self.hyperelliptic_polynomials()
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        return invariants.clebsch_invariants(4*f + h**2)
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    def igusa_clebsch_invariants(self):
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        r"""
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        Return the Igusa-Clebsch invariants `I_2, I_4, I_6, I_{10}` of Igusa and Clebsch [I]_.
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        SEEALSO::
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        .. sage.schemes.hyperelliptic_curves.invariants
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        EXAMPLES::
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            sage: R.<x> = QQ[]
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            sage: f = x^5 - x + 2
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            sage: HyperellipticCurve(f).igusa_clebsch_invariants()
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            (-640, -20480, 1310720, 52160364544)
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            sage: HyperellipticCurve(f(2*x)).igusa_clebsch_invariants()
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            (-40960, -83886080, 343597383680, 56006764965979488256)
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            sage: HyperellipticCurve(f, x).igusa_clebsch_invariants()
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            (-640, 17920, -1966656, 52409511936)
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            sage: HyperellipticCurve(f(2*x), 2*x).igusa_clebsch_invariants()
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            (-40960, 73400320, -515547070464, 56274284941110411264)
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        TESTS::
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            sage: magma(HyperellipticCurve(f)).IgusaClebschInvariants() # optional - magma
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            [ -640, -20480, 1310720, 52160364544 ]
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            sage: magma(HyperellipticCurve(f(2*x))).IgusaClebschInvariants() # optional - magma
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            [ -40960, -83886080, 343597383680, 56006764965979488256 ]
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            sage: magma(HyperellipticCurve(f, x)).IgusaClebschInvariants() # optional - magma
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            [ -640, 17920, -1966656, 52409511936 ]
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            sage: magma(HyperellipticCurve(f(2*x), 2*x)).IgusaClebschInvariants() # optional - magma
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            [ -40960, 73400320, -515547070464, 56274284941110411264 ]
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        """
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        f, h = self.hyperelliptic_polynomials()
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        return invariants.igusa_clebsch_invariants(4*f + h**2)
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    def absolute_igusa_invariants_wamelen(self):
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        r"""
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        Return the three absolute Igusa invariants used by van Wamelen [W]_.
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        EXAMPLES::
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            sage: R.<x> = QQ[]
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            sage: HyperellipticCurve(x^5 - 1).absolute_igusa_invariants_wamelen()
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            (0, 0, 0)
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            sage: HyperellipticCurve((x^5 - 1)(x - 2), (x^2)(x - 2)).absolute_igusa_invariants_wamelen()
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            (0, 0, 0)
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        """
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        f, h = self.hyperelliptic_polynomials()
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        return invariants.absolute_igusa_invariants_wamelen(4*f + h**2)
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    def absolute_igusa_invariants_kohel(self):
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        r"""
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        Return the three absolute Igusa invariants used by Kohel [K]_.
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        SEEALSO::
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        .. sage.schemes.hyperelliptic_curves.invariants
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        EXAMPLES::
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            sage: R.<x> = QQ[]
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            sage: HyperellipticCurve(x^5 - 1).absolute_igusa_invariants_kohel()
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            (0, 0, 0)
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            sage: HyperellipticCurve(x^5 - x + 1, x^2).absolute_igusa_invariants_kohel()
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            (-1030567/178769, 259686400/178769, 20806400/178769)
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            sage: HyperellipticCurve((x^5 - x + 1)(3*x + 1), (x^2)(3*x + 1)).absolute_igusa_invariants_kohel()
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            (-1030567/178769, 259686400/178769, 20806400/178769)
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        """
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        f, h = self.hyperelliptic_polynomials()
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        return invariants.absolute_igusa_invariants_kohel(4*f + h**2)
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    def clebsch_invariants(self):
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        r"""
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        Return the Clebsch invariants `(A, B, C, D)` of Mestre, p 317, [M]_.
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        SEEALSO::
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        .. sage.schemes.hyperelliptic_curves.invariants
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        EXAMPLES::
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            sage: R.<x> = QQ[]
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            sage: f = x^5 - x^4 + 3
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            sage: HyperellipticCurve(f).clebsch_invariants()
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            (0, -2048/375, -4096/25, -4881645568/84375)
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            sage: HyperellipticCurve(f(2*x)).clebsch_invariants()
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            (0, -8388608/375, -1073741824/25, -5241627016305836032/84375)
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            sage: HyperellipticCurve(f, x).clebsch_invariants()
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            (-8/15, 17504/5625, -23162896/140625, -420832861216768/7119140625)
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            sage: HyperellipticCurve(f(2*x), 2*x).clebsch_invariants()
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            (-512/15, 71696384/5625, -6072014209024/140625, -451865844002031331704832/7119140625)
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        TESTS::
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            sage: magma(HyperellipticCurve(f)).ClebschInvariants() # optional - magma
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            [ 0, -2048/375, -4096/25, -4881645568/84375 ]
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            sage: magma(HyperellipticCurve(f(2*x))).ClebschInvariants() # optional - magma
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            [ 0, -8388608/375, -1073741824/25, -5241627016305836032/84375 ]
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            sage: magma(HyperellipticCurve(f, x)).ClebschInvariants() # optional - magma
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            [ -8/15, 17504/5625, -23162896/140625, -420832861216768/7119140625 ]
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            sage: magma(HyperellipticCurve(f(2*x), 2*x)).ClebschInvariants() # optional - magma
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            [ -512/15, 71696384/5625, -6072014209024/140625, -451865844002031331704832/7119140625 ]
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        """
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        f, h = self.hyperelliptic_polynomials()
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        return invariants.clebsch_invariants(4*f + h**2)
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    def igusa_clebsch_invariants(self):
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        r"""
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        Return the Igusa-Clebsch invariants `I_2, I_4, I_6, I_{10}` of Igusa and Clebsch [I]_.
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        SEEALSO::
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        .. sage.schemes.hyperelliptic_curves.invariants
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        EXAMPLES::
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            sage: R.<x> = QQ[]
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            sage: f = x^5 - x + 2
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            sage: HyperellipticCurve(f).igusa_clebsch_invariants()
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            (-640, -20480, 1310720, 52160364544)
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            sage: HyperellipticCurve(f(2*x)).igusa_clebsch_invariants()
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            (-40960, -83886080, 343597383680, 56006764965979488256)
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            sage: HyperellipticCurve(f, x).igusa_clebsch_invariants()
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            (-640, 17920, -1966656, 52409511936)
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            sage: HyperellipticCurve(f(2*x), 2*x).igusa_clebsch_invariants()
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            (-40960, 73400320, -515547070464, 56274284941110411264)
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        TESTS::
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            sage: magma(HyperellipticCurve(f)).IgusaClebschInvariants() # optional - magma
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            [ -640, -20480, 1310720, 52160364544 ]
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            sage: magma(HyperellipticCurve(f(2*x))).IgusaClebschInvariants() # optional - magma
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            [ -40960, -83886080, 343597383680, 56006764965979488256 ]
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            sage: magma(HyperellipticCurve(f, x)).IgusaClebschInvariants() # optional - magma
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            [ -640, 17920, -1966656, 52409511936 ]
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            sage: magma(HyperellipticCurve(f(2*x), 2*x)).IgusaClebschInvariants() # optional - magma
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            [ -40960, 73400320, -515547070464, 56274284941110411264 ]
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        """
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        f, h = self.hyperelliptic_polynomials()
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        return invariants.igusa_clebsch_invariants(4*f + h**2)
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    def absolute_igusa_invariants_wamelen(self):
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        r"""
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        Return the three absolute Igusa invariants used by van Wamelen [W]_.
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        EXAMPLES::
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            sage: R.<x> = QQ[]
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            sage: HyperellipticCurve(x^5 - 1).absolute_igusa_invariants_wamelen()
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            (0, 0, 0)
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            sage: HyperellipticCurve((x^5 - 1)(x - 2), (x^2)(x - 2)).absolute_igusa_invariants_wamelen()
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            (0, 0, 0)
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        """
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        f, h = self.hyperelliptic_polynomials()
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        return invariants.absolute_igusa_invariants_wamelen(4*f + h**2)
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    def absolute_igusa_invariants_kohel(self):
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        r"""
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        Return the three absolute Igusa invariants used by Kohel [K]_.
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        SEEALSO::
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        .. sage.schemes.hyperelliptic_curves.invariants
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        EXAMPLES::
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            sage: R.<x> = QQ[]
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            sage: HyperellipticCurve(x^5 - 1).absolute_igusa_invariants_kohel()
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            (0, 0, 0)
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            sage: HyperellipticCurve(x^5 - x + 1, x^2).absolute_igusa_invariants_kohel()
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            (-1030567/178769, 259686400/178769, 20806400/178769)
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            sage: HyperellipticCurve((x^5 - x + 1)(3*x + 1), (x^2)(3*x + 1)).absolute_igusa_invariants_kohel()
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            (-1030567/178769, 259686400/178769, 20806400/178769)
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        """
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        f, h = self.hyperelliptic_polynomials()
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        return invariants.absolute_igusa_invariants_kohel(4*f + h**2)
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