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"""

Hyperelliptic curves of genus 2 over a general ring

"""

from __future__ import absolute_import

#*****************************************************************************

# Distributed under the terms of the GNU General Public License (GPL)

# http://www.gnu.org/licenses/

#*****************************************************************************

from . import hyperelliptic_generic

from . import jacobian_g2

from . import invariants

class HyperellipticCurve_g2_generic(hyperelliptic_generic.HyperellipticCurve_generic):

def is_odd_degree(self):

"""

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

EXAMPLES::

sage: R.<x> = QQ[]

sage: f = x^5 - x^4 + 3

sage: HyperellipticCurve(f).is_odd_degree()

True

"""

f, h = self.hyperelliptic_polynomials()

df = f.degree()

if h.degree() < 3:

return df%2 == 1

elif df < 6:

return False

else:

a0 = f.leading_coefficient()

c0 = h.leading_coefficient()

return (c0**2 + 4*a0) == 0

def jacobian(self):

"""

Return the Jacobian of the hyperelliptic curve.

EXAMPLES::

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

"""

return jacobian_g2.HyperellipticJacobian_g2(self)

def kummer_morphism(self):

"""

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: R.<x> = QQ[]

sage: f = x^5 - x^4 + 3

sage: HyperellipticCurve(f).kummer_morphism() # not tested

"""

try:

return self._kummer_morphism

except AttributeError:

pass

if not self.is_odd_degree():

raise TypeError("Kummer embedding not determined for even degree model curves.")

J = self.jacobian()

K = J.kummer_surface()

return self._kummer_morphism

def clebsch_invariants(self):

r"""

Return the Clebsch invariants `(A, B, C, D)` of Mestre, p 317, [M]_.

.. SEEALSO::

:meth:`sage.schemes.hyperelliptic_curves.invariants`

EXAMPLES::

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)

TESTS::

sage: magma(HyperellipticCurve(f)).ClebschInvariants() # optional - magma

[ 0, -2048/375, -4096/25, -4881645568/84375 ]

sage: magma(HyperellipticCurve(f(2*x))).ClebschInvariants() # optional - magma

[ 0, -8388608/375, -1073741824/25, -5241627016305836032/84375 ]

sage: magma(HyperellipticCurve(f, x)).ClebschInvariants() # optional - magma

[ -8/15, 17504/5625, -23162896/140625, -420832861216768/7119140625 ]

sage: magma(HyperellipticCurve(f(2*x), 2*x)).ClebschInvariants() # optional - magma

[ -512/15, 71696384/5625, -6072014209024/140625, -451865844002031331704832/7119140625 ]

"""

f, h = self.hyperelliptic_polynomials()

return invariants.clebsch_invariants(4*f + h**2)

def igusa_clebsch_invariants(self):

r"""

Return the Igusa-Clebsch invariants `I_2, I_4, I_6, I_{10}` of Igusa and Clebsch [I]_.

.. SEEALSO::

:meth:`sage.schemes.hyperelliptic_curves.invariants`

EXAMPLES::

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)

TESTS::

sage: magma(HyperellipticCurve(f)).IgusaClebschInvariants() # optional - magma

[ -640, -20480, 1310720, 52160364544 ]

sage: magma(HyperellipticCurve(f(2*x))).IgusaClebschInvariants() # optional - magma

[ -40960, -83886080, 343597383680, 56006764965979488256 ]

sage: magma(HyperellipticCurve(f, x)).IgusaClebschInvariants() # optional - magma

[ -640, 17920, -1966656, 52409511936 ]

sage: magma(HyperellipticCurve(f(2*x), 2*x)).IgusaClebschInvariants() # optional - magma

[ -40960, 73400320, -515547070464, 56274284941110411264 ]

"""

f, h = self.hyperelliptic_polynomials()

return invariants.igusa_clebsch_invariants(4*f + h**2)

def absolute_igusa_invariants_wamelen(self):

r"""

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

EXAMPLES::

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)

"""

f, h = self.hyperelliptic_polynomials()

return invariants.absolute_igusa_invariants_wamelen(4*f + h**2)

def absolute_igusa_invariants_kohel(self):

r"""

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

.. SEEALSO::

:meth:`sage.schemes.hyperelliptic_curves.invariants`

EXAMPLES::

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)

"""

f, h = self.hyperelliptic_polynomials()

return invariants.absolute_igusa_invariants_kohel(4*f + h**2)

def clebsch_invariants(self):

r"""

Return the Clebsch invariants `(A, B, C, D)` of Mestre, p 317, [M]_.

.. SEEALSO::

:meth:`sage.schemes.hyperelliptic_curves.invariants`

EXAMPLES::

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)

TESTS::

sage: magma(HyperellipticCurve(f)).ClebschInvariants() # optional - magma

[ 0, -2048/375, -4096/25, -4881645568/84375 ]

sage: magma(HyperellipticCurve(f(2*x))).ClebschInvariants() # optional - magma

[ 0, -8388608/375, -1073741824/25, -5241627016305836032/84375 ]

sage: magma(HyperellipticCurve(f, x)).ClebschInvariants() # optional - magma

[ -8/15, 17504/5625, -23162896/140625, -420832861216768/7119140625 ]

sage: magma(HyperellipticCurve(f(2*x), 2*x)).ClebschInvariants() # optional - magma

[ -512/15, 71696384/5625, -6072014209024/140625, -451865844002031331704832/7119140625 ]

"""

f, h = self.hyperelliptic_polynomials()

return invariants.clebsch_invariants(4*f + h**2)

def igusa_clebsch_invariants(self):

r"""

Return the Igusa-Clebsch invariants `I_2, I_4, I_6, I_{10}` of Igusa and Clebsch [I]_.

.. SEEALSO::

:meth:`sage.schemes.hyperelliptic_curves.invariants`

EXAMPLES::

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)

TESTS::

sage: magma(HyperellipticCurve(f)).IgusaClebschInvariants() # optional - magma

[ -640, -20480, 1310720, 52160364544 ]

sage: magma(HyperellipticCurve(f(2*x))).IgusaClebschInvariants() # optional - magma

[ -40960, -83886080, 343597383680, 56006764965979488256 ]

sage: magma(HyperellipticCurve(f, x)).IgusaClebschInvariants() # optional - magma

[ -640, 17920, -1966656, 52409511936 ]

sage: magma(HyperellipticCurve(f(2*x), 2*x)).IgusaClebschInvariants() # optional - magma

[ -40960, 73400320, -515547070464, 56274284941110411264 ]

"""

f, h = self.hyperelliptic_polynomials()

return invariants.igusa_clebsch_invariants(4*f + h**2)

def absolute_igusa_invariants_wamelen(self):

r"""

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

EXAMPLES::

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)

"""

f, h = self.hyperelliptic_polynomials()

return invariants.absolute_igusa_invariants_wamelen(4*f + h**2)

def absolute_igusa_invariants_kohel(self):

r"""

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

.. SEEALSO::

:meth:`sage.schemes.hyperelliptic_curves.invariants`

EXAMPLES::

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)

"""

f, h = self.hyperelliptic_polynomials()

return invariants.absolute_igusa_invariants_kohel(4*f + h**2)

Coverage for local/lib/python2.7/site-packages/sage/schemes/hyperelliptic_curves/hyperelliptic_g2_generic.py : 25%

51 statements 13 run 38 missing 0 excluded