# 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)[source]#
absolute_igusa_invariants_kohel()[source]#

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

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)

>>> 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()[source]#

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

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)

>>> 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()[source]#

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

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)

>>> 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()[source]#

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

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)

>>> 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()[source]#

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

>>> 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()[source]#

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

>>> 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()[source]#

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

>>> 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