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 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)