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Returns a hyperelliptic curve with the given Igusa-Clebsch invariants up to scaling.
The output is a curve over the field in which the Igusa-Clebsch invariants are given. The output curve is unique up to isomorphism over the algebraic closure. If no such curve exists over the given field, then raise a ValueError.
INPUT:
OUTPUT:
A hyperelliptic curve object.
EXAMPLE:
Examples over the rationals:
sage: HyperellipticCurve_from_invariants([3840,414720,491028480,2437709561856])
Traceback (most recent call last):
...
NotImplementedError: Reduction of hyperelliptic curves not yet implemented. See trac #14755 and #14756.
sage: HyperellipticCurve_from_invariants([3840,414720,491028480,2437709561856],reduced = False)
Hyperelliptic Curve over Rational Field defined by y^2 = -x^6 + 4410*x^5 - 540*x^4 + 4320*x^3 - 19440*x^2 + 46656*x - 46656
sage: HyperellipticCurve_from_invariants([21, 225/64, 22941/512, 1])
Traceback (most recent call last):
...
NotImplementedError: Reduction of hyperelliptic curves not yet implemented. See trac #14755 and #14756.
An example over a finite field:
sage: HyperellipticCurve_from_invariants([GF(13)(1),3,7,5])
Hyperelliptic Curve over Finite Field of size 13 defined by y^2 = 8*x^5 + 5*x^4 + 5*x^2 + 9*x + 3
An example over a number field:
sage: K = QuadraticField(353, 'a')
sage: H = HyperellipticCurve_from_invariants([21, 225/64, 22941/512, 1], reduced = false)
sage: f = K['x'](H.hyperelliptic_polynomials()[0])
If the Mestre Conic defined by the Igusa-Clebsch invariants has no rational points, then there exists no hyperelliptic curve over the base field with the given invariants.:
sage: HyperellipticCurve_from_invariants([1,2,3,4])
Traceback (most recent call last):
...
ValueError: No such curve exists over Rational Field as there are no rational points on Projective Conic Curve over Rational Field defined by -2572155000*u^2 - 317736000*u*v + 1250755459200*v^2 + 2501510918400*u*w + 39276887040*v*w + 2736219686912*w^2
Mestre’s algorithm only works for generic curves of genus two, so another algorithm is needed for those curves with extra automorphism. See also trac ticket #12199:
sage: P.<x> = QQ[]
sage: C = HyperellipticCurve(x^6+1)
sage: i = C.igusa_clebsch_invariants()
sage: HyperellipticCurve_from_invariants(i)
Traceback (most recent call last):
...
TypeError: F (=0) must have degree 2
Igusa-Clebsch invariants also only work over fields of charateristic different from 2, 3, and 5, so another algorithm will be needed for fields of those characteristics. See also trac ticket #12200:
sage: P.<x> = GF(3)[]
sage: HyperellipticCurve(x^6+x+1).igusa_clebsch_invariants()
Traceback (most recent call last):
...
NotImplementedError: Invariants of binary sextics/genus 2 hyperelliptic curves not implemented in characteristics 2, 3, and 5
sage: HyperellipticCurve_from_invariants([GF(5)(1),1,0,1])
Traceback (most recent call last):
...
ZeroDivisionError: Inverse does not exist.
ALGORITHM:
This is Mestre’s algorithm [M1991]. Our implementation is based on the formulae on page 957 of [LY2001], cross-referenced with [W1999] to correct typos.
First construct Mestre’s conic using the Mestre_conic() function.
Parametrize the conic if possible.
Let be the three coordinates of the parametrization of the
conic by the projective line, and change them into one variable by letting
. Note that each
has degree at most 2.
Then construct a sextic polynomial
,
where
are defined as rational functions in the invariants
(see the source code for detailed formulae for
).
The output is the hyperelliptic curve
.
REFERENCES:
[LY2001] | (1, 2, 3) K. Lauter and T. Yang, “Computing genus 2 curves from invariants on the Hilbert moduli space”, Journal of Number Theory 131 (2011), pages 936 - 958 |
[M1991] | (1, 2) J.-F. Mestre, “Construction de courbes de genre 2 a partir de leurs modules”, in Effective methods in algebraic geometry (Castiglioncello, 1990), volume 94 of Progr. Math., pages 313 - 334 |
[W1999] | P. van Wamelen, Pari-GP code, section “thecubic” https://www.math.lsu.edu/~wamelen/Genus2/FindCurve/igusa2curve.gp |
Return the conic equation from Mestre’s algorithm given the Igusa-Clebsch invariants.
It has a rational point if and only if a hyperelliptic curve corresponding to the invariants exists.
INPUT:
OUTPUT:
A Conic object
EXAMPLES:
A standard example:
sage: Mestre_conic([1,2,3,4])
Projective Conic Curve over Rational Field defined by -2572155000*u^2 - 317736000*u*v + 1250755459200*v^2 + 2501510918400*u*w + 39276887040*v*w + 2736219686912*w^2
Note that the algorithm works over number fields as well:
sage: k = NumberField(x^2-41,'a')
sage: a = k.an_element()
sage: Mestre_conic([1,2+a,a,4+a])
Projective Conic Curve over Number Field in a with defining polynomial x^2 - 41 defined by (-801900000*a + 343845000)*u^2 + (855360000*a + 15795864000)*u*v + (312292800000*a + 1284808579200)*v^2 + (624585600000*a + 2569617158400)*u*w + (15799910400*a + 234573143040)*v*w + (2034199306240*a + 16429854656512)*w^2
And over finite fields:
sage: Mestre_conic([GF(7)(10),GF(7)(1),GF(7)(2),GF(7)(3)])
Projective Conic Curve over Finite Field of size 7 defined by -2*u*v - v^2 - 2*u*w + 2*v*w - 3*w^2
An example with xyz:
sage: Mestre_conic([5,6,7,8], xyz=True)
(Projective Conic Curve over Rational Field defined by -415125000*u^2 + 608040000*u*v + 33065136000*v^2 + 66130272000*u*w + 240829440*v*w + 10208835584*w^2, 232/1125, -1072/16875, 14695616/2109375)
ALGORITHM:
The formulas are taken from pages 956 - 957 of [LY2001] and based on pages 321 and 332 of [M1991].
See the code or [LY2001] for the detailed formulae defining x, y, z and L.