Mestre’s algorithm¶
This file contains functions that:
create hyperelliptic curves from the Igusa-Clebsch invariants (over \(\QQ\) and finite fields)
create Mestre’s conic from the Igusa-Clebsch invariants
AUTHORS:
Florian Bouyer
Marco Streng
- sage.schemes.hyperelliptic_curves.mestre.HyperellipticCurve_from_invariants(i, reduced=True, precision=None, algorithm='default')[source]¶
Return 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:
i– list or tuple of length 4 containing the four Igusa-Clebsch invariants: I2,I4,I6,I10reduced– boolean (default:True); ifTrue, tries to reduce the polynomial defining the hyperelliptic curve using the functionreduce_polynomial()(see thereduce_polynomial()documentation for more details).precision– integer (default:None); which precision for real and complex numbers should the reduction use. This only affects the reduction, not the correctness. IfNone, the algorithm uses the default 53 bit precision.algorithm–'default'or'magma'. If set to'magma', uses Magma to parameterize Mestre’s conic (needs Magma to be installed)
OUTPUT: a hyperelliptic curve object
EXAMPLES:
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 issues #14755 and #14756. sage: HyperellipticCurve_from_invariants([3840,414720,491028480,2437709561856], reduced=False) Hyperelliptic Curve over Rational Field defined by y^2 = -46656*x^6 + 46656*x^5 - 19440*x^4 + 4320*x^3 - 540*x^2 + 4410*x - 1 sage: HyperellipticCurve_from_invariants([21, 225/64, 22941/512, 1]) Traceback (most recent call last): ... NotImplementedError: Reduction of hyperelliptic curves not yet implemented. See issues #14755 and #14756.
>>> from sage.all import * >>> HyperellipticCurve_from_invariants([Integer(3840),Integer(414720),Integer(491028480),Integer(2437709561856)]) Traceback (most recent call last): ... NotImplementedError: Reduction of hyperelliptic curves not yet implemented. See issues #14755 and #14756. >>> HyperellipticCurve_from_invariants([Integer(3840),Integer(414720),Integer(491028480),Integer(2437709561856)], reduced=False) Hyperelliptic Curve over Rational Field defined by y^2 = -46656*x^6 + 46656*x^5 - 19440*x^4 + 4320*x^3 - 540*x^2 + 4410*x - 1 >>> HyperellipticCurve_from_invariants([Integer(21), Integer(225)/Integer(64), Integer(22941)/Integer(512), Integer(1)]) Traceback (most recent call last): ... NotImplementedError: Reduction of hyperelliptic curves not yet implemented. See issues #14755 and #14756.
HyperellipticCurve_from_invariants([3840,414720,491028480,2437709561856]) HyperellipticCurve_from_invariants([3840,414720,491028480,2437709561856], reduced=False) HyperellipticCurve_from_invariants([21, 225/64, 22941/512, 1])
An example over a finite field:
sage: H = HyperellipticCurve_from_invariants([GF(13)(1), 3, 7, 5]); H Hyperelliptic Curve over Finite Field of size 13 defined by ... sage: H.igusa_clebsch_invariants() (4, 9, 6, 11)
>>> from sage.all import * >>> H = HyperellipticCurve_from_invariants([GF(Integer(13))(Integer(1)), Integer(3), Integer(7), Integer(5)]); H Hyperelliptic Curve over Finite Field of size 13 defined by ... >>> H.igusa_clebsch_invariants() (4, 9, 6, 11)
H = HyperellipticCurve_from_invariants([GF(13)(1), 3, 7, 5]); H H.igusa_clebsch_invariants()
An example over a number field:
sage: K = QuadraticField(353, 'a') # needs sage.rings.number_field sage: H = HyperellipticCurve_from_invariants([21, 225/64, 22941/512, 1], # needs sage.rings.number_field ....: reduced=false) sage: f = K['x'](H.hyperelliptic_polynomials()[0]) # needs sage.rings.number_field
>>> from sage.all import * >>> K = QuadraticField(Integer(353), 'a') # needs sage.rings.number_field >>> H = HyperellipticCurve_from_invariants([Integer(21), Integer(225)/Integer(64), Integer(22941)/Integer(512), Integer(1)], # needs sage.rings.number_field ... reduced=false) >>> f = K['x'](H.hyperelliptic_polynomials()[Integer(0)]) # needs sage.rings.number_field
K = QuadraticField(353, 'a') # needs sage.rings.number_field H = HyperellipticCurve_from_invariants([21, 225/64, 22941/512, 1], # needs sage.rings.number_field reduced=false) f = K['x'](H.hyperelliptic_polynomials()[0]) # needs sage.rings.number_fieldIf 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
>>> from sage.all import * >>> HyperellipticCurve_from_invariants([Integer(1),Integer(2),Integer(3),Integer(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
HyperellipticCurve_from_invariants([1,2,3,4])
Mestre’s algorithm only works for generic curves of genus two, so another algorithm is needed for those curves with extra automorphism. See also Issue #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
>>> from sage.all import * >>> P = QQ['x']; (x,) = P._first_ngens(1) >>> C = HyperellipticCurve(x**Integer(6) + Integer(1)) >>> i = C.igusa_clebsch_invariants() >>> HyperellipticCurve_from_invariants(i) Traceback (most recent call last): ... TypeError: F (=0) must have degree 2
P.<x> = QQ[] C = HyperellipticCurve(x^6 + 1) i = C.igusa_clebsch_invariants() HyperellipticCurve_from_invariants(i)
Igusa-Clebsch invariants also only work over fields of characteristic different from 2, 3, and 5, so another algorithm will be needed for fields of those characteristics. See also Issue #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 of Mod(0, 5) does not exist
>>> from sage.all import * >>> P = GF(Integer(3))['x']; (x,) = P._first_ngens(1) >>> HyperellipticCurve(x**Integer(6) + x + Integer(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 >>> HyperellipticCurve_from_invariants([GF(Integer(5))(Integer(1)), Integer(1), Integer(0), Integer(1)]) Traceback (most recent call last): ... ZeroDivisionError: inverse of Mod(0, 5) does not exist
P.<x> = GF(3)[] HyperellipticCurve(x^6 + x + 1).igusa_clebsch_invariants() HyperellipticCurve_from_invariants([GF(5)(1), 1, 0, 1])
ALGORITHM:
This is Mestre’s algorithm [Mes1991]. Our implementation is based on the formulae on page 957 of [LY2001], cross-referenced with [Wam1999b] to correct typos.
First construct Mestre’s conic using the
Mestre_conic()function. Parametrize the conic if possible. Let \(f_1, f_2, f_3\) be the three coordinates of the parametrization of the conic by the projective line, and change them into one variable by letting \(F_i = f_i(t, 1)\). Note that each \(F_i\) has degree at most 2.Then construct a sextic polynomial \(f = \sum_{0<=i,j,k<=3}{c_{ijk}*F_i*F_j*F_k}\), where \(c_{ijk}\) are defined as rational functions in the invariants (see the source code for detailed formulae for \(c_{ijk}\)). The output is the hyperelliptic curve \(y^2 = f\).
- sage.schemes.hyperelliptic_curves.mestre.Mestre_conic(i, xyz=False, names='u,v,w')[source]¶
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:
i– list or tuple of length 4 containing the four Igusa-Clebsch invariants: I2, I4, I6, I10xyz– boolean (default:False); ifTrue, the algorithm also returns three invariants \(x\),`y`,`z` used in Mestre’s algorithmnames– (default:'u,v,w') the variable names for the conic
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
>>> from sage.all import * >>> Mestre_conic([Integer(1),Integer(2),Integer(3),Integer(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
Mestre_conic([1,2,3,4])
Note that the algorithm works over number fields as well:
sage: x = polygen(ZZ, 'x') sage: k = NumberField(x^2 - 41, 'a') # needs sage.rings.number_field sage: a = k.an_element() # needs sage.rings.number_field sage: Mestre_conic([1, 2 + a, a, 4 + a]) # needs sage.rings.number_field 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
>>> from sage.all import * >>> x = polygen(ZZ, 'x') >>> k = NumberField(x**Integer(2) - Integer(41), 'a') # needs sage.rings.number_field >>> a = k.an_element() # needs sage.rings.number_field >>> Mestre_conic([Integer(1), Integer(2) + a, a, Integer(4) + a]) # needs sage.rings.number_field 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
x = polygen(ZZ, 'x') k = NumberField(x^2 - 41, 'a') # needs sage.rings.number_field a = k.an_element() # needs sage.rings.number_field Mestre_conic([1, 2 + a, a, 4 + a]) # needs sage.rings.number_field
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
>>> from sage.all import * >>> Mestre_conic([GF(Integer(7))(Integer(10)), GF(Integer(7))(Integer(1)), GF(Integer(7))(Integer(2)), GF(Integer(7))(Integer(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
Mestre_conic([GF(7)(10), GF(7)(1), GF(7)(2), GF(7)(3)])
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)
>>> from sage.all import * >>> Mestre_conic([Integer(5),Integer(6),Integer(7),Integer(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)
Mestre_conic([5,6,7,8], xyz=True)
ALGORITHM:
The formulas are taken from pages 956 - 957 of [LY2001] and based on pages 321 and 332 of [Mes1991].
See the code or [LY2001] for the detailed formulae defining x, y, z and L.