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,I10
- reduced– boolean (default:- True); if- True, tries to reduce the polynomial defining the hyperelliptic curve using the function- reduce_polynomial()(see the- reduce_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. If- None, 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. - 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) - 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 - 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 - >>> 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 - 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 - 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 - 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, I10
- xyz– boolean (default:- False); if- True, the algorithm also returns three invariants \(x\),`y`,`z` used in Mestre’s algorithm
- names– (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 - 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 - 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 - 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) - 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.