Mestre’s algorithm - Elliptic curves

Skip to content
Back to top
View this page
Edit this page

# 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')

  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:

  - `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

      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.

  Python

      >>> 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

      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)

  [Python]

      >>> 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

      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

  Python

      >>> 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

      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

  [Python]

      >>> 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

      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

  Python

      >>> 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

      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

  [Python]

      >>> 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')

  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

      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

  Python

      >>> 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

      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

  [Python]

      >>> 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

      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

  Python

      >>> 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

      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)

  [Python]

      >>> 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.