Projective plane conics over \(\QQ\)

AUTHORS:

  • Marco Streng (2010-07-20)

  • Nick Alexander (2008-01-08)

class sage.schemes.plane_conics.con_rational_field.ProjectiveConic_rational_field(A, f)[source]

Bases: ProjectiveConic_number_field

Create a projective plane conic curve over \(\QQ\).

See Conic for full documentation.

EXAMPLES:

sage: P.<X, Y, Z> = QQ[]
sage: Conic(X^2 + Y^2 - 3*Z^2)
Projective Conic Curve over Rational Field defined by X^2 + Y^2 - 3*Z^2

>>> from sage.all import *
>>> P = QQ['X, Y, Z']; (X, Y, Z,) = P._first_ngens(3)
>>> Conic(X**Integer(2) + Y**Integer(2) - Integer(3)*Z**Integer(2))
Projective Conic Curve over Rational Field defined by X^2 + Y^2 - 3*Z^2
has_rational_point(point=False, obstruction=False, algorithm='default', read_cache=True)[source]

Return True if and only if self has a point defined over \(\QQ\).

If point and obstruction are both False (default), then the output is a boolean out saying whether self has a rational point.

If point or obstruction is True, then the output is a pair (out, S), where out is as above and the following holds:

  • if point is True and self has a rational point, then S is a rational point,

  • if obstruction is True and self has no rational point, then S is a prime such that no rational point exists over the completion at S or \(-1\) if no point exists over \(\RR\).

Points and obstructions are cached, whenever they are found. Cached information is used if and only if read_cache is True.

ALGORITHM:

The parameter algorithm specifies the algorithm to be used:

  • 'qfsolve' – use PARI/GP function pari:qfsolve

  • 'rnfisnorm' – use PARI’s function pari:rnfisnorm (cannot be combined with obstruction = True)

  • 'local' – check if a local solution exists for all primes and infinite places of \(\QQ\) and apply the Hasse principle (cannot be combined with point = True)

  • 'default' – use 'qfsolve'

  • 'magma' (requires Magma to be installed) – delegates the task to the Magma computer algebra system.

EXAMPLES:

sage: # needs sage.libs.pari
sage: C = Conic(QQ, [1, 2, -3])
sage: C.has_rational_point(point=True)
(True, (1 : 1 : 1))
sage: D = Conic(QQ, [1, 3, -5])
sage: D.has_rational_point(point=True)
(False, 3)
sage: P.<X,Y,Z> = QQ[]
sage: E = Curve(X^2 + Y^2 + Z^2); E
Projective Conic Curve over Rational Field defined by X^2 + Y^2 + Z^2
sage: E.has_rational_point(obstruction=True)
(False, -1)

>>> from sage.all import *
>>> # needs sage.libs.pari
>>> C = Conic(QQ, [Integer(1), Integer(2), -Integer(3)])
>>> C.has_rational_point(point=True)
(True, (1 : 1 : 1))
>>> D = Conic(QQ, [Integer(1), Integer(3), -Integer(5)])
>>> D.has_rational_point(point=True)
(False, 3)
>>> P = QQ['X, Y, Z']; (X, Y, Z,) = P._first_ngens(3)
>>> E = Curve(X**Integer(2) + Y**Integer(2) + Z**Integer(2)); E
Projective Conic Curve over Rational Field defined by X^2 + Y^2 + Z^2
>>> E.has_rational_point(obstruction=True)
(False, -1)

The following would not terminate quickly with algorithm = 'rnfisnorm'

sage: C = Conic(QQ, [1, 113922743, -310146482690273725409])
sage: C.has_rational_point(point=True)                                      # needs sage.libs.pari
(True, (-76842858034579/5424 : -5316144401/5424 : 1))
sage: C.has_rational_point(algorithm='local', read_cache=False)
True
sage: C.has_rational_point(point=True, algorithm='magma',       # optional - magma
....:                      read_cache=False)
(True, (30106379962113/7913 : 12747947692/7913 : 1))

>>> from sage.all import *
>>> C = Conic(QQ, [Integer(1), Integer(113922743), -Integer(310146482690273725409)])
>>> C.has_rational_point(point=True)                                      # needs sage.libs.pari
(True, (-76842858034579/5424 : -5316144401/5424 : 1))
>>> C.has_rational_point(algorithm='local', read_cache=False)
True
>>> C.has_rational_point(point=True, algorithm='magma',       # optional - magma
...                      read_cache=False)
(True, (30106379962113/7913 : 12747947692/7913 : 1))
is_locally_solvable(p)[source]

Return True if and only if self has a solution over the \(p\)-adic numbers.

Here \(p\) is a prime number or equals \(-1\), infinity, or \(\RR\) to denote the infinite place.

EXAMPLES:

sage: # needs sage.libs.pari
sage: C = Conic(QQ, [1,2,3])
sage: C.is_locally_solvable(-1)
False
sage: C.is_locally_solvable(2)
False
sage: C.is_locally_solvable(3)
True
sage: C.is_locally_solvable(QQ.hom(RR))
False
sage: D = Conic(QQ, [1, 2, -3])
sage: D.is_locally_solvable(infinity)
True
sage: D.is_locally_solvable(RR)
True

>>> from sage.all import *
>>> # needs sage.libs.pari
>>> C = Conic(QQ, [Integer(1),Integer(2),Integer(3)])
>>> C.is_locally_solvable(-Integer(1))
False
>>> C.is_locally_solvable(Integer(2))
False
>>> C.is_locally_solvable(Integer(3))
True
>>> C.is_locally_solvable(QQ.hom(RR))
False
>>> D = Conic(QQ, [Integer(1), Integer(2), -Integer(3)])
>>> D.is_locally_solvable(infinity)
True
>>> D.is_locally_solvable(RR)
True
local_obstructions(finite=True, infinite=True, read_cache=True)[source]

Return the sequence of finite primes and/or infinite places such that self is locally solvable at those primes and places.

The infinite place is denoted \(-1\).

The parameters finite and infinite (both True by default) are used to specify whether to look at finite and/or infinite places.

Note that finite = True involves factorization of the determinant of self, hence may be slow.

Local obstructions are cached. The parameter read_cache specifies whether to look at the cache before computing anything.

EXAMPLES:

sage: # needs sage.libs.pari
sage: Conic(QQ, [1, 1, 1]).local_obstructions()
[2, -1]
sage: Conic(QQ, [1, 2, -3]).local_obstructions()
[]
sage: Conic(QQ, [1, 2, 3, 4, 5, 6]).local_obstructions()
[41, -1]

>>> from sage.all import *
>>> # needs sage.libs.pari
>>> Conic(QQ, [Integer(1), Integer(1), Integer(1)]).local_obstructions()
[2, -1]
>>> Conic(QQ, [Integer(1), Integer(2), -Integer(3)]).local_obstructions()
[]
>>> Conic(QQ, [Integer(1), Integer(2), Integer(3), Integer(4), Integer(5), Integer(6)]).local_obstructions()
[41, -1]
parametrization(point=None, morphism=True)[source]

Return a parametrization \(f\) of self together with the inverse of \(f\).

If point is specified, then that point is used for the parametrization. Otherwise, use self.rational_point() to find a point.

If morphism is True, then \(f\) is returned in the form of a Scheme morphism. Otherwise, it is a tuple of polynomials that gives the parametrization.

ALGORITHM:

Uses the PARI/GP function pari:qfparam.

EXAMPLES:

sage: # needs sage.libs.pari
sage: c = Conic([1,1,-1])
sage: c.parametrization()
(Scheme morphism:
  From: Projective Space of dimension 1 over Rational Field
  To:   Projective Conic Curve over Rational Field defined by x^2 + y^2 - z^2
  Defn: Defined on coordinates by sending (x : y) to
        (2*x*y : x^2 - y^2 : x^2 + y^2),
 Scheme morphism:
  From: Projective Conic Curve over Rational Field defined by x^2 + y^2 - z^2
  To:   Projective Space of dimension 1 over Rational Field
  Defn: Defined on coordinates by sending (x : y : z) to
        (1/2*x : -1/2*y + 1/2*z))

>>> from sage.all import *
>>> # needs sage.libs.pari
>>> c = Conic([Integer(1),Integer(1),-Integer(1)])
>>> c.parametrization()
(Scheme morphism:
  From: Projective Space of dimension 1 over Rational Field
  To:   Projective Conic Curve over Rational Field defined by x^2 + y^2 - z^2
  Defn: Defined on coordinates by sending (x : y) to
        (2*x*y : x^2 - y^2 : x^2 + y^2),
 Scheme morphism:
  From: Projective Conic Curve over Rational Field defined by x^2 + y^2 - z^2
  To:   Projective Space of dimension 1 over Rational Field
  Defn: Defined on coordinates by sending (x : y : z) to
        (1/2*x : -1/2*y + 1/2*z))

An example with morphism = False

sage: # needs sage.libs.pari
sage: R.<x,y,z> = QQ[]
sage: C = Curve(7*x^2 + 2*y*z + z^2)
sage: (p, i) = C.parametrization(morphism=False); (p, i)
([-2*x*y, x^2 + 7*y^2, -2*x^2], [-1/2*x, 1/7*y + 1/14*z])
sage: C.defining_polynomial()(p)
0
sage: i[0](p) / i[1](p)
x/y

>>> from sage.all import *
>>> # needs sage.libs.pari
>>> R = QQ['x, y, z']; (x, y, z,) = R._first_ngens(3)
>>> C = Curve(Integer(7)*x**Integer(2) + Integer(2)*y*z + z**Integer(2))
>>> (p, i) = C.parametrization(morphism=False); (p, i)
([-2*x*y, x^2 + 7*y^2, -2*x^2], [-1/2*x, 1/7*y + 1/14*z])
>>> C.defining_polynomial()(p)
0
>>> i[Integer(0)](p) / i[Integer(1)](p)
x/y

A ValueError is raised if self has no rational point

sage: # needs sage.libs.pari
sage: C = Conic(x^2 + 2*y^2 + z^2)
sage: C.parametrization()
Traceback (most recent call last):
...
ValueError: Conic Projective Conic Curve over Rational Field defined
by x^2 + 2*y^2 + z^2 has no rational points over Rational Field!

>>> from sage.all import *
>>> # needs sage.libs.pari
>>> C = Conic(x**Integer(2) + Integer(2)*y**Integer(2) + z**Integer(2))
>>> C.parametrization()
Traceback (most recent call last):
...
ValueError: Conic Projective Conic Curve over Rational Field defined
by x^2 + 2*y^2 + z^2 has no rational points over Rational Field!

A ValueError is raised if self is not smooth

sage: # needs sage.libs.pari
sage: C = Conic(x^2 + y^2)
sage: C.parametrization()
Traceback (most recent call last):
...
ValueError: The conic self (=Projective Conic Curve over Rational Field defined
by x^2 + y^2) is not smooth, hence does not have a parametrization.

>>> from sage.all import *
>>> # needs sage.libs.pari
>>> C = Conic(x**Integer(2) + y**Integer(2))
>>> C.parametrization()
Traceback (most recent call last):
...
ValueError: The conic self (=Projective Conic Curve over Rational Field defined
by x^2 + y^2) is not smooth, hence does not have a parametrization.