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)

Bases: sage.schemes.plane_conics.con_number_field.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

TESTS:

sage: Conic([2, 1, -1])._test_pickling()
has_rational_point(point=False, obstruction=False, algorithm='default', read_cache=True)

Returns 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 Denis Simon’s GP script qfsolve (see sage.quadratic_forms.qfsolve.qfsolve)
  • 'rnfisnorm' – Use PARI’s function 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: 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)

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

sage: C = Conic(QQ, [1, 113922743, -310146482690273725409])
sage: C.has_rational_point(point = True)
(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', read_cache=False) # optional - magma
(True, (30106379962113/7913 : 12747947692/7913 : 1))

TESTS:

Create a bunch of conics over \(\QQ\), check if has_rational_point runs without errors and returns consistent answers for all algorithms. Check if all points returned are valid.

sage: l = Sequence(cartesian_product_iterator([[-1, 0, 1] for i in range(6)]))
sage: c = [Conic(QQ, a) for a in l if a != [0,0,0] and a != (0,0,0,0,0,0)]
sage: d = []
sage: d = [[C]+[C.has_rational_point(algorithm = algorithm, read_cache = False, obstruction = (algorithm != 'rnfisnorm'), point = (algorithm != 'local')) for algorithm in ['local', 'qfsolve', 'rnfisnorm']] for C in c[::10]] # long time: 7 seconds
sage: assert all([e[1][0] == e[2][0] and e[1][0] == e[3][0] for e in d])
sage: assert all([e[0].defining_polynomial()(Sequence(e[i][1])) == 0 for e in d for i in [2,3] if e[1][0]])
is_locally_solvable(p)

Returns 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: 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
local_obstructions(finite=True, infinite=True, read_cache=True)

Returns 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: 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]
parametrization(point=None, morphism=True)

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 Denis Simon’s GP script qfparam. See sage.quadratic_forms.qfsolve.qfparam.

EXAMPLES

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))

An example with morphism = False

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, 7*x^2 + y^2, -2*y^2], [-1/2*x, -1/2*z])
sage: C.defining_polynomial()(p)
0
sage: i[0](p) / i[1](p)
x/y

A ValueError is raised if self has no rational point

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!

A ValueError is raised if self is not smooth

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.

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