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:
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
- has_rational_point(point=False, obstruction=False, algorithm='default', read_cache=True)#
Return
True
if and only ifself
has a point defined over \(\QQ\).If
point
andobstruction
are bothFalse
(default), then the output is a booleanout
saying whetherself
has a rational point.If
point
orobstruction
isTrue
, then the output is a pair(out, S)
, whereout
is as above and the following holds:if
point
isTrue
andself
has a rational point, thenS
is a rational point,if
obstruction
isTrue
andself
has no rational point, thenS
is a prime such that no rational point exists over the completion atS
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
isTrue
.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 withobstruction = True
)'local'
– Check if a local solution exists for all primes and infinite places of \(\QQ\) and apply the Hasse principle (cannot be combined withpoint = 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)
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))
- is_locally_solvable(p)#
Return
True
if and only ifself
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
- local_obstructions(finite=True, infinite=True, read_cache=True)#
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
andinfinite
(bothTrue
by default) are used to specify whether to look at finite and/or infinite places.Note that
finite = True
involves factorization of the determinant ofself
, 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]
- 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, useself.rational_point()
to find a point.If
morphism
isTrue
, 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))
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
A
ValueError
is raised ifself
has no rational pointsage: # 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!
A
ValueError
is raised ifself
is not smoothsage: # 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.