Projective plane conics over a number field#
AUTHORS:
Marco Streng (2010-07-20)
- class sage.schemes.plane_conics.con_number_field.ProjectiveConic_number_field(A, f)#
Bases:
sage.schemes.plane_conics.con_field.ProjectiveConic_field
Create a projective plane conic curve over a number field. See
Conic
for full documentation.EXAMPLES:
sage: K.<a> = NumberField(x^3 - 2, 'a') sage: P.<X, Y, Z> = K[] sage: Conic(X^2 + Y^2 - a*Z^2) Projective Conic Curve over Number Field in a with defining polynomial x^3 - 2 defined by X^2 + Y^2 + (-a)*Z^2
- has_rational_point(point=False, obstruction=False, algorithm='default', read_cache=True)#
Returns
True
if and only ifself
has a point defined over its base field \(B\).If
point
andobstruction
are both False (default), then the output is a booleanout
saying whetherself
has a rational point.If
point
orobstruction
is True, then the output is a pair(out, S)
, whereout
is as above and:if
point
is True andself
has a rational point, thenS
is a rational point,if
obstruction
is True,self
has no rational point, thenS
is a prime or infinite place of \(B\) such that no rational point exists over the completion atS
.
Points and obstructions are cached whenever they are found. Cached information is used for the output if available, but only if
read_cache
is True.ALGORITHM:
The parameter
algorithm
specifies the algorithm to be used:'rnfisnorm'
– Use PARI’s rnfisnorm (cannot be combined withobstruction = True
)'local'
– Check if a local solution exists for all primes and infinite places of \(B\) and apply the Hasse principle. (Cannot be combined withpoint = True
.)'default'
– Use algorithm'rnfisnorm'
first. Then, if no point exists and obstructions are requested, use algorithm'local'
to find an obstruction.'magma'
(requires Magma to be installed) – delegates the task to the Magma computer algebra system.
EXAMPLES:
An example over \(\QQ\)
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
Examples over number fields:
sage: K.<i> = QuadraticField(-1) sage: C = Conic(K, [1, 3, -5]) sage: C.has_rational_point(point = True, obstruction = True) (False, Fractional ideal (-i - 2)) sage: C.has_rational_point(algorithm = "rnfisnorm") False sage: C.has_rational_point(algorithm = "rnfisnorm", obstruction = True, read_cache=False) Traceback (most recent call last): ... ValueError: Algorithm rnfisnorm cannot be combined with obstruction = True in has_rational_point sage: P.<x> = QQ[] sage: L.<b> = NumberField(x^3-5) sage: C = Conic(L, [1, 2, -3]) sage: C.has_rational_point(point = True, algorithm = 'rnfisnorm') (True, (5/3 : -1/3 : 1)) sage: K.<a> = NumberField(x^4+2) sage: Conic(QQ, [4,5,6]).has_rational_point() False sage: Conic(K, [4,5,6]).has_rational_point() True sage: Conic(K, [4,5,6]).has_rational_point(algorithm='magma', read_cache=False) # optional - magma True sage: P.<a> = QuadraticField(2) sage: C = Conic(P, [1,1,1]) sage: C.has_rational_point() False sage: C.has_rational_point(point=True) (False, None) sage: C.has_rational_point(obstruction=True) (False, Ring morphism: From: Number Field in a with defining polynomial x^2 - 2 with a = 1.414213562373095? To: Algebraic Real Field Defn: a |--> -1.414213562373095?) sage: C.has_rational_point(point=True, obstruction=True) (False, Ring morphism: From: Number Field in a with defining polynomial x^2 - 2 with a = 1.414213562373095? To: Algebraic Real Field Defn: a |--> -1.414213562373095?)
- is_locally_solvable(p)#
Returns
True
if and only ifself
has a solution over the completion of the base field \(B\) ofself
atp
. Herep
is a finite prime or infinite place of \(B\).EXAMPLES:
sage: P.<x> = QQ[] sage: K.<a> = NumberField(x^3 + 5) sage: C = Conic(K, [1, 2, 3 - a]) sage: [p1, p2] = K.places() sage: C.is_locally_solvable(p1) False sage: C.is_locally_solvable(p2) True sage: O = K.maximal_order() sage: f = (2*O).factor() sage: C.is_locally_solvable(f[0][0]) True sage: C.is_locally_solvable(f[1][0]) False
- 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.If the base field is \(\QQ\), then the infinite place is denoted \(-1\).
The parameters
finite
andinfinite
(both True by default) are used to specify whether to look at finite and/or infinite places. Note thatfinite = 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: K.<i> = QuadraticField(-1) sage: Conic(K, [1, 2, 3]).local_obstructions() [] sage: L.<a> = QuadraticField(5) sage: Conic(L, [1, 2, 3]).local_obstructions() [Ring morphism: From: Number Field in a with defining polynomial x^2 - 5 with a = 2.236067977499790? To: Algebraic Real Field Defn: a |--> -2.236067977499790?, Ring morphism: From: Number Field in a with defining polynomial x^2 - 5 with a = 2.236067977499790? To: Algebraic Real Field Defn: a |--> 2.236067977499790?]