Sage Reference Manual: Elliptic and Plane Curves - Mirrors

CHAPTERSIXPROJECTIVE PLANE CONICS OVER ANUMBER FIELDAUTHORS:• 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_fieldCreate a projective plane conic curve over a number field. See Conic for full documentation.EXAMPLES:sage: K. = NumberField(x^3 - 2, ’a’)sage: P. = 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 +TESTS:sage: K. = NumberField(x^3 - 3, ’a’)sage: Conic([a, 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 its base field B.If point and obstruction are both False (default), then the output is a boolean out saying whetherself has a rational point.If point or obstruction is True, then the output is a pair (out, S), where out is as above and:•if point is True and self has a rational point, then S is a rational point,•if obstruction is True, self has no rational point, then S is a prime or infinite place of B suchthat no rational point exists over the completion at S.Points and obstructions are cached whenever they are found. Cached information is used for the output ifavailable, 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 with obstruction = True)•’local’ – Check if a local solution exists for all primes and infinite places of B and apply the Hasseprinciple. (Cannot be combined with point = True.)29