Sage Reference Manual: Elliptic and Plane Curves - Mirrors

CHAPTERSEVENPROJECTIVE PLANE CONICS OVER QAUTHORS:• 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_fieldCreate a projective plane conic curve over Q. See Conic for full documentation.EXAMPLES:sage: P. = QQ[]sage: Conic(X^2 + Y^2 - 3*Z^2)Projective Conic Curve over Rational Field defined by X^2 + Y^2 - 3*Z^2TESTS: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 Q.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 thefollowing 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 rationalpoint exists over the completion at S or −1 if no point exists over R.Points and obstructions are cached, whenever they are found. Cached information is used if and only ifread_cache is True.ALGORITHM:The parameter algorithm specifies the algorithm to be used:•’qfsolve’ – Use Denis Simon’s pari script Qfsolve (seesage.quadratic_forms.qfsolve.qfsolve)•’rnfisnorm’ – Use PARI’s function rnfisnorm (cannot be combined with obstruction =True)33