Sage Reference Manual: Elliptic and Plane Curves - Mirrors

CHAPTERFOURPLANE CONIC CONSTRUCTORAUTHORS:• Marco Streng (2010-07-20)• Nick Alexander (2008-01-08)sage.schemes.plane_conics.constructor.Conic(base_field, F=None, names=None,unique=True)Return the plane projective conic curve defined by F over base_field.The input form Conic(F, names=None) is also accepted, in which case the fraction field of the base ringof F is used as base field.INPUT:•base_field – The base field of the conic.•names – a list, tuple, or comma separated string of three variable names specifying the names of thecoordinate functions of the ambient space P 3 . If not specified or read off from F, then this defaults to’x,y,z’.•F – a polynomial, list, matrix, ternary quadratic form, or list or tuple of 5 points in the plane.If F is a polynomial or quadratic form, then the output is the curve in the projective plane definedby F = 0.If F is a polynomial, then it must be a polynomial of degree at most 2 in 2 variables, or a homogeneouspolynomial in of degree 2 in 3 variables.If F is a matrix, then the output is the zero locus of (x, y, z)F (x, y, z) t .If F is a list of coefficients, then it has length 3 or 6 and gives the coefficients of the monomialsx 2 , y 2 , z 2 or all 6 monomials x 2 , xy, xz, y 2 , yz, z 2 in lexicographic order.If F is a list of 5 points in the plane, then the output is a conic through those points.•unique – Used only if F is a list of points in the plane. If the conic through the points is not unique, thenraise ValueError if and only if unique is TrueOUTPUT:A plane projective conic curve defined by F over a field.EXAMPLES:Conic curves given by polynomials15