Sage Reference Manual: Elliptic and Plane Curves - Mirrors

Sage Reference Manual: Elliptic and Plane Curves, Release 6.1.1EXAMPLE: We can also make non-reduced non-irreducible curves.sage: x,y,z = QQ[’x,y,z’].gens()sage: Curve((x-y)*(x+y))Projective Conic Curve over Rational Field defined by x^2 - y^2sage: Curve((x-y)^2*(x+y)^2)Projective Curve over Rational Field defined by x^4 - 2*x^2*y^2 + y^4EXAMPLE: A union of curves is a curve.sage: x,y,z = QQ[’x,y,z’].gens()sage: C = Curve(x^3 + y^3 + z^3)sage: D = Curve(x^4 + y^4 + z^4)sage: C.union(D)Projective Curve over Rational Field defined byx^7 + x^4*y^3 + x^3*y^4 + y^7 + x^4*z^3 + y^4*z^3 + x^3*z^4 + y^3*z^4 + z^7The intersection is not a curve, though it is a scheme.sage: X = C.intersection(D); XClosed subscheme of Projective Space of dimension 2 over Rational Field defined by:x^3 + y^3 + z^3,x^4 + y^4 + z^4Note that the intersection has dimension 0.sage: X.dimension()0sage: I = X.defining_ideal(); IIdeal (x^3 + y^3 + z^3, x^4 + y^4 + z^4) of Multivariate Polynomial Ring in x, y, z over RationaEXAMPLE: In three variables, the defining equation must be homogeneous.If the parent polynomial ring is in three variables, then the defining ideal must be homogeneous.sage: x,y,z = QQ[’x,y,z’].gens()sage: Curve(x^2+y^2)Projective Conic Curve over Rational Field defined by x^2 + y^2sage: Curve(x^2+y^2+z)Traceback (most recent call last):...TypeError: x^2 + y^2 + z is not a homogeneous polynomial!The defining polynomial must always be nonzero:sage: P1. = ProjectiveSpace(1,GF(5))sage: Curve(0*x)Traceback (most recent call last):...ValueError: defining polynomial of curve must be nonzero2 Chapter 1. Plane curve constructors