Sage Reference Manual: Elliptic and Plane Curves - Mirrors

Sage Reference Manual: Elliptic and Plane Curves, Release 6.1.1Truesage: E = Curve(Y^2*Z-X^3+Z^3)sage: E.is_singular()FalseShowing that ticket #12187 is fixed:sage: F. = GF(2)[]sage: G = Curve(X^2+Y*Z)sage: G.is_singular()Falselocal_coordinates(pt, n)Return local coordinates to precision n at the given point.INPUT:Behaviour is flaky - some choices of n are worst that others.•pt - an F-rational point on X which is not a point of ramification for the projection (x,y) - x.•n - the number of terms desiredOUTPUT: x = x0 + t y = y0 + power series in tEXAMPLES:sage: FF = FiniteField(5)sage: P2 = ProjectiveSpace(2, FF, names = [’x’,’y’,’z’])sage: x, y, z = P2.coordinate_ring().gens()sage: C = Curve(y^2*z^7-x^9-x*z^8)sage: pt = C([2,3,1])sage: C.local_coordinates(pt,9) # todo: not implemented !!!![2 + t, 3 + 3*t^2 + t^3 + 3*t^4 + 3*t^6 + 3*t^7 + t^8 + 2*t^9 + 3*t^11 + 3*t^12]plot(*args, **kwds)Plot the real points of an affine patch of this projective plane curve.INPUT:•self - an affine plane curve•patch - (optional) the affine patch to be plotted; if not specified, the patch corresponding to the lastprojective coordinate being nonzero•*args - optional tuples (variable, minimum, maximum) for plotting dimensions•**kwds - optional keyword arguments passed on to implicit_plotEXAMPLES:A cuspidal curve:sage: R. = QQ[]sage: C = Curve(x^3 - y^2*z)sage: C.plot()The other affine patches of the same curve:sage: C.plot(patch=0)sage: C.plot(patch=1)An elliptic curve:11