{{{id=1| PP.=ProjectiveSpace(QQ,2) /// }}} {{{id=3| C1=PP.subscheme([x^2+y^2-z^2]) C1.dimension(),C1.is_projective(),C1.is_smooth(),C1.irreducible_components() /// (1, True, True, [ Closed subscheme of Projective Space of dimension 2 over Rational Field defined by: x^2 + y^2 - z^2 ]) }}} {{{id=9| plot(Curve(C1)) ///

}}} {{{id=5| f2=y^2*z-x^3-x^2*z C2=Curve(f2) C2.dimension(),C2.is_projective(),C2.is_smooth(),C2.irreducible_components() /// (1, True, False, [ Closed subscheme of Projective Space of dimension 2 over Rational Field defined by: x^3 + x^2*z - y^2*z ]) }}} {{{id=11| plot(C2) ///

}}} {{{id=10| f3=(x^2+y^2-z^2)^3+27*x^2*y^2*z^2 C3=Curve(f3) C3.dimension(),C3.is_projective(),C3.is_smooth(),C3.irreducible_components() /// (1, True, False, [ Closed subscheme of Projective Space of dimension 2 over Rational Field defined by: x^6 + 3*x^4*y^2 + 3*x^2*y^4 + y^6 - 3*x^4*z^2 + 21*x^2*y^2*z^2 - 3*y^4*z^2 + 3*x^2*z^4 + 3*y^2*z^4 - z^6 ]) }}} {{{id=12| plot(C3) ///

}}} {{{id=13| f4=x^2+2*x*y+y^2-2*z^2 C4=Curve(f4) C4.dimension(),C4.is_projective(),C4.is_smooth(),C4.irreducible_components() /// (1, True, False, [ Closed subscheme of Projective Space of dimension 2 over Rational Field defined by: x^2 + 2*x*y + y^2 - 2*z^2 ]) }}} {{{id=14| plot(C4) ///

}}} {{{id=6| F.=QuadraticField(2) C4b=C4.change_ring(F) C4b.irreducible_components() /// [ Closed subscheme of Projective Space of dimension 2 over Number Field in a with defining polynomial x^2 - 2 defined by: x + y + (-a)*z, Closed subscheme of Projective Space of dimension 2 over Number Field in a with defining polynomial x^2 - 2 defined by: x + y + (a)*z ] }}} {{{id=7| PP3.=ProjectiveSpace(QQ,3) C5=PP3.subscheme([x0^2+x1^2+x2^2+x3^2,x0^3+x1^3+x2^3+x3^3]) C5.dimension(),C5.is_projective(),C5.is_smooth(),C5.irreducible_components() /// (1, True, True, [ Closed subscheme of Projective Space of dimension 3 over Rational Field defined by: x0^2 + x1^2 + x2^2 + x3^2, x0*x1^2 - x1^3 + x0*x2^2 - x2^3 + x0*x3^2 - x3^3, 2*x1^4 - x0*x1*x2^2 + 2*x1^2*x2^2 + x0*x2^3 + x1*x2^3 + x2^4 - x0*x1*x3^2 + 2*x1^2*x3^2 + 2*x2^2*x3^2 + x0*x3^3 + x1*x3^3 + x3^4 ]) }}} {{{id=15| C6=Curve(x^4+y^4+z^4) C6b=C6.change_ring(GF(11)) C6b=Curve(C6b) /// }}} {{{id=16| pts = C6b.rational_points() pts /// [(1 : 5 : 1), (1 : 6 : 1), (2 : 2 : 1), (2 : 9 : 1), (5 : 1 : 1), (5 : 10 : 1), (6 : 1 : 1), (6 : 10 : 1), (9 : 2 : 1), (9 : 9 : 1), (10 : 5 : 1), (10 : 6 : 1)] }}} {{{id=17| Dl = [C6b.divisor([ (n, pts[0])]) for n in range(10)] RR=[C6b.riemann_roch_basis(D) for D in Dl] [len(B) for B in RR] /// [1, 1, 1, 1, 2, 3, 4, 5, 6, 7] }}} {{{id=18| C6b.genus() /// 3 }}} {{{id=20| P.=PolynomialRing(QQ,2) F=FractionField(P) E=EllipticCurve(F,[0,0,0,A,B]) E /// Elliptic Curve defined by y^2 = x^3 + A*x + B over Fraction Field of Multivariate Polynomial Ring in A, B over Rational Field }}} {{{id=22| E.j_invariant() == 1728*(4*A^3)/(4*A^3+27*B^2) /// True }}} {{{id=24| P.=PolynomialRing(QQ,2) F=4*(x^2+y^2+2*y)^2+(2*y+3)*(y+1)*(y^2-x^2)+1/360 R=implicit_plot(F,(x,-2,2),(y,-2,1)) R ///

}}} {{{id=25| secant=implicit_plot(y-x+1/2,(x,-2,2),(y,-2,1),color="red") R+secant ///

}}} {{{id=26| tangent=implicit_plot(-0.4285*(x-0.85)-1.217*(y+1.4954),(x,-2,2),(y,-2,1),color="red") R+tangent ///

}}} {{{id=27| bitangent=implicit_plot(y-0.0996170277553054,(x,-2,2),(y,-2,1),color="red") R+bitangent ///

}}} {{{id=28| hessian=implicit_plot(-9264*x^6-14796*y^6-38/5-(142404/5)*x^2*y-103806*x^4*y-247068*x^2*y^3-(1199556/5)*y^2*x^2-65736*x^2*y^4-58428*x^4*y^2-(71994/5)*x^4-(60926/15)*x^2-(749466/5)*y^4-84582*y^5-(332364/5)*y^3+(130426/5)*y^2+(8/3)*y,(x,-2,2),(y,-2,1),color="red") R+hessian ///

}}} {{{id=29| flex=implicit_plot(0.55641*(x+0.084)-1.13615*(y+0.033),(x,-2,2),(y,-2,1),color="red") flex+R ///

}}} {{{id=30| /// }}}