Rational points of curves¶
We can create points on projective curves:
sage: P.<x,y,z,w> = ProjectiveSpace(QQ, 3)
sage: C = Curve([x^3 - 2*x*z^2 - y^3, z^3 - w^3 - x*y*z], P)
sage: Q = C([1,1,0,0])
sage: Q.parent()
Set of rational points of Projective Curve over Rational Field
 defined by x^3 - y^3 - 2*x*z^2, -x*y*z + z^3 - w^3
>>> from sage.all import *
>>> P = ProjectiveSpace(QQ, Integer(3), names=('x', 'y', 'z', 'w',)); (x, y, z, w,) = P._first_ngens(4)
>>> C = Curve([x**Integer(3) - Integer(2)*x*z**Integer(2) - y**Integer(3), z**Integer(3) - w**Integer(3) - x*y*z], P)
>>> Q = C([Integer(1),Integer(1),Integer(0),Integer(0)])
>>> Q.parent()
Set of rational points of Projective Curve over Rational Field
 defined by x^3 - y^3 - 2*x*z^2, -x*y*z + z^3 - w^3
or on affine curves:
sage: A.<x,y> = AffineSpace(GF(23), 2)
sage: C = Curve([y - y^4 + 17*x^2 - 2*x + 22], A)
sage: Q = C([22,21])
sage: Q.parent()
Set of rational points of Affine Plane Curve over Finite Field of size 23
 defined by -y^4 - 6*x^2 - 2*x + y - 1
>>> from sage.all import *
>>> A = AffineSpace(GF(Integer(23)), Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
>>> C = Curve([y - y**Integer(4) + Integer(17)*x**Integer(2) - Integer(2)*x + Integer(22)], A)
>>> Q = C([Integer(22),Integer(21)])
>>> Q.parent()
Set of rational points of Affine Plane Curve over Finite Field of size 23
 defined by -y^4 - 6*x^2 - 2*x + y - 1
AUTHORS:
- Grayson Jorgenson (2016-6): initial version 
- class sage.schemes.curves.point.AffineCurvePoint_field(X, v, check=True)[source]¶
- Bases: - SchemeMorphism_point_affine_field- is_singular()[source]¶
- Return whether this point is a singular point of the affine curve it is on. - EXAMPLES: - sage: # needs sage.rings.number_field sage: K = QuadraticField(-1) sage: A.<x,y,z> = AffineSpace(K, 3) sage: C = Curve([(x^4 + 2*z + 2)*y, z - y + 1]) sage: Q1 = C([0,0,-1]) sage: Q1.is_singular() True sage: Q2 = C([-K.gen(),0,-1]) sage: Q2.is_singular() False - >>> from sage.all import * >>> # needs sage.rings.number_field >>> K = QuadraticField(-Integer(1)) >>> A = AffineSpace(K, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3) >>> C = Curve([(x**Integer(4) + Integer(2)*z + Integer(2))*y, z - y + Integer(1)]) >>> Q1 = C([Integer(0),Integer(0),-Integer(1)]) >>> Q1.is_singular() True >>> Q2 = C([-K.gen(),Integer(0),-Integer(1)]) >>> Q2.is_singular() False 
 
- class sage.schemes.curves.point.AffinePlaneCurvePoint_field(X, v, check=True)[source]¶
- Bases: - AffineCurvePoint_field- Point of an affine plane curve over a field. - is_ordinary_singularity()[source]¶
- Return whether this point is an ordinary singularity of the affine plane curve it is on. - EXAMPLES: - sage: A.<x,y> = AffineSpace(QQ, 2) sage: C = A.curve([x^5 - x^3*y^2 + 5*x^4 - x^3*y - 3*x^2*y^2 + ....: x*y^3 + 10*x^3 - 3*x^2*y - 3*x*y^2 + y^3 + 10*x^2 - 3*x*y - y^2 + ....: 5*x - y + 1]) sage: Q = C([-1,0]) sage: Q.is_ordinary_singularity() True - >>> from sage.all import * >>> A = AffineSpace(QQ, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2) >>> C = A.curve([x**Integer(5) - x**Integer(3)*y**Integer(2) + Integer(5)*x**Integer(4) - x**Integer(3)*y - Integer(3)*x**Integer(2)*y**Integer(2) + ... x*y**Integer(3) + Integer(10)*x**Integer(3) - Integer(3)*x**Integer(2)*y - Integer(3)*x*y**Integer(2) + y**Integer(3) + Integer(10)*x**Integer(2) - Integer(3)*x*y - y**Integer(2) + ... Integer(5)*x - y + Integer(1)]) >>> Q = C([-Integer(1),Integer(0)]) >>> Q.is_ordinary_singularity() True - sage: A.<x,y> = AffineSpace(GF(7), 2) sage: C = A.curve([y^2 - x^7 - 6*x^3]) sage: Q = C([0,0]) sage: Q.is_ordinary_singularity() False - >>> from sage.all import * >>> A = AffineSpace(GF(Integer(7)), Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2) >>> C = A.curve([y**Integer(2) - x**Integer(7) - Integer(6)*x**Integer(3)]) >>> Q = C([Integer(0),Integer(0)]) >>> Q.is_ordinary_singularity() False 
 - is_transverse(D)[source]¶
- Return whether the intersection of the curve - Dat this point with the curve this point is on is transverse or not.- INPUT: - D– a curve in the same ambient space as the curve this point is on
 - EXAMPLES: - sage: A.<x,y> = AffineSpace(QQ, 2) sage: C = Curve([y - x^2], A) sage: D = Curve([y], A) sage: Q = C([0,0]) sage: Q.is_transverse(D) False - >>> from sage.all import * >>> A = AffineSpace(QQ, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2) >>> C = Curve([y - x**Integer(2)], A) >>> D = Curve([y], A) >>> Q = C([Integer(0),Integer(0)]) >>> Q.is_transverse(D) False - sage: # needs sage.rings.number_field sage: R.<a> = QQ[] sage: K.<b> = NumberField(a^2 - 2) sage: A.<x,y> = AffineSpace(K, 2) sage: C = Curve([y^2 + x^2 - 1], A) sage: D = Curve([y - x], A) sage: Q = C([-1/2*b, -1/2*b]) sage: Q.is_transverse(D) True - >>> from sage.all import * >>> # needs sage.rings.number_field >>> R = QQ['a']; (a,) = R._first_ngens(1) >>> K = NumberField(a**Integer(2) - Integer(2), names=('b',)); (b,) = K._first_ngens(1) >>> A = AffineSpace(K, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2) >>> C = Curve([y**Integer(2) + x**Integer(2) - Integer(1)], A) >>> D = Curve([y - x], A) >>> Q = C([-Integer(1)/Integer(2)*b, -Integer(1)/Integer(2)*b]) >>> Q.is_transverse(D) True 
 - multiplicity()[source]¶
- Return the multiplicity of this point with respect to the affine curve it is on. - EXAMPLES: - sage: A.<x,y> = AffineSpace(QQ, 2) sage: C = A.curve([2*x^7 - 3*x^6*y + x^5*y^2 + 31*x^6 - 40*x^5*y + ....: 13*x^4*y^2 - x^3*y^3 + 207*x^5 - 228*x^4*y + 70*x^3*y^2 - 7*x^2*y^3 ....: + 775*x^4 - 713*x^3*y + 193*x^2*y^2 - 19*x*y^3 + y^4 + 1764*x^3 - ....: 1293*x^2*y + 277*x*y^2 - 22*y^3 + 2451*x^2 - 1297*x*y + 172*y^2 + ....: 1935*x - 570*y + 675]) sage: Q = C([-2,1]) sage: Q.multiplicity() 4 - >>> from sage.all import * >>> A = AffineSpace(QQ, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2) >>> C = A.curve([Integer(2)*x**Integer(7) - Integer(3)*x**Integer(6)*y + x**Integer(5)*y**Integer(2) + Integer(31)*x**Integer(6) - Integer(40)*x**Integer(5)*y + ... Integer(13)*x**Integer(4)*y**Integer(2) - x**Integer(3)*y**Integer(3) + Integer(207)*x**Integer(5) - Integer(228)*x**Integer(4)*y + Integer(70)*x**Integer(3)*y**Integer(2) - Integer(7)*x**Integer(2)*y**Integer(3) ... + Integer(775)*x**Integer(4) - Integer(713)*x**Integer(3)*y + Integer(193)*x**Integer(2)*y**Integer(2) - Integer(19)*x*y**Integer(3) + y**Integer(4) + Integer(1764)*x**Integer(3) - ... Integer(1293)*x**Integer(2)*y + Integer(277)*x*y**Integer(2) - Integer(22)*y**Integer(3) + Integer(2451)*x**Integer(2) - Integer(1297)*x*y + Integer(172)*y**Integer(2) + ... Integer(1935)*x - Integer(570)*y + Integer(675)]) >>> Q = C([-Integer(2),Integer(1)]) >>> Q.multiplicity() 4 
 - tangents()[source]¶
- Return the tangents at this point of the affine plane curve this point is on. - OUTPUT: list of polynomials in the coordinate ring of the ambient space of the curve this point is on - EXAMPLES: - sage: A.<x,y> = AffineSpace(QQ, 2) sage: C = A.curve([x^5 - x^3*y^2 + 5*x^4 - x^3*y - 3*x^2*y^2 + ....: x*y^3 + 10*x^3 - 3*x^2*y - 3*x*y^2 + y^3 + 10*x^2 - 3*x*y - y^2 + ....: 5*x - y + 1]) sage: Q = C([-1,0]) sage: Q.tangents() [y, x - y + 1, x + 1, x + y + 1] - >>> from sage.all import * >>> A = AffineSpace(QQ, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2) >>> C = A.curve([x**Integer(5) - x**Integer(3)*y**Integer(2) + Integer(5)*x**Integer(4) - x**Integer(3)*y - Integer(3)*x**Integer(2)*y**Integer(2) + ... x*y**Integer(3) + Integer(10)*x**Integer(3) - Integer(3)*x**Integer(2)*y - Integer(3)*x*y**Integer(2) + y**Integer(3) + Integer(10)*x**Integer(2) - Integer(3)*x*y - y**Integer(2) + ... Integer(5)*x - y + Integer(1)]) >>> Q = C([-Integer(1),Integer(0)]) >>> Q.tangents() [y, x - y + 1, x + 1, x + y + 1] 
 
- class sage.schemes.curves.point.AffinePlaneCurvePoint_finite_field(X, v, check=True)[source]¶
- Bases: - AffinePlaneCurvePoint_field,- SchemeMorphism_point_affine_finite_field- Point of an affine plane curve over a finite field. 
- class sage.schemes.curves.point.IntegralAffineCurvePoint(X, v, check=True)[source]¶
- Bases: - AffineCurvePoint_field- Point of an integral affine curve. - closed_point()[source]¶
- Return the closed point that corresponds to this rational point. - EXAMPLES: - sage: # needs sage.rings.finite_rings sage: A.<x,y> = AffineSpace(GF(8), 2) sage: C = Curve(x^5 + y^5 + x*y + 1) sage: p = C([1,1]) sage: p.closed_point() Point (x + 1, y + 1) - >>> from sage.all import * >>> # needs sage.rings.finite_rings >>> A = AffineSpace(GF(Integer(8)), Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2) >>> C = Curve(x**Integer(5) + y**Integer(5) + x*y + Integer(1)) >>> p = C([Integer(1),Integer(1)]) >>> p.closed_point() Point (x + 1, y + 1) 
 - place()[source]¶
- Return a place on this point. - EXAMPLES: - sage: A.<x,y> = AffineSpace(GF(2), 2) sage: C = Curve(x^5 + y^5 + x*y + 1) sage: p = C(-1,-1) sage: p (1, 1) sage: p.closed_point() Point (x + 1, y + 1) sage: _.place() Place (x + 1, (1/(x^5 + 1))*y^4 + ((x^5 + x^4 + 1)/(x^5 + 1))*y^3 + ((x^5 + x^3 + 1)/(x^5 + 1))*y^2 + (x^2/(x^5 + 1))*y) - >>> from sage.all import * >>> A = AffineSpace(GF(Integer(2)), Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2) >>> C = Curve(x**Integer(5) + y**Integer(5) + x*y + Integer(1)) >>> p = C(-Integer(1),-Integer(1)) >>> p (1, 1) >>> p.closed_point() Point (x + 1, y + 1) >>> _.place() Place (x + 1, (1/(x^5 + 1))*y^4 + ((x^5 + x^4 + 1)/(x^5 + 1))*y^3 + ((x^5 + x^3 + 1)/(x^5 + 1))*y^2 + (x^2/(x^5 + 1))*y) 
 - places()[source]¶
- Return all places on this point. - EXAMPLES: - sage: A.<x,y> = AffineSpace(GF(2), 2) sage: C = Curve(x^5 + y^5 + x*y + 1) sage: p = C(-1,-1) sage: p (1, 1) sage: p.closed_point() Point (x + 1, y + 1) sage: _.places() [Place (x + 1, (1/(x^5 + 1))*y^4 + ((x^5 + x^4 + 1)/(x^5 + 1))*y^3 + ((x^5 + x^3 + 1)/(x^5 + 1))*y^2 + (x^2/(x^5 + 1))*y), Place (x + 1, (1/(x^5 + 1))*y^4 + ((x^5 + x^4 + 1)/(x^5 + 1))*y^3 + (x^3/(x^5 + 1))*y^2 + (x^2/(x^5 + 1))*y + x + 1)] - >>> from sage.all import * >>> A = AffineSpace(GF(Integer(2)), Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2) >>> C = Curve(x**Integer(5) + y**Integer(5) + x*y + Integer(1)) >>> p = C(-Integer(1),-Integer(1)) >>> p (1, 1) >>> p.closed_point() Point (x + 1, y + 1) >>> _.places() [Place (x + 1, (1/(x^5 + 1))*y^4 + ((x^5 + x^4 + 1)/(x^5 + 1))*y^3 + ((x^5 + x^3 + 1)/(x^5 + 1))*y^2 + (x^2/(x^5 + 1))*y), Place (x + 1, (1/(x^5 + 1))*y^4 + ((x^5 + x^4 + 1)/(x^5 + 1))*y^3 + (x^3/(x^5 + 1))*y^2 + (x^2/(x^5 + 1))*y + x + 1)] 
 
- class sage.schemes.curves.point.IntegralAffineCurvePoint_finite_field(X, v, check=True)[source]¶
- Bases: - IntegralAffineCurvePoint- Point of an integral affine curve over a finite field. 
- class sage.schemes.curves.point.IntegralAffinePlaneCurvePoint(X, v, check=True)[source]¶
- Bases: - IntegralAffineCurvePoint,- AffinePlaneCurvePoint_field- Point of an integral affine plane curve. 
- class sage.schemes.curves.point.IntegralAffinePlaneCurvePoint_finite_field(X, v, check=True)[source]¶
- Bases: - AffinePlaneCurvePoint_finite_field,- IntegralAffineCurvePoint_finite_field- Point of an integral affine plane curve over a finite field. 
- class sage.schemes.curves.point.IntegralProjectiveCurvePoint(X, v, check=True)[source]¶
- Bases: - ProjectiveCurvePoint_field- closed_point()[source]¶
- Return the closed point corresponding to this rational point. - EXAMPLES: - sage: P.<x,y,z> = ProjectiveSpace(GF(17), 2) sage: C = Curve([x^4 - 16*y^3*z], P) sage: C.singular_points() [(0 : 0 : 1)] sage: p = _[0] sage: p.closed_point() Point (x, y) - >>> from sage.all import * >>> P = ProjectiveSpace(GF(Integer(17)), Integer(2), names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3) >>> C = Curve([x**Integer(4) - Integer(16)*y**Integer(3)*z], P) >>> C.singular_points() [(0 : 0 : 1)] >>> p = _[Integer(0)] >>> p.closed_point() Point (x, y) 
 - place()[source]¶
- Return a place on this point. - EXAMPLES: - sage: P.<x,y,z> = ProjectiveSpace(GF(17), 2) sage: C = Curve([x^4 - 16*y^3*z], P) sage: C.singular_points() [(0 : 0 : 1)] sage: p = _[0] sage: p.place() Place (y) - >>> from sage.all import * >>> P = ProjectiveSpace(GF(Integer(17)), Integer(2), names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3) >>> C = Curve([x**Integer(4) - Integer(16)*y**Integer(3)*z], P) >>> C.singular_points() [(0 : 0 : 1)] >>> p = _[Integer(0)] >>> p.place() Place (y) 
 - places()[source]¶
- Return all places on this point. - EXAMPLES: - sage: P.<x,y,z> = ProjectiveSpace(GF(17), 2) sage: C = Curve([x^4 - 16*y^3*z], P) sage: C.singular_points() [(0 : 0 : 1)] sage: p = _[0] sage: p.places() [Place (y)] - >>> from sage.all import * >>> P = ProjectiveSpace(GF(Integer(17)), Integer(2), names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3) >>> C = Curve([x**Integer(4) - Integer(16)*y**Integer(3)*z], P) >>> C.singular_points() [(0 : 0 : 1)] >>> p = _[Integer(0)] >>> p.places() [Place (y)] 
 
- class sage.schemes.curves.point.IntegralProjectiveCurvePoint_finite_field(X, v, check=True)[source]¶
- Bases: - IntegralProjectiveCurvePoint- Point of an integral projective curve over a finite field. 
- class sage.schemes.curves.point.IntegralProjectivePlaneCurvePoint(X, v, check=True)[source]¶
- Bases: - IntegralProjectiveCurvePoint,- ProjectivePlaneCurvePoint_field- Point of an integral projective plane curve over a field. 
- class sage.schemes.curves.point.IntegralProjectivePlaneCurvePoint_finite_field(X, v, check=True)[source]¶
- Bases: - ProjectivePlaneCurvePoint_finite_field,- IntegralProjectiveCurvePoint_finite_field- Point of an integral projective plane curve over a finite field. 
- class sage.schemes.curves.point.ProjectiveCurvePoint_field(X, v, check=True)[source]¶
- Bases: - SchemeMorphism_point_projective_field- Point of a projective curve over a field. - is_singular()[source]¶
- Return whether this point is a singular point of the projective curve it is on. - EXAMPLES: - sage: P.<x,y,z,w> = ProjectiveSpace(QQ, 3) sage: C = Curve([x^2 - y^2, z - w], P) sage: Q1 = C([0,0,1,1]) sage: Q1.is_singular() True sage: Q2 = C([1,1,1,1]) sage: Q2.is_singular() False - >>> from sage.all import * >>> P = ProjectiveSpace(QQ, Integer(3), names=('x', 'y', 'z', 'w',)); (x, y, z, w,) = P._first_ngens(4) >>> C = Curve([x**Integer(2) - y**Integer(2), z - w], P) >>> Q1 = C([Integer(0),Integer(0),Integer(1),Integer(1)]) >>> Q1.is_singular() True >>> Q2 = C([Integer(1),Integer(1),Integer(1),Integer(1)]) >>> Q2.is_singular() False 
 
- class sage.schemes.curves.point.ProjectivePlaneCurvePoint_field(X, v, check=True)[source]¶
- Bases: - ProjectiveCurvePoint_field- Point of a projective plane curve over a field. - is_ordinary_singularity()[source]¶
- Return whether this point is an ordinary singularity of the projective plane curve it is on. - EXAMPLES: - sage: P.<x,y,z> = ProjectiveSpace(QQ, 2) sage: C = Curve([z^6 - x^6 - x^3*z^3 - x^3*y^3]) sage: Q = C([0,1,0]) sage: Q.is_ordinary_singularity() False - >>> from sage.all import * >>> P = ProjectiveSpace(QQ, Integer(2), names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3) >>> C = Curve([z**Integer(6) - x**Integer(6) - x**Integer(3)*z**Integer(3) - x**Integer(3)*y**Integer(3)]) >>> Q = C([Integer(0),Integer(1),Integer(0)]) >>> Q.is_ordinary_singularity() False - sage: # needs sage.rings.number_field sage: R.<a> = QQ[] sage: K.<b> = NumberField(a^2 - 3) sage: P.<x,y,z> = ProjectiveSpace(K, 2) sage: C = P.curve([x^2*y^3*z^4 - y^6*z^3 - 4*x^2*y^4*z^3 - ....: 4*x^4*y^2*z^3 + 3*y^7*z^2 + 10*x^2*y^5*z^2 + 9*x^4*y^3*z^2 + ....: 5*x^6*y*z^2 - 3*y^8*z - 9*x^2*y^6*z - 11*x^4*y^4*z - 7*x^6*y^2*z - ....: 2*x^8*z + y^9 + 2*x^2*y^7 + 3*x^4*y^5 + 4*x^6*y^3 + 2*x^8*y]) sage: Q = C([-1/2, 1/2, 1]) sage: Q.is_ordinary_singularity() True - >>> from sage.all import * >>> # needs sage.rings.number_field >>> R = QQ['a']; (a,) = R._first_ngens(1) >>> K = NumberField(a**Integer(2) - Integer(3), names=('b',)); (b,) = K._first_ngens(1) >>> P = ProjectiveSpace(K, Integer(2), names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3) >>> C = P.curve([x**Integer(2)*y**Integer(3)*z**Integer(4) - y**Integer(6)*z**Integer(3) - Integer(4)*x**Integer(2)*y**Integer(4)*z**Integer(3) - ... Integer(4)*x**Integer(4)*y**Integer(2)*z**Integer(3) + Integer(3)*y**Integer(7)*z**Integer(2) + Integer(10)*x**Integer(2)*y**Integer(5)*z**Integer(2) + Integer(9)*x**Integer(4)*y**Integer(3)*z**Integer(2) + ... Integer(5)*x**Integer(6)*y*z**Integer(2) - Integer(3)*y**Integer(8)*z - Integer(9)*x**Integer(2)*y**Integer(6)*z - Integer(11)*x**Integer(4)*y**Integer(4)*z - Integer(7)*x**Integer(6)*y**Integer(2)*z - ... Integer(2)*x**Integer(8)*z + y**Integer(9) + Integer(2)*x**Integer(2)*y**Integer(7) + Integer(3)*x**Integer(4)*y**Integer(5) + Integer(4)*x**Integer(6)*y**Integer(3) + Integer(2)*x**Integer(8)*y]) >>> Q = C([-Integer(1)/Integer(2), Integer(1)/Integer(2), Integer(1)]) >>> Q.is_ordinary_singularity() True 
 - is_transverse(D)[source]¶
- Return whether the intersection of the curve - Dat this point with the curve this point is on is transverse or not.- INPUT: - D– a curve in the same ambient space as the curve this point is on
 - EXAMPLES: - sage: P.<x,y,z> = ProjectiveSpace(QQ, 2) sage: C = Curve([x^2 - 2*y^2 - 2*z^2], P) sage: D = Curve([y - z], P) sage: Q = C([2,1,1]) sage: Q.is_transverse(D) True - >>> from sage.all import * >>> P = ProjectiveSpace(QQ, Integer(2), names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3) >>> C = Curve([x**Integer(2) - Integer(2)*y**Integer(2) - Integer(2)*z**Integer(2)], P) >>> D = Curve([y - z], P) >>> Q = C([Integer(2),Integer(1),Integer(1)]) >>> Q.is_transverse(D) True - sage: P.<x,y,z> = ProjectiveSpace(GF(17), 2) sage: C = Curve([x^4 - 16*y^3*z], P) sage: D = Curve([y^2 - z*x], P) sage: Q = C([0,0,1]) sage: Q.is_transverse(D) False - >>> from sage.all import * >>> P = ProjectiveSpace(GF(Integer(17)), Integer(2), names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3) >>> C = Curve([x**Integer(4) - Integer(16)*y**Integer(3)*z], P) >>> D = Curve([y**Integer(2) - z*x], P) >>> Q = C([Integer(0),Integer(0),Integer(1)]) >>> Q.is_transverse(D) False 
 - multiplicity()[source]¶
- Return the multiplicity of this point with respect to the projective curve it is on. - EXAMPLES: - sage: P.<x,y,z> = ProjectiveSpace(GF(17), 2) sage: C = Curve([y^3*z - 16*x^4], P) sage: Q = C([0,0,1]) sage: Q.multiplicity() 3 - >>> from sage.all import * >>> P = ProjectiveSpace(GF(Integer(17)), Integer(2), names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3) >>> C = Curve([y**Integer(3)*z - Integer(16)*x**Integer(4)], P) >>> Q = C([Integer(0),Integer(0),Integer(1)]) >>> Q.multiplicity() 3 
 - tangents()[source]¶
- Return the tangents at this point of the projective plane curve this point is on. - OUTPUT: - A list of polynomials in the coordinate ring of the ambient space of the curve this point is on. - EXAMPLES: - sage: P.<x,y,z> = ProjectiveSpace(QQ, 2) sage: C = Curve([y^2*z^3 - x^5 + 18*y*x*z^3]) sage: Q = C([0,0,1]) sage: Q.tangents() [y, 18*x + y] - >>> from sage.all import * >>> P = ProjectiveSpace(QQ, Integer(2), names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3) >>> C = Curve([y**Integer(2)*z**Integer(3) - x**Integer(5) + Integer(18)*y*x*z**Integer(3)]) >>> Q = C([Integer(0),Integer(0),Integer(1)]) >>> Q.tangents() [y, 18*x + y] 
 
- class sage.schemes.curves.point.ProjectivePlaneCurvePoint_finite_field(X, v, check=True)[source]¶
- Bases: - ProjectivePlaneCurvePoint_field,- SchemeMorphism_point_projective_finite_field- Point of a projective plane curve over a finite field.