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
or on affine curves:
sage: A.<x,y> = AffineSpace(GF(23), 2) # optional - sage.rings.finite_rings
sage: C = Curve([y - y^4 + 17*x^2 - 2*x + 22], A) # optional - sage.rings.finite_rings
sage: Q = C([22,21]) # optional - sage.rings.finite_rings
sage: Q.parent() # optional - sage.rings.finite_rings
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)#
Bases:
SchemeMorphism_point_affine_field- is_singular()#
Return whether this point is a singular point of the affine curve it is on.
EXAMPLES:
sage: K = QuadraticField(-1) # optional - sage.rings.number_field sage: A.<x,y,z> = AffineSpace(K, 3) # optional - sage.rings.number_field sage: C = Curve([(x^4 + 2*z + 2)*y, z - y + 1]) # optional - sage.rings.number_field sage: Q1 = C([0,0,-1]) # optional - sage.rings.number_field sage: Q1.is_singular() # optional - sage.rings.number_field True sage: Q2 = C([-K.gen(),0,-1]) # optional - sage.rings.number_field sage: Q2.is_singular() # optional - sage.rings.number_field False
- class sage.schemes.curves.point.AffinePlaneCurvePoint_field(X, v, check=True)#
Bases:
AffineCurvePoint_fieldPoint of an affine plane curve over a field.
- is_ordinary_singularity()#
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
sage: A.<x,y> = AffineSpace(GF(7), 2) # optional - sage.rings.finite_rings sage: C = A.curve([y^2 - x^7 - 6*x^3]) # optional - sage.rings.finite_rings sage: Q = C([0,0]) # optional - sage.rings.finite_rings sage: Q.is_ordinary_singularity() # optional - sage.rings.finite_rings False
- is_transverse(D)#
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
sage: R.<a> = QQ[] sage: K.<b> = NumberField(a^2 - 2) # optional - sage.rings.number_field sage: A.<x,y> = AffineSpace(K, 2) # optional - sage.rings.number_field sage: C = Curve([y^2 + x^2 - 1], A) # optional - sage.rings.number_field sage: D = Curve([y - x], A) # optional - sage.rings.number_field sage: Q = C([-1/2*b, -1/2*b]) # optional - sage.rings.number_field sage: Q.is_transverse(D) # optional - sage.rings.number_field True
- multiplicity()#
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
- tangents()#
Return the tangents at this point of the affine 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: 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 + 1, x - y + 1, x + y + 1]
- class sage.schemes.curves.point.AffinePlaneCurvePoint_finite_field(X, v, check=True)#
Bases:
AffinePlaneCurvePoint_field,SchemeMorphism_point_affine_finite_fieldPoint of an affine plane curve over a finite field.
- class sage.schemes.curves.point.IntegralAffineCurvePoint(X, v, check=True)#
Bases:
AffineCurvePoint_fieldPoint of an integral affine curve.
- closed_point()#
Return the closed point that corresponds to this rational point.
EXAMPLES:
sage: A.<x,y> = AffineSpace(GF(8), 2) # optional - sage.rings.finite_rings sage: C = Curve(x^5 + y^5 + x*y + 1) # optional - sage.rings.finite_rings sage: p = C([1,1]) # optional - sage.rings.finite_rings sage: p.closed_point() # optional - sage.rings.finite_rings Point (x + 1, y + 1)
- place()#
Return a place on this point.
EXAMPLES:
sage: A.<x,y> = AffineSpace(GF(2), 2) # optional - sage.rings.finite_rings sage: C = Curve(x^5 + y^5 + x*y + 1) # optional - sage.rings.finite_rings sage: p = C(-1,-1) # optional - sage.rings.finite_rings sage: p # optional - sage.rings.finite_rings (1, 1) sage: p.closed_point() # optional - sage.rings.finite_rings Point (x + 1, y + 1) sage: _.place() # optional - sage.rings.finite_rings 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()#
Return all places on this point.
EXAMPLES:
sage: A.<x,y> = AffineSpace(GF(2), 2) # optional - sage.rings.finite_rings sage: C = Curve(x^5 + y^5 + x*y + 1) # optional - sage.rings.finite_rings sage: p = C(-1,-1) # optional - sage.rings.finite_rings sage: p # optional - sage.rings.finite_rings (1, 1) sage: p.closed_point() # optional - sage.rings.finite_rings Point (x + 1, y + 1) sage: _.places() # optional - sage.rings.finite_rings [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)#
Bases:
IntegralAffineCurvePointPoint of an integral affine curve over a finite field.
- class sage.schemes.curves.point.IntegralAffinePlaneCurvePoint(X, v, check=True)#
Bases:
IntegralAffineCurvePoint,AffinePlaneCurvePoint_fieldPoint of an integral affine plane curve over a finite field.
- class sage.schemes.curves.point.IntegralAffinePlaneCurvePoint_finite_field(X, v, check=True)#
Bases:
AffinePlaneCurvePoint_finite_field,IntegralAffineCurvePoint_finite_fieldPoint of an integral affine plane curve over a finite field.
- class sage.schemes.curves.point.IntegralProjectiveCurvePoint(X, v, check=True)#
Bases:
ProjectiveCurvePoint_field- closed_point()#
Return the closed point corresponding to this rational point.
EXAMPLES:
sage: P.<x,y,z> = ProjectiveSpace(GF(17), 2) # optional - sage.rings.finite_rings sage: C = Curve([x^4 - 16*y^3*z], P) # optional - sage.rings.finite_rings sage: C.singular_points() # optional - sage.rings.finite_rings [(0 : 0 : 1)] sage: p = _[0] # optional - sage.rings.finite_rings sage: p.closed_point() # optional - sage.rings.finite_rings Point (x, y)
- place()#
Return a place on this point.
EXAMPLES:
sage: P.<x,y,z> = ProjectiveSpace(GF(17), 2) # optional - sage.rings.finite_rings sage: C = Curve([x^4 - 16*y^3*z], P) # optional - sage.rings.finite_rings sage: C.singular_points() # optional - sage.rings.finite_rings [(0 : 0 : 1)] sage: p = _[0] # optional - sage.rings.finite_rings sage: p.place() # optional - sage.rings.finite_rings Place (y)
- places()#
Return all places on this point.
EXAMPLES:
sage: P.<x,y,z> = ProjectiveSpace(GF(17), 2) # optional - sage.rings.finite_rings sage: C = Curve([x^4 - 16*y^3*z], P) # optional - sage.rings.finite_rings sage: C.singular_points() # optional - sage.rings.finite_rings [(0 : 0 : 1)] sage: p = _[0] # optional - sage.rings.finite_rings sage: p.places() # optional - sage.rings.finite_rings [Place (y)]
- class sage.schemes.curves.point.IntegralProjectiveCurvePoint_finite_field(X, v, check=True)#
Bases:
IntegralProjectiveCurvePointPoint of an integral projective curve over a finite field.
- class sage.schemes.curves.point.IntegralProjectivePlaneCurvePoint(X, v, check=True)#
Bases:
IntegralProjectiveCurvePoint,ProjectivePlaneCurvePoint_fieldPoint of an integral projective plane curve over a field.
- class sage.schemes.curves.point.IntegralProjectivePlaneCurvePoint_finite_field(X, v, check=True)#
Bases:
ProjectivePlaneCurvePoint_finite_field,IntegralProjectiveCurvePoint_finite_fieldPoint of an integral projective plane curve over a finite field.
- class sage.schemes.curves.point.ProjectiveCurvePoint_field(X, v, check=True)#
Bases:
SchemeMorphism_point_projective_fieldPoint of a projective curve over a field.
- is_singular()#
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
- class sage.schemes.curves.point.ProjectivePlaneCurvePoint_field(X, v, check=True)#
Bases:
ProjectiveCurvePoint_fieldPoint of a projective plane curve over a field.
- is_ordinary_singularity()#
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
sage: R.<a> = QQ[] sage: K.<b> = NumberField(a^2 - 3) # optional - sage.rings.number_field sage: P.<x,y,z> = ProjectiveSpace(K, 2) # optional - sage.rings.number_field sage: C = P.curve([x^2*y^3*z^4 - y^6*z^3 - 4*x^2*y^4*z^3 - # optional - sage.rings.number_field ....: 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]) # optional - sage.rings.number_field sage: Q.is_ordinary_singularity() # optional - sage.rings.number_field True
- is_transverse(D)#
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
sage: P.<x,y,z> = ProjectiveSpace(GF(17), 2) # optional - sage.rings.finite_rings sage: C = Curve([x^4 - 16*y^3*z], P) # optional - sage.rings.finite_rings sage: D = Curve([y^2 - z*x], P) # optional - sage.rings.finite_rings sage: Q = C([0,0,1]) # optional - sage.rings.finite_rings sage: Q.is_transverse(D) # optional - sage.rings.finite_rings False
- multiplicity()#
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) # optional - sage.rings.finite_rings sage: C = Curve([y^3*z - 16*x^4], P) # optional - sage.rings.finite_rings sage: Q = C([0,0,1]) # optional - sage.rings.finite_rings sage: Q.multiplicity() # optional - sage.rings.finite_rings 3
- tangents()#
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]
- class sage.schemes.curves.point.ProjectivePlaneCurvePoint_finite_field(X, v, check=True)#
Bases:
ProjectivePlaneCurvePoint_field,SchemeMorphism_point_projective_finite_fieldPoint of a projective plane curve over a finite field.