Closed points of integral curves#
A rational point of a curve in Sage is represented by its coordinates. If the curve is defined over finite field and integral, that is reduced and irreducible, then it is empowered by the global function field machinery of Sage. Thus closed points of the curve are computable, as represented by maximal ideals of the coordinate ring of the ambient space.
EXAMPLES:
sage: F.<a> = GF(2) # optional - sage.rings.finite_rings
sage: P.<x,y> = AffineSpace(F, 2) # optional - sage.rings.finite_rings
sage: C = Curve(y^2 + y - x^3) # optional - sage.rings.finite_rings
sage: C.closed_points() # optional - sage.rings.finite_rings
[Point (x, y), Point (x, y + 1)]
sage: C.closed_points(2) # optional - sage.rings.finite_rings
[Point (y^2 + y + 1, x + 1),
Point (y^2 + y + 1, x + y),
Point (y^2 + y + 1, x + y + 1)]
sage: C.closed_points(3) # optional - sage.rings.finite_rings
[Point (x^2 + x + y, x*y + 1, y^2 + x + 1),
Point (x^2 + x + y + 1, x*y + x + 1, y^2 + x)]
Closed points of projective curves are represented by homogeneous maximal ideals:
sage: F.<a> = GF(2) # optional - sage.rings.finite_rings
sage: P.<x,y,z> = ProjectiveSpace(F, 2) # optional - sage.rings.finite_rings
sage: C = Curve(x^3*y + y^3*z + x*z^3) # optional - sage.rings.finite_rings
sage: C.closed_points() # optional - sage.rings.finite_rings
[Point (x, z), Point (x, y), Point (y, z)]
sage: C.closed_points(2) # optional - sage.rings.finite_rings
[Point (y^2 + y*z + z^2, x + y + z)]
sage: C.closed_points(3) # optional - sage.rings.finite_rings
[Point (y^3 + y^2*z + z^3, x + y),
Point (y^3 + y*z^2 + z^3, x + z),
Point (x^2 + x*z + y*z + z^2, x*y + x*z + z^2, y^2 + x*z),
Point (x^2 + y*z, x*y + x*z + z^2, y^2 + x*z + y*z),
Point (x^3 + x*z^2 + z^3, y + z),
Point (x^2 + y*z + z^2, x*y + x*z + y*z, y^2 + x*z + y*z + z^2),
Point (x^2 + y*z + z^2, x*y + z^2, y^2 + x*z + y*z)]
Rational points are easily converted to closed points and vice versa if the closed point is of degree one:
sage: F.<a> = GF(2) # optional - sage.rings.finite_rings
sage: P.<x,y,z> = ProjectiveSpace(F, 2) # optional - sage.rings.finite_rings
sage: C = Curve(x^3*y + y^3*z + x*z^3) # optional - sage.rings.finite_rings
sage: p1, p2, p3 = C.closed_points() # optional - sage.rings.finite_rings
sage: p1.rational_point() # optional - sage.rings.finite_rings
(0 : 1 : 0)
sage: p2.rational_point() # optional - sage.rings.finite_rings
(0 : 0 : 1)
sage: p3.rational_point() # optional - sage.rings.finite_rings
(1 : 0 : 0)
sage: _.closed_point() # optional - sage.rings.finite_rings
Point (y, z)
sage: _ == p3 # optional - sage.rings.finite_rings
True
AUTHORS:
Kwankyu Lee (2019-03): initial version
- class sage.schemes.curves.closed_point.CurveClosedPoint(S, P, check=False)#
Bases:
SchemeTopologicalPoint_prime_idealBase class of closed points of curves.
- class sage.schemes.curves.closed_point.IntegralAffineCurveClosedPoint(curve, prime_ideal, degree)#
Bases:
IntegralCurveClosedPointClosed points of affine curves.
- projective(i=0)#
Return the point in the projective closure of the curve, of which this curve is the
i-th affine patch.INPUT:
i– an integer
EXAMPLES:
sage: F.<a> = GF(2) # optional - sage.rings.finite_rings sage: A.<x,y> = AffineSpace(F, 2) # optional - sage.rings.finite_rings sage: C = Curve(y^2 + y - x^3, A) # optional - sage.rings.finite_rings sage: p1, p2 = C.closed_points() # optional - sage.rings.finite_rings sage: p1 # optional - sage.rings.finite_rings Point (x, y) sage: p2 # optional - sage.rings.finite_rings Point (x, y + 1) sage: p1.projective() # optional - sage.rings.finite_rings Point (x1, x2) sage: p2.projective(0) # optional - sage.rings.finite_rings Point (x1, x0 + x2) sage: p2.projective(1) # optional - sage.rings.finite_rings Point (x0, x1 + x2) sage: p2.projective(2) # optional - sage.rings.finite_rings Point (x0, x1 + x2)
- rational_point()#
Return the rational point if this closed point is of degree \(1\).
EXAMPLES:
sage: A.<x,y> = AffineSpace(GF(3^2), 2) # optional - sage.rings.finite_rings sage: C = Curve(y^2 - x^5 - x^4 - 2*x^3 - 2*x - 2) # optional - sage.rings.finite_rings sage: C.closed_points() # optional - sage.rings.finite_rings [Point (x, y + (z2 + 1)), Point (x, y + (-z2 - 1)), Point (x + (z2 + 1), y + (z2 - 1)), Point (x + (z2 + 1), y + (-z2 + 1)), Point (x - 1, y + (z2 + 1)), Point (x - 1, y + (-z2 - 1)), Point (x + (-z2 - 1), y + z2), Point (x + (-z2 - 1), y + (-z2)), Point (x + 1, y + 1), Point (x + 1, y - 1)] sage: [p.rational_point() for p in _] # optional - sage.rings.finite_rings [(0, 2*z2 + 2), (0, z2 + 1), (2*z2 + 2, 2*z2 + 1), (2*z2 + 2, z2 + 2), (1, 2*z2 + 2), (1, z2 + 1), (z2 + 1, 2*z2), (z2 + 1, z2), (2, 2), (2, 1)] sage: set(_) == set(C.rational_points()) # optional - sage.rings.finite_rings True
- class sage.schemes.curves.closed_point.IntegralCurveClosedPoint(curve, prime_ideal, degree)#
Bases:
CurveClosedPointClosed points of integral curves.
INPUT:
curve– the curve to which the closed point belongsprime_ideal– a prime idealdegree– degree of the closed point
EXAMPLES:
sage: F.<a> = GF(4) # optional - sage.rings.finite_rings sage: P.<x,y> = AffineSpace(F, 2) # optional - sage.rings.finite_rings sage: C = Curve(y^2 + y - x^3) # optional - sage.rings.finite_rings sage: C.closed_points() # optional - sage.rings.finite_rings [Point (x, y), Point (x, y + 1), Point (x + a, y + a), Point (x + a, y + (a + 1)), Point (x + (a + 1), y + a), Point (x + (a + 1), y + (a + 1)), Point (x + 1, y + a), Point (x + 1, y + (a + 1))]
- curve()#
Return the curve to which this point belongs.
EXAMPLES:
sage: F.<a> = GF(4) # optional - sage.rings.finite_rings sage: P.<x,y> = AffineSpace(F, 2) # optional - sage.rings.finite_rings sage: C = Curve(y^2 + y - x^3) # optional - sage.rings.finite_rings sage: pts = C.closed_points() # optional - sage.rings.finite_rings sage: p = pts[0] # optional - sage.rings.finite_rings sage: p.curve() # optional - sage.rings.finite_rings Affine Plane Curve over Finite Field in a of size 2^2 defined by x^3 + y^2 + y
- degree()#
Return the degree of the point.
EXAMPLES:
sage: F.<a> = GF(4) # optional - sage.rings.finite_rings sage: P.<x,y> = AffineSpace(F, 2) # optional - sage.rings.finite_rings sage: C = Curve(y^2 + y - x^3) # optional - sage.rings.finite_rings sage: pts = C.closed_points() # optional - sage.rings.finite_rings sage: p = pts[0] # optional - sage.rings.finite_rings sage: p.degree() # optional - sage.rings.finite_rings 1
- place()#
Return a place on this closed point.
If there are more than one, arbitrary one is chosen.
EXAMPLES:
sage: F.<a> = GF(4) # optional - sage.rings.finite_rings sage: P.<x,y> = AffineSpace(F, 2) # optional - sage.rings.finite_rings sage: C = Curve(y^2 + y - x^3) # optional - sage.rings.finite_rings sage: pts = C.closed_points() # optional - sage.rings.finite_rings sage: p = pts[0] # optional - sage.rings.finite_rings sage: p.place() # optional - sage.rings.finite_rings Place (x, y)
- places()#
Return all places on this closed point.
EXAMPLES:
sage: F.<a> = GF(4) # optional - sage.rings.finite_rings sage: P.<x,y> = AffineSpace(F, 2) # optional - sage.rings.finite_rings sage: C = Curve(y^2 + y - x^3) # optional - sage.rings.finite_rings sage: pts = C.closed_points() # optional - sage.rings.finite_rings sage: p = pts[0] # optional - sage.rings.finite_rings sage: p.places() # optional - sage.rings.finite_rings [Place (x, y)]
- class sage.schemes.curves.closed_point.IntegralProjectiveCurveClosedPoint(curve, prime_ideal, degree)#
Bases:
IntegralCurveClosedPointClosed points of projective plane curves.
- affine(i=None)#
Return the point in the
i-th affine patch of the curve.INPUT:
i– an integer; if not specified, it is chosen automatically.
EXAMPLES:
sage: F.<a> = GF(2) # optional - sage.rings.finite_rings sage: P.<x,y,z> = ProjectiveSpace(F, 2) # optional - sage.rings.finite_rings sage: C = Curve(x^3*y + y^3*z + x*z^3) # optional - sage.rings.finite_rings sage: p1, p2, p3 = C.closed_points() # optional - sage.rings.finite_rings sage: p1.affine() # optional - sage.rings.finite_rings Point (x, z) sage: p2.affine() # optional - sage.rings.finite_rings Point (x, y) sage: p3.affine() # optional - sage.rings.finite_rings Point (y, z) sage: p3.affine(0) # optional - sage.rings.finite_rings Point (y, z) sage: p3.affine(1) # optional - sage.rings.finite_rings Traceback (most recent call last): ... ValueError: not in the affine patch
- rational_point()#
Return the rational point if this closed point is of degree \(1\).
EXAMPLES:
sage: F.<a> = GF(4) # optional - sage.rings.finite_rings sage: P.<x,y,z> = ProjectiveSpace(F, 2) # optional - sage.rings.finite_rings sage: C = Curve(x^3*y + y^3*z + x*z^3) # optional - sage.rings.finite_rings sage: C.closed_points() # optional - sage.rings.finite_rings [Point (x, z), Point (x, y), Point (y, z), Point (x + a*z, y + (a + 1)*z), Point (x + (a + 1)*z, y + a*z)] sage: [p.rational_point() for p in _] # optional - sage.rings.finite_rings [(0 : 1 : 0), (0 : 0 : 1), (1 : 0 : 0), (a : a + 1 : 1), (a + 1 : a : 1)] sage: set(_) == set(C.rational_points()) # optional - sage.rings.finite_rings True