# 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)#

Base class of closed points of curves.

class sage.schemes.curves.closed_point.IntegralAffineCurveClosedPoint(curve, prime_ideal, degree)#

Closed 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)#

Closed points of integral curves.

INPUT:

• curve – the curve to which the closed point belongs

• prime_ideal – a prime ideal

• degree – 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)#

Closed 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