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)
sage: P.<x,y> = AffineSpace(F, 2)
sage: C = Curve(y^2 + y - x^3)
sage: C.closed_points()
[Point (x, y), Point (x, y + 1)]
sage: C.closed_points(2)
[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)
[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)]

>>> from sage.all import *
>>> F = GF(Integer(2), names=('a',)); (a,) = F._first_ngens(1)
>>> P = AffineSpace(F, Integer(2), names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> C = Curve(y**Integer(2) + y - x**Integer(3))
>>> C.closed_points()
[Point (x, y), Point (x, y + 1)]
>>> C.closed_points(Integer(2))
[Point (y^2 + y + 1, x + 1),
 Point (y^2 + y + 1, x + y),
 Point (y^2 + y + 1, x + y + 1)]
>>> C.closed_points(Integer(3))
[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)]

F.<a> = GF(2)
P.<x,y> = AffineSpace(F, 2)
C = Curve(y^2 + y - x^3)
C.closed_points()
C.closed_points(2)
C.closed_points(3)

Closed points of projective curves are represented by homogeneous maximal ideals:

sage: F.<a> = GF(2)
sage: P.<x,y,z> = ProjectiveSpace(F, 2)
sage: C = Curve(x^3*y + y^3*z + x*z^3)
sage: C.closed_points()
[Point (x, z), Point (x, y), Point (y, z)]
sage: C.closed_points(2)
[Point (y^2 + y*z + z^2, x + y + z)]
sage: C.closed_points(3)
[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)]

>>> from sage.all import *
>>> F = GF(Integer(2), names=('a',)); (a,) = F._first_ngens(1)
>>> P = ProjectiveSpace(F, Integer(2), names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)
>>> C = Curve(x**Integer(3)*y + y**Integer(3)*z + x*z**Integer(3))
>>> C.closed_points()
[Point (x, z), Point (x, y), Point (y, z)]
>>> C.closed_points(Integer(2))
[Point (y^2 + y*z + z^2, x + y + z)]
>>> C.closed_points(Integer(3))
[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)]

F.<a> = GF(2)
P.<x,y,z> = ProjectiveSpace(F, 2)
C = Curve(x^3*y + y^3*z + x*z^3)
C.closed_points()
C.closed_points(2)
C.closed_points(3)

Rational points are easily converted to closed points and vice versa if the closed point is of degree one:

sage: F.<a> = GF(2)
sage: P.<x,y,z> = ProjectiveSpace(F, 2)
sage: C = Curve(x^3*y + y^3*z + x*z^3)
sage: p1, p2, p3 = C.closed_points()
sage: p1.rational_point()
(0 : 1 : 0)
sage: p2.rational_point()
(0 : 0 : 1)
sage: p3.rational_point()
(1 : 0 : 0)
sage: _.closed_point()
Point (y, z)
sage: _ == p3
True

>>> from sage.all import *
>>> F = GF(Integer(2), names=('a',)); (a,) = F._first_ngens(1)
>>> P = ProjectiveSpace(F, Integer(2), names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)
>>> C = Curve(x**Integer(3)*y + y**Integer(3)*z + x*z**Integer(3))
>>> p1, p2, p3 = C.closed_points()
>>> p1.rational_point()
(0 : 1 : 0)
>>> p2.rational_point()
(0 : 0 : 1)
>>> p3.rational_point()
(1 : 0 : 0)
>>> _.closed_point()
Point (y, z)
>>> _ == p3
True

F.<a> = GF(2)
P.<x,y,z> = ProjectiveSpace(F, 2)
C = Curve(x^3*y + y^3*z + x*z^3)
p1, p2, p3 = C.closed_points()
p1.rational_point()
p2.rational_point()
p3.rational_point()
_.closed_point()
_ == p3

AUTHORS:

  • Kwankyu Lee (2019-03): initial version

class sage.schemes.curves.closed_point.CurveClosedPoint(S, P, check=False)[source]

Bases: SchemeTopologicalPoint_prime_ideal

Base class of closed points of curves.

class sage.schemes.curves.closed_point.IntegralAffineCurveClosedPoint(curve, prime_ideal, degree)[source]

Bases: IntegralCurveClosedPoint

Closed points of affine curves.

projective(i=0)[source]

Return the point in the projective closure of the curve, of which this curve is the i-th affine patch.

INPUT:

  • i – integer

EXAMPLES:

sage: F.<a> = GF(2)
sage: A.<x,y> = AffineSpace(F, 2)
sage: C = Curve(y^2 + y - x^3, A)
sage: p1, p2 = C.closed_points()
sage: p1
Point (x, y)
sage: p2
Point (x, y + 1)
sage: p1.projective()
Point (x1, x2)
sage: p2.projective(0)
Point (x1, x0 + x2)
sage: p2.projective(1)
Point (x0, x1 + x2)
sage: p2.projective(2)
Point (x0, x1 + x2)

>>> from sage.all import *
>>> F = GF(Integer(2), names=('a',)); (a,) = F._first_ngens(1)
>>> A = AffineSpace(F, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
>>> C = Curve(y**Integer(2) + y - x**Integer(3), A)
>>> p1, p2 = C.closed_points()
>>> p1
Point (x, y)
>>> p2
Point (x, y + 1)
>>> p1.projective()
Point (x1, x2)
>>> p2.projective(Integer(0))
Point (x1, x0 + x2)
>>> p2.projective(Integer(1))
Point (x0, x1 + x2)
>>> p2.projective(Integer(2))
Point (x0, x1 + x2)

F.<a> = GF(2)
A.<x,y> = AffineSpace(F, 2)
C = Curve(y^2 + y - x^3, A)
p1, p2 = C.closed_points()
p1
p2
p1.projective()
p2.projective(0)
p2.projective(1)
p2.projective(2)
rational_point()[source]

Return the rational point if this closed point is of degree \(1\).

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: A.<x,y> = AffineSpace(GF(3^2), 2)
sage: C = Curve(y^2 - x^5 - x^4 - 2*x^3 - 2*x - 2)
sage: C.closed_points()
[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 _]
[(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())
True

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> A = AffineSpace(GF(Integer(3)**Integer(2)), Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
>>> C = Curve(y**Integer(2) - x**Integer(5) - x**Integer(4) - Integer(2)*x**Integer(3) - Integer(2)*x - Integer(2))
>>> C.closed_points()
[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)]
>>> [p.rational_point() for p in _]
[(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)]
>>> set(_) == set(C.rational_points())
True

# needs sage.rings.finite_rings
A.<x,y> = AffineSpace(GF(3^2), 2)
C = Curve(y^2 - x^5 - x^4 - 2*x^3 - 2*x - 2)
C.closed_points()
[p.rational_point() for p in _]
set(_) == set(C.rational_points())
class sage.schemes.curves.closed_point.IntegralCurveClosedPoint(curve, prime_ideal, degree)[source]

Bases: CurveClosedPoint

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: # needs sage.rings.finite_rings
sage: F.<a> = GF(4)
sage: P.<x,y> = AffineSpace(F, 2)
sage: C = Curve(y^2 + y - x^3)
sage: C.closed_points()
[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))]

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> F = GF(Integer(4), names=('a',)); (a,) = F._first_ngens(1)
>>> P = AffineSpace(F, Integer(2), names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> C = Curve(y**Integer(2) + y - x**Integer(3))
>>> C.closed_points()
[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))]

# needs sage.rings.finite_rings
F.<a> = GF(4)
P.<x,y> = AffineSpace(F, 2)
C = Curve(y^2 + y - x^3)
C.closed_points()
curve()[source]

Return the curve to which this point belongs.

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: F.<a> = GF(4)
sage: P.<x,y> = AffineSpace(F, 2)
sage: C = Curve(y^2 + y - x^3)
sage: pts = C.closed_points()
sage: p = pts[0]
sage: p.curve()
Affine Plane Curve over Finite Field in a of size 2^2 defined by x^3 + y^2 + y

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> F = GF(Integer(4), names=('a',)); (a,) = F._first_ngens(1)
>>> P = AffineSpace(F, Integer(2), names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> C = Curve(y**Integer(2) + y - x**Integer(3))
>>> pts = C.closed_points()
>>> p = pts[Integer(0)]
>>> p.curve()
Affine Plane Curve over Finite Field in a of size 2^2 defined by x^3 + y^2 + y

# needs sage.rings.finite_rings
F.<a> = GF(4)
P.<x,y> = AffineSpace(F, 2)
C = Curve(y^2 + y - x^3)
pts = C.closed_points()
p = pts[0]
p.curve()
degree()[source]

Return the degree of the point.

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: F.<a> = GF(4)
sage: P.<x,y> = AffineSpace(F, 2)
sage: C = Curve(y^2 + y - x^3)
sage: pts = C.closed_points()
sage: p = pts[0]
sage: p.degree()
1

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> F = GF(Integer(4), names=('a',)); (a,) = F._first_ngens(1)
>>> P = AffineSpace(F, Integer(2), names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> C = Curve(y**Integer(2) + y - x**Integer(3))
>>> pts = C.closed_points()
>>> p = pts[Integer(0)]
>>> p.degree()
1

# needs sage.rings.finite_rings
F.<a> = GF(4)
P.<x,y> = AffineSpace(F, 2)
C = Curve(y^2 + y - x^3)
pts = C.closed_points()
p = pts[0]
p.degree()
place()[source]

Return a place on this closed point.

If there are more than one, arbitrary one is chosen.

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: F.<a> = GF(4)
sage: P.<x,y> = AffineSpace(F, 2)
sage: C = Curve(y^2 + y - x^3)
sage: pts = C.closed_points()
sage: p = pts[0]
sage: p.place()
Place (x, y)

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> F = GF(Integer(4), names=('a',)); (a,) = F._first_ngens(1)
>>> P = AffineSpace(F, Integer(2), names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> C = Curve(y**Integer(2) + y - x**Integer(3))
>>> pts = C.closed_points()
>>> p = pts[Integer(0)]
>>> p.place()
Place (x, y)

# needs sage.rings.finite_rings
F.<a> = GF(4)
P.<x,y> = AffineSpace(F, 2)
C = Curve(y^2 + y - x^3)
pts = C.closed_points()
p = pts[0]
p.place()
places()[source]

Return all places on this closed point.

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: F.<a> = GF(4)
sage: P.<x,y> = AffineSpace(F, 2)
sage: C = Curve(y^2 + y - x^3)
sage: pts = C.closed_points()
sage: p = pts[0]
sage: p.places()
[Place (x, y)]

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> F = GF(Integer(4), names=('a',)); (a,) = F._first_ngens(1)
>>> P = AffineSpace(F, Integer(2), names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> C = Curve(y**Integer(2) + y - x**Integer(3))
>>> pts = C.closed_points()
>>> p = pts[Integer(0)]
>>> p.places()
[Place (x, y)]

# needs sage.rings.finite_rings
F.<a> = GF(4)
P.<x,y> = AffineSpace(F, 2)
C = Curve(y^2 + y - x^3)
pts = C.closed_points()
p = pts[0]
p.places()
class sage.schemes.curves.closed_point.IntegralProjectiveCurveClosedPoint(curve, prime_ideal, degree)[source]

Bases: IntegralCurveClosedPoint

Closed points of projective plane curves.

affine(i=None)[source]

Return the point in the i-th affine patch of the curve.

INPUT:

  • i – integer; if not specified, it is chosen automatically

EXAMPLES:

sage: F.<a> = GF(2)
sage: P.<x,y,z> = ProjectiveSpace(F, 2)
sage: C = Curve(x^3*y + y^3*z + x*z^3)
sage: p1, p2, p3 = C.closed_points()
sage: p1.affine()
Point (x, z)
sage: p2.affine()
Point (x, y)
sage: p3.affine()
Point (y, z)
sage: p3.affine(0)
Point (y, z)
sage: p3.affine(1)
Traceback (most recent call last):
...
ValueError: not in the affine patch

>>> from sage.all import *
>>> F = GF(Integer(2), names=('a',)); (a,) = F._first_ngens(1)
>>> P = ProjectiveSpace(F, Integer(2), names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)
>>> C = Curve(x**Integer(3)*y + y**Integer(3)*z + x*z**Integer(3))
>>> p1, p2, p3 = C.closed_points()
>>> p1.affine()
Point (x, z)
>>> p2.affine()
Point (x, y)
>>> p3.affine()
Point (y, z)
>>> p3.affine(Integer(0))
Point (y, z)
>>> p3.affine(Integer(1))
Traceback (most recent call last):
...
ValueError: not in the affine patch

F.<a> = GF(2)
P.<x,y,z> = ProjectiveSpace(F, 2)
C = Curve(x^3*y + y^3*z + x*z^3)
p1, p2, p3 = C.closed_points()
p1.affine()
p2.affine()
p3.affine()
p3.affine(0)
p3.affine(1)
rational_point()[source]

Return the rational point if this closed point is of degree \(1\).

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: F.<a> = GF(4)
sage: P.<x,y,z> = ProjectiveSpace(F, 2)
sage: C = Curve(x^3*y + y^3*z + x*z^3)
sage: C.closed_points()
[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 _]
[(0 : 1 : 0), (0 : 0 : 1), (1 : 0 : 0), (a : a + 1 : 1), (a + 1 : a : 1)]
sage: set(_) == set(C.rational_points())
True

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> F = GF(Integer(4), names=('a',)); (a,) = F._first_ngens(1)
>>> P = ProjectiveSpace(F, Integer(2), names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)
>>> C = Curve(x**Integer(3)*y + y**Integer(3)*z + x*z**Integer(3))
>>> C.closed_points()
[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)]
>>> [p.rational_point() for p in _]
[(0 : 1 : 0), (0 : 0 : 1), (1 : 0 : 0), (a : a + 1 : 1), (a + 1 : a : 1)]
>>> set(_) == set(C.rational_points())
True

# needs sage.rings.finite_rings
F.<a> = GF(4)
P.<x,y,z> = ProjectiveSpace(F, 2)
C = Curve(x^3*y + y^3*z + x*z^3)
C.closed_points()
[p.rational_point() for p in _]
set(_) == set(C.rational_points())