Projective plane conics over finite fields

AUTHORS:

  • Marco Streng (2010-07-20)

class sage.schemes.plane_conics.con_finite_field.ProjectiveConic_finite_field(A, f)[source]

Bases: ProjectiveConic_field, ProjectivePlaneCurve_finite_field

Create a projective plane conic curve over a finite field.

See Conic for full documentation.

EXAMPLES:

sage: K.<a> = FiniteField(9, 'a')
sage: P.<X, Y, Z> = K[]
sage: Conic(X^2 + Y^2 - a*Z^2)
Projective Conic Curve over Finite Field in a of size 3^2
 defined by X^2 + Y^2 + (-a)*Z^2

>>> from sage.all import *
>>> K = FiniteField(Integer(9), 'a', names=('a',)); (a,) = K._first_ngens(1)
>>> P = K['X, Y, Z']; (X, Y, Z,) = P._first_ngens(3)
>>> Conic(X**Integer(2) + Y**Integer(2) - a*Z**Integer(2))
Projective Conic Curve over Finite Field in a of size 3^2
 defined by X^2 + Y^2 + (-a)*Z^2

K.<a> = FiniteField(9, 'a')
P.<X, Y, Z> = K[]
Conic(X^2 + Y^2 - a*Z^2)
sage: P.<X, Y, Z> = FiniteField(5)[]
sage: Conic(X^2 + Y^2 - 2*Z^2)
Projective Conic Curve over Finite Field of size 5 defined by X^2 + Y^2 - 2*Z^2

>>> from sage.all import *
>>> P = FiniteField(Integer(5))['X, Y, Z']; (X, Y, Z,) = P._first_ngens(3)
>>> Conic(X**Integer(2) + Y**Integer(2) - Integer(2)*Z**Integer(2))
Projective Conic Curve over Finite Field of size 5 defined by X^2 + Y^2 - 2*Z^2

P.<X, Y, Z> = FiniteField(5)[]
Conic(X^2 + Y^2 - 2*Z^2)
>>> from sage.all import *
>>> P = FiniteField(Integer(5))['X, Y, Z']; (X, Y, Z,) = P._first_ngens(3)
>>> Conic(X**Integer(2) + Y**Integer(2) - Integer(2)*Z**Integer(2))
Projective Conic Curve over Finite Field of size 5 defined by X^2 + Y^2 - 2*Z^2

P.<X, Y, Z> = FiniteField(5)[]
Conic(X^2 + Y^2 - 2*Z^2)
count_points(n)[source]

If the base field \(B\) of self is finite of order \(q\), then returns the number of points over \(\GF{q}, ..., \GF{q^n}\).

EXAMPLES:

sage: P.<x,y,z> = GF(3)[]
sage: c = Curve(x^2+y^2+z^2); c
Projective Conic Curve over Finite Field of size 3 defined by x^2 + y^2 + z^2
sage: c.count_points(4)
[4, 10, 28, 82]

>>> from sage.all import *
>>> P = GF(Integer(3))['x, y, z']; (x, y, z,) = P._first_ngens(3)
>>> c = Curve(x**Integer(2)+y**Integer(2)+z**Integer(2)); c
Projective Conic Curve over Finite Field of size 3 defined by x^2 + y^2 + z^2
>>> c.count_points(Integer(4))
[4, 10, 28, 82]

P.<x,y,z> = GF(3)[]
c = Curve(x^2+y^2+z^2); c
c.count_points(4)
has_rational_point(point=False, read_cache=True, algorithm='default')[source]

Always returns True because self has a point defined over its finite base field \(B\).

If point is True, then returns a second output \(S\), which is a rational point if one exists.

Points are cached. If read_cache is True, then cached information is used for the output if available. If no cached point is available or read_cache is False, then random \(y\)-coordinates are tried if self is smooth and a singular point is returned otherwise.

EXAMPLES:

sage: Conic(FiniteField(37), [1, 2, 3, 4, 5, 6]).has_rational_point()
True

sage: C = Conic(FiniteField(2), [1, 1, 1, 1, 1, 0]); C
Projective Conic Curve over Finite Field of size 2
 defined by x^2 + x*y + y^2 + x*z + y*z
sage: C.has_rational_point(point = True)  # output is random
(True, (0 : 0 : 1))

sage: p = next_prime(10^50)
sage: F = FiniteField(p)
sage: C = Conic(F, [1, 2, 3]); C
Projective Conic Curve over Finite Field
 of size 100000000000000000000000000000000000000000000000151
 defined by x^2 + 2*y^2 + 3*z^2
sage: C.has_rational_point(point = True)  # output is random
(True,
 (14971942941468509742682168602989039212496867586852
  : 75235465708017792892762202088174741054630437326388 : 1)

sage: F.<a> = FiniteField(7^20)
sage: C = Conic([1, a, -5]); C
Projective Conic Curve over Finite Field in a of size 7^20
 defined by x^2 + a*y^2 + 2*z^2
sage: C.has_rational_point(point = True)  # output is random
(True,
 (a^18 + 2*a^17 + 4*a^16 + 6*a^13 + a^12 + 6*a^11 + 3*a^10 + 4*a^9 + 2*a^8
    + 4*a^7 + a^6 + 4*a^4 + 6*a^2 + 3*a + 6
  : 5*a^19 + 5*a^18 + 5*a^17 + a^16 + 2*a^15 + 3*a^14 + 4*a^13 + 5*a^12
    + a^11 + 3*a^10 + 2*a^8 + 3*a^7 + 4*a^6 + 4*a^5 + 6*a^3 + 5*a^2 + 2*a + 4
  : 1))

>>> from sage.all import *
>>> Conic(FiniteField(Integer(37)), [Integer(1), Integer(2), Integer(3), Integer(4), Integer(5), Integer(6)]).has_rational_point()
True

>>> C = Conic(FiniteField(Integer(2)), [Integer(1), Integer(1), Integer(1), Integer(1), Integer(1), Integer(0)]); C
Projective Conic Curve over Finite Field of size 2
 defined by x^2 + x*y + y^2 + x*z + y*z
>>> C.has_rational_point(point = True)  # output is random
(True, (0 : 0 : 1))

>>> p = next_prime(Integer(10)**Integer(50))
>>> F = FiniteField(p)
>>> C = Conic(F, [Integer(1), Integer(2), Integer(3)]); C
Projective Conic Curve over Finite Field
 of size 100000000000000000000000000000000000000000000000151
 defined by x^2 + 2*y^2 + 3*z^2
>>> C.has_rational_point(point = True)  # output is random
(True,
 (14971942941468509742682168602989039212496867586852
  : 75235465708017792892762202088174741054630437326388 : 1)

>>> F = FiniteField(Integer(7)**Integer(20), names=('a',)); (a,) = F._first_ngens(1)
>>> C = Conic([Integer(1), a, -Integer(5)]); C
Projective Conic Curve over Finite Field in a of size 7^20
 defined by x^2 + a*y^2 + 2*z^2
>>> C.has_rational_point(point = True)  # output is random
(True,
 (a^18 + 2*a^17 + 4*a^16 + 6*a^13 + a^12 + 6*a^11 + 3*a^10 + 4*a^9 + 2*a^8
    + 4*a^7 + a^6 + 4*a^4 + 6*a^2 + 3*a + 6
  : 5*a^19 + 5*a^18 + 5*a^17 + a^16 + 2*a^15 + 3*a^14 + 4*a^13 + 5*a^12
    + a^11 + 3*a^10 + 2*a^8 + 3*a^7 + 4*a^6 + 4*a^5 + 6*a^3 + 5*a^2 + 2*a + 4
  : 1))

Conic(FiniteField(37), [1, 2, 3, 4, 5, 6]).has_rational_point()
C = Conic(FiniteField(2), [1, 1, 1, 1, 1, 0]); C
C.has_rational_point(point = True)  # output is random
p = next_prime(10^50)
F = FiniteField(p)
C = Conic(F, [1, 2, 3]); C
C.has_rational_point(point = True)  # output is random
F.<a> = FiniteField(7^20)
C = Conic([1, a, -5]); C
C.has_rational_point(point = True)  # output is random