Coverage for local/lib/python2.7/site-packages/sage/schemes/plane_conics/con_finite_field.py : 63%

r""" 

Projective plane conics over finite fields 

AUTHORS: 

- Marco Streng (2010-07-20) 

""" 

from __future__ import absolute_import 

#***************************************************************************** 

# Copyright (C) 2009/2010 Marco Streng <marco.streng@gmail.com> 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# 

# This code is distributed in the hope that it will be useful, 

# but WITHOUT ANY WARRANTY; without even the implied warranty of 

# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU 

# General Public License for more details. 

# 

# The full text of the GPL is available at: 

# 

# http://www.gnu.org/licenses/ 

#***************************************************************************** 

from sage.rings.all import PolynomialRing 

from sage.schemes.curves.projective_curve import ProjectivePlaneCurve_finite_field 

from .con_field import ProjectiveConic_field 

class ProjectiveConic_finite_field(ProjectiveConic_field, ProjectivePlaneCurve_finite_field): 

r""" 

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 

TESTS:: 

sage: K.<a> = FiniteField(4, 'a') 

sage: Conic([a, 1, -1])._test_pickling() 

""" 

def __init__(self, A, f): 

r""" 

See ``Conic`` for full documentation. 

EXAMPLES :: 

sage: Conic([GF(3)(1), 1, 1]) 

Projective Conic Curve over Finite Field of size 3 defined by x^2 + y^2 + z^2 

""" 

ProjectiveConic_field.__init__(self, A, f) 

def count_points(self, n): 

r""" 

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] 

""" 

F = self.base_ring() 

q = F.cardinality() 

return [q**i+1 for i in range(1, n+1)] 

def has_rational_point(self, point = False, read_cache = True, \ 

algorithm = 'default'): 

r""" 

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)) 

TESTS:: 

sage: l = Sequence(cartesian_product_iterator([[0, 1] for i in range(6)])) 

sage: bigF = GF(next_prime(2^100)) 

sage: bigF2 = GF(next_prime(2^50)^2, 'b') 

sage: m = [[F(b) for b in a] for a in l for F in [GF(2), GF(4, 'a'), GF(5), GF(9, 'a'), bigF, bigF2]] 

sage: m += [[F.random_element() for i in range(6)] for j in range(20) for F in [GF(5), bigF]] 

sage: c = [Conic(a) for a in m if a != [0,0,0,0,0,0]] 

sage: assert all([C.has_rational_point() for C in c]) 

sage: r = randrange(0, 5) 

sage: assert all([C.defining_polynomial()(Sequence(C.has_rational_point(point = True)[1])) == 0 for C in c[r::5]]) # long time (1.4s on sage.math, 2013) 

""" 

if not point: 

return True 

if read_cache: 

if self._rational_point is not None: 

return True, self._rational_point 

B = self.base_ring() 

s, pt = self.has_singular_point(point = True) 

if s: 

return True, pt 

while True: 

x = B.random_element() 

Y = PolynomialRing(B,'Y').gen() 

r = self.defining_polynomial()([x,Y,1]).roots() 

if len(r) > 0: 

return True, self.point([x,r[0][0],B(1)])