Projective plane conics over a field

AUTHORS:

  • Marco Streng (2010-07-20)

  • Nick Alexander (2008-01-08)

class sage.schemes.plane_conics.con_field.ProjectiveConic_field(A, f)[source]

Bases: ProjectivePlaneCurve_field

Create a projective plane conic curve over a field. See Conic for full documentation.

EXAMPLES:

sage: K = FractionField(PolynomialRing(QQ, 't'))
sage: P.<X, Y, Z> = K[]
sage: Conic(X^2 + Y^2 - Z^2)
Projective Conic Curve over Fraction Field of Univariate Polynomial Ring in t
 over Rational Field defined by X^2 + Y^2 - Z^2

>>> from sage.all import *
>>> K = FractionField(PolynomialRing(QQ, 't'))
>>> P = K['X, Y, Z']; (X, Y, Z,) = P._first_ngens(3)
>>> Conic(X**Integer(2) + Y**Integer(2) - Z**Integer(2))
Projective Conic Curve over Fraction Field of Univariate Polynomial Ring in t
 over Rational Field defined by X^2 + Y^2 - Z^2

K = FractionField(PolynomialRing(QQ, 't'))
P.<X, Y, Z> = K[]
Conic(X^2 + Y^2 - Z^2)
base_extend(S)[source]

Return the conic over S given by the same equation as self.

EXAMPLES:

sage: c = Conic([1, 1, 1]); c
Projective Conic Curve over Rational Field defined by x^2 + y^2 + z^2
sage: c.has_rational_point()                                                # needs sage.libs.pari
False
sage: d = c.base_extend(QuadraticField(-1, 'i')); d                         # needs sage.rings.number_field
Projective Conic Curve over Number Field in i
 with defining polynomial x^2 + 1 with i = 1*I defined by x^2 + y^2 + z^2
sage: d.rational_point(algorithm='rnfisnorm')                               # needs sage.rings.number_field
(i : 1 : 0)

>>> from sage.all import *
>>> c = Conic([Integer(1), Integer(1), Integer(1)]); c
Projective Conic Curve over Rational Field defined by x^2 + y^2 + z^2
>>> c.has_rational_point()                                                # needs sage.libs.pari
False
>>> d = c.base_extend(QuadraticField(-Integer(1), 'i')); d                         # needs sage.rings.number_field
Projective Conic Curve over Number Field in i
 with defining polynomial x^2 + 1 with i = 1*I defined by x^2 + y^2 + z^2
>>> d.rational_point(algorithm='rnfisnorm')                               # needs sage.rings.number_field
(i : 1 : 0)

c = Conic([1, 1, 1]); c
c.has_rational_point()                                                # needs sage.libs.pari
d = c.base_extend(QuadraticField(-1, 'i')); d                         # needs sage.rings.number_field
d.rational_point(algorithm='rnfisnorm')                               # needs sage.rings.number_field
cache_point(p)[source]

Replace the point in the cache of self by p for use by rational_point() and parametrization().

EXAMPLES:

sage: c = Conic([1, -1, 1])
sage: c.point([15, 17, 8])
(15/8 : 17/8 : 1)
sage: c.rational_point()
(15/8 : 17/8 : 1)

sage: # needs sage.libs.pari
sage: c.cache_point(c.rational_point(read_cache=False))
sage: c.rational_point()
(-1 : 1 : 0)

>>> from sage.all import *
>>> c = Conic([Integer(1), -Integer(1), Integer(1)])
>>> c.point([Integer(15), Integer(17), Integer(8)])
(15/8 : 17/8 : 1)
>>> c.rational_point()
(15/8 : 17/8 : 1)

>>> # needs sage.libs.pari
>>> c.cache_point(c.rational_point(read_cache=False))
>>> c.rational_point()
(-1 : 1 : 0)

c = Conic([1, -1, 1])
c.point([15, 17, 8])
c.rational_point()
# needs sage.libs.pari
c.cache_point(c.rational_point(read_cache=False))
c.rational_point()
coefficients()[source]

Give a the \(6\) coefficients of the conic self in lexicographic order.

EXAMPLES:

sage: Conic(QQ, [1,2,3,4,5,6]).coefficients()
[1, 2, 3, 4, 5, 6]

sage: P.<x,y,z> = GF(13)[]
sage: a = Conic(x^2 + 5*x*y + y^2 + z^2).coefficients(); a
[1, 5, 0, 1, 0, 1]
sage: Conic(a)
Projective Conic Curve over Finite Field of size 13
defined by x^2 + 5*x*y + y^2 + z^2

>>> from sage.all import *
>>> Conic(QQ, [Integer(1),Integer(2),Integer(3),Integer(4),Integer(5),Integer(6)]).coefficients()
[1, 2, 3, 4, 5, 6]

>>> P = GF(Integer(13))['x, y, z']; (x, y, z,) = P._first_ngens(3)
>>> a = Conic(x**Integer(2) + Integer(5)*x*y + y**Integer(2) + z**Integer(2)).coefficients(); a
[1, 5, 0, 1, 0, 1]
>>> Conic(a)
Projective Conic Curve over Finite Field of size 13
defined by x^2 + 5*x*y + y^2 + z^2

Conic(QQ, [1,2,3,4,5,6]).coefficients()
P.<x,y,z> = GF(13)[]
a = Conic(x^2 + 5*x*y + y^2 + z^2).coefficients(); a
Conic(a)
derivative_matrix()[source]

Give the derivative of the defining polynomial of the conic self, which is a linear map, as a \(3 \times 3\) matrix.

EXAMPLES:

In characteristic different from \(2\), the derivative matrix is twice the symmetric matrix:

sage: c = Conic(QQ, [1,1,1,1,1,0])
sage: c.symmetric_matrix()
[  1 1/2 1/2]
[1/2   1 1/2]
[1/2 1/2   0]
sage: c.derivative_matrix()
[2 1 1]
[1 2 1]
[1 1 0]

>>> from sage.all import *
>>> c = Conic(QQ, [Integer(1),Integer(1),Integer(1),Integer(1),Integer(1),Integer(0)])
>>> c.symmetric_matrix()
[  1 1/2 1/2]
[1/2   1 1/2]
[1/2 1/2   0]
>>> c.derivative_matrix()
[2 1 1]
[1 2 1]
[1 1 0]

c = Conic(QQ, [1,1,1,1,1,0])
c.symmetric_matrix()
c.derivative_matrix()

An example in characteristic \(2\):

sage: P.<t> = GF(2)[]
sage: c = Conic([t, 1, t^2, 1, 1, 0]); c                                    # needs sage.libs.ntl
Projective Conic Curve over Fraction Field of Univariate
 Polynomial Ring in t over Finite Field of size 2 (using GF2X)
 defined by t*x^2 + x*y + y^2 + (t^2)*x*z + y*z
sage: c.is_smooth()
True
sage: c.derivative_matrix()
[  0   1 t^2]
[  1   0   1]
[t^2   1   0]

>>> from sage.all import *
>>> P = GF(Integer(2))['t']; (t,) = P._first_ngens(1)
>>> c = Conic([t, Integer(1), t**Integer(2), Integer(1), Integer(1), Integer(0)]); c                                    # needs sage.libs.ntl
Projective Conic Curve over Fraction Field of Univariate
 Polynomial Ring in t over Finite Field of size 2 (using GF2X)
 defined by t*x^2 + x*y + y^2 + (t^2)*x*z + y*z
>>> c.is_smooth()
True
>>> c.derivative_matrix()
[  0   1 t^2]
[  1   0   1]
[t^2   1   0]

P.<t> = GF(2)[]
c = Conic([t, 1, t^2, 1, 1, 0]); c                                    # needs sage.libs.ntl
c.is_smooth()
c.derivative_matrix()
determinant()[source]

Return the determinant of the symmetric matrix that defines the conic self.

This is defined only if the base field has characteristic different from \(2\).

EXAMPLES:

sage: C = Conic([1,2,3,4,5,6])
sage: C.determinant()
41/4
sage: C.symmetric_matrix().determinant()
41/4

>>> from sage.all import *
>>> C = Conic([Integer(1),Integer(2),Integer(3),Integer(4),Integer(5),Integer(6)])
>>> C.determinant()
41/4
>>> C.symmetric_matrix().determinant()
41/4

C = Conic([1,2,3,4,5,6])
C.determinant()
C.symmetric_matrix().determinant()

Determinants are only defined in characteristic different from \(2\):

sage: C = Conic(GF(2), [1, 1, 1, 1, 1, 0])
sage: C.is_smooth()
True
sage: C.determinant()
Traceback (most recent call last):
...
ValueError: The conic self (= Projective Conic Curve over Finite Field
of size 2 defined by x^2 + x*y + y^2 + x*z + y*z) has no symmetric matrix
because the base field has characteristic 2

>>> from sage.all import *
>>> C = Conic(GF(Integer(2)), [Integer(1), Integer(1), Integer(1), Integer(1), Integer(1), Integer(0)])
>>> C.is_smooth()
True
>>> C.determinant()
Traceback (most recent call last):
...
ValueError: The conic self (= Projective Conic Curve over Finite Field
of size 2 defined by x^2 + x*y + y^2 + x*z + y*z) has no symmetric matrix
because the base field has characteristic 2

C = Conic(GF(2), [1, 1, 1, 1, 1, 0])
C.is_smooth()
C.determinant()
diagonal_matrix()[source]

Return a diagonal matrix \(D\) and a matrix \(T\) such that \(T^t A T = D\) holds, where \((x, y, z) A (x, y, z)^t\) is the defining polynomial of the conic self.

EXAMPLES:

sage: c = Conic(QQ, [1,2,3,4,5,6])
sage: d, t = c.diagonal_matrix(); d, t
(
[    1     0     0]  [   1   -1 -7/6]
[    0     3     0]  [   0    1 -1/3]
[    0     0 41/12], [   0    0    1]
)
sage: t.transpose()*c.symmetric_matrix()*t
[    1     0     0]
[    0     3     0]
[    0     0 41/12]

>>> from sage.all import *
>>> c = Conic(QQ, [Integer(1),Integer(2),Integer(3),Integer(4),Integer(5),Integer(6)])
>>> d, t = c.diagonal_matrix(); d, t
(
[    1     0     0]  [   1   -1 -7/6]
[    0     3     0]  [   0    1 -1/3]
[    0     0 41/12], [   0    0    1]
)
>>> t.transpose()*c.symmetric_matrix()*t
[    1     0     0]
[    0     3     0]
[    0     0 41/12]

c = Conic(QQ, [1,2,3,4,5,6])
d, t = c.diagonal_matrix(); d, t
t.transpose()*c.symmetric_matrix()*t

Diagonal matrices are only defined in characteristic different from \(2\):

sage: # needs sage.rings.finite_rings
sage: c = Conic(GF(4, 'a'), [0, 1, 1, 1, 1, 1])
sage: c.is_smooth()
True
sage: c.diagonal_matrix()
Traceback (most recent call last):
...
ValueError: The conic self (= Projective Conic Curve over Finite Field
in a of size 2^2 defined by x*y + y^2 + x*z + y*z + z^2) has
no symmetric matrix because the base field has characteristic 2

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> c = Conic(GF(Integer(4), 'a'), [Integer(0), Integer(1), Integer(1), Integer(1), Integer(1), Integer(1)])
>>> c.is_smooth()
True
>>> c.diagonal_matrix()
Traceback (most recent call last):
...
ValueError: The conic self (= Projective Conic Curve over Finite Field
in a of size 2^2 defined by x*y + y^2 + x*z + y*z + z^2) has
no symmetric matrix because the base field has characteristic 2

# needs sage.rings.finite_rings
c = Conic(GF(4, 'a'), [0, 1, 1, 1, 1, 1])
c.is_smooth()
c.diagonal_matrix()
diagonalization(names=None)[source]

Return a diagonal conic \(C\), an isomorphism of schemes \(M: C\) -> self and the inverse \(N\) of \(M\).

EXAMPLES:

sage: Conic(GF(5), [1,0,1,1,0,1]).diagonalization()
(Projective Conic Curve over Finite Field of size 5
  defined by x^2 + y^2 + 2*z^2,
 Scheme morphism:
  From: Projective Conic Curve over Finite Field of size 5
        defined by x^2 + y^2 + 2*z^2
  To:   Projective Conic Curve over Finite Field of size 5
        defined by x^2 + y^2 + x*z + z^2
  Defn: Defined on coordinates by sending (x : y : z) to (x + 2*z : y : z),
 Scheme morphism:
  From: Projective Conic Curve over Finite Field of size 5
        defined by x^2 + y^2 + x*z + z^2
  To:   Projective Conic Curve over Finite Field of size 5
        defined by x^2 + y^2 + 2*z^2
  Defn: Defined on coordinates by sending (x : y : z) to (x - 2*z : y : z))

>>> from sage.all import *
>>> Conic(GF(Integer(5)), [Integer(1),Integer(0),Integer(1),Integer(1),Integer(0),Integer(1)]).diagonalization()
(Projective Conic Curve over Finite Field of size 5
  defined by x^2 + y^2 + 2*z^2,
 Scheme morphism:
  From: Projective Conic Curve over Finite Field of size 5
        defined by x^2 + y^2 + 2*z^2
  To:   Projective Conic Curve over Finite Field of size 5
        defined by x^2 + y^2 + x*z + z^2
  Defn: Defined on coordinates by sending (x : y : z) to (x + 2*z : y : z),
 Scheme morphism:
  From: Projective Conic Curve over Finite Field of size 5
        defined by x^2 + y^2 + x*z + z^2
  To:   Projective Conic Curve over Finite Field of size 5
        defined by x^2 + y^2 + 2*z^2
  Defn: Defined on coordinates by sending (x : y : z) to (x - 2*z : y : z))

Conic(GF(5), [1,0,1,1,0,1]).diagonalization()

The diagonalization is only defined in characteristic different from 2:

sage: Conic(GF(2), [1,1,1,1,1,0]).diagonalization()
Traceback (most recent call last):
...
ValueError: The conic self (= Projective Conic Curve over Finite Field
of size 2 defined by x^2 + x*y + y^2 + x*z + y*z) has no symmetric matrix
because the base field has characteristic 2

>>> from sage.all import *
>>> Conic(GF(Integer(2)), [Integer(1),Integer(1),Integer(1),Integer(1),Integer(1),Integer(0)]).diagonalization()
Traceback (most recent call last):
...
ValueError: The conic self (= Projective Conic Curve over Finite Field
of size 2 defined by x^2 + x*y + y^2 + x*z + y*z) has no symmetric matrix
because the base field has characteristic 2

Conic(GF(2), [1,1,1,1,1,0]).diagonalization()

An example over a global function field:

sage: K = FractionField(PolynomialRing(GF(7), 't'))
sage: (t,) = K.gens()
sage: C = Conic(K, [t/2,0, 1, 2, 0, 3])
sage: C.diagonalization()
(Projective Conic Curve over Fraction Field of Univariate
  Polynomial Ring in t over Finite Field of size 7
  defined by (-3*t)*x^2 + 2*y^2 + (3*t + 3)/t*z^2,
 Scheme morphism:
   From: Projective Conic Curve over Fraction Field of Univariate
         Polynomial Ring in t over Finite Field of size 7
         defined by (-3*t)*x^2 + 2*y^2 + (3*t + 3)/t*z^2
   To:   Projective Conic Curve over Fraction Field of Univariate
         Polynomial Ring in t over Finite Field of size 7
         defined by (-3*t)*x^2 + 2*y^2 + x*z + 3*z^2
   Defn: Defined on coordinates by sending (x : y : z) to (x - 1/t*z : y : z),
 Scheme morphism:
   From: Projective Conic Curve over Fraction Field of Univariate
         Polynomial Ring in t over Finite Field of size 7
         defined by (-3*t)*x^2 + 2*y^2 + x*z + 3*z^2
   To:   Projective Conic Curve over Fraction Field of Univariate
         Polynomial Ring in t over Finite Field of size 7
         defined by (-3*t)*x^2 + 2*y^2 + (3*t + 3)/t*z^2
   Defn: Defined on coordinates by sending (x : y : z) to (x + 1/t*z : y : z))

>>> from sage.all import *
>>> K = FractionField(PolynomialRing(GF(Integer(7)), 't'))
>>> (t,) = K.gens()
>>> C = Conic(K, [t/Integer(2),Integer(0), Integer(1), Integer(2), Integer(0), Integer(3)])
>>> C.diagonalization()
(Projective Conic Curve over Fraction Field of Univariate
  Polynomial Ring in t over Finite Field of size 7
  defined by (-3*t)*x^2 + 2*y^2 + (3*t + 3)/t*z^2,
 Scheme morphism:
   From: Projective Conic Curve over Fraction Field of Univariate
         Polynomial Ring in t over Finite Field of size 7
         defined by (-3*t)*x^2 + 2*y^2 + (3*t + 3)/t*z^2
   To:   Projective Conic Curve over Fraction Field of Univariate
         Polynomial Ring in t over Finite Field of size 7
         defined by (-3*t)*x^2 + 2*y^2 + x*z + 3*z^2
   Defn: Defined on coordinates by sending (x : y : z) to (x - 1/t*z : y : z),
 Scheme morphism:
   From: Projective Conic Curve over Fraction Field of Univariate
         Polynomial Ring in t over Finite Field of size 7
         defined by (-3*t)*x^2 + 2*y^2 + x*z + 3*z^2
   To:   Projective Conic Curve over Fraction Field of Univariate
         Polynomial Ring in t over Finite Field of size 7
         defined by (-3*t)*x^2 + 2*y^2 + (3*t + 3)/t*z^2
   Defn: Defined on coordinates by sending (x : y : z) to (x + 1/t*z : y : z))

K = FractionField(PolynomialRing(GF(7), 't'))
(t,) = K.gens()
C = Conic(K, [t/2,0, 1, 2, 0, 3])
C.diagonalization()
gens()[source]

Return the generators of the coordinate ring of self.

EXAMPLES:

sage: P.<x,y,z> = QQ[]
sage: c = Conic(x^2 + y^2 + z^2)
sage: c.gens()                                                              # needs sage.libs.singular
(xbar, ybar, zbar)
sage: c.defining_polynomial()(c.gens())                                     # needs sage.libs.singular
0

>>> from sage.all import *
>>> P = QQ['x, y, z']; (x, y, z,) = P._first_ngens(3)
>>> c = Conic(x**Integer(2) + y**Integer(2) + z**Integer(2))
>>> c.gens()                                                              # needs sage.libs.singular
(xbar, ybar, zbar)
>>> c.defining_polynomial()(c.gens())                                     # needs sage.libs.singular
0

P.<x,y,z> = QQ[]
c = Conic(x^2 + y^2 + z^2)
c.gens()                                                              # needs sage.libs.singular
c.defining_polynomial()(c.gens())                                     # needs sage.libs.singular

The function gens() is required for the following construction:

sage: C.<a,b,c> = Conic(GF(3), [1, 1, 1]); C                                # needs sage.libs.singular
Projective Conic Curve over
 Finite Field of size 3 defined by a^2 + b^2 + c^2

>>> from sage.all import *
>>> C = Conic(GF(Integer(3)), [Integer(1), Integer(1), Integer(1)], names=('a', 'b', 'c',)); (a, b, c,) = C._first_ngens(3); C                                # needs sage.libs.singular
Projective Conic Curve over
 Finite Field of size 3 defined by a^2 + b^2 + c^2

C.<a,b,c> = Conic(GF(3), [1, 1, 1]); C                                # needs sage.libs.singular
has_rational_point(point=False, algorithm='default', read_cache=True)[source]

Return True if and only if the conic self has a point over its base field \(B\).

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

Points are cached whenever they are found. Cached information is used if and only if read_cache is True.

ALGORITHM:

The parameter algorithm specifies the algorithm to be used:

  • 'default' – if the base field is real or complex, use an elementary native Sage implementation

  • 'magma' (requires Magma to be installed) – delegates the task to the Magma computer algebra system

EXAMPLES:

sage: Conic(RR, [1, 1, 1]).has_rational_point()
False
sage: Conic(CC, [1, 1, 1]).has_rational_point()
True

sage: Conic(RR, [1, 2, -3]).has_rational_point(point = True)
(True, (1.73205080756888 : 0.000000000000000 : 1.00000000000000))

>>> from sage.all import *
>>> Conic(RR, [Integer(1), Integer(1), Integer(1)]).has_rational_point()
False
>>> Conic(CC, [Integer(1), Integer(1), Integer(1)]).has_rational_point()
True

>>> Conic(RR, [Integer(1), Integer(2), -Integer(3)]).has_rational_point(point = True)
(True, (1.73205080756888 : 0.000000000000000 : 1.00000000000000))

Conic(RR, [1, 1, 1]).has_rational_point()
Conic(CC, [1, 1, 1]).has_rational_point()
Conic(RR, [1, 2, -3]).has_rational_point(point = True)

Conics over polynomial rings can be solved internally:

sage: R.<t> = QQ[]
sage: C = Conic([-2, t^2 + 1, t^2 - 1])
sage: C.has_rational_point()                                                # needs sage.libs.pari
True

>>> from sage.all import *
>>> R = QQ['t']; (t,) = R._first_ngens(1)
>>> C = Conic([-Integer(2), t**Integer(2) + Integer(1), t**Integer(2) - Integer(1)])
>>> C.has_rational_point()                                                # needs sage.libs.pari
True

R.<t> = QQ[]
C = Conic([-2, t^2 + 1, t^2 - 1])
C.has_rational_point()                                                # needs sage.libs.pari

And they can also be solved with Magma:

sage: C.has_rational_point(algorithm='magma')               # optional - magma
True
sage: C.has_rational_point(algorithm='magma', point=True)   # optional - magma
(True, (-t : 1 : 1))

sage: D = Conic([t, 1, t^2])
sage: D.has_rational_point(algorithm='magma')               # optional - magma
False

>>> from sage.all import *
>>> C.has_rational_point(algorithm='magma')               # optional - magma
True
>>> C.has_rational_point(algorithm='magma', point=True)   # optional - magma
(True, (-t : 1 : 1))

>>> D = Conic([t, Integer(1), t**Integer(2)])
>>> D.has_rational_point(algorithm='magma')               # optional - magma
False

C.has_rational_point(algorithm='magma')               # optional - magma
C.has_rational_point(algorithm='magma', point=True)   # optional - magma
D = Conic([t, 1, t^2])
D.has_rational_point(algorithm='magma')               # optional - magma
has_singular_point(point=False)[source]

Return True if and only if the conic self has a rational singular point.

If point is True, then also return a rational singular point (or None if no such point exists).

EXAMPLES:

sage: c = Conic(QQ, [1,0,1]); c
Projective Conic Curve over Rational Field defined by x^2 + z^2
sage: c.has_singular_point(point = True)
(True, (0 : 1 : 0))

sage: P.<x,y,z> = GF(7)[]
sage: e = Conic((x+y+z)*(x-y+2*z)); e
Projective Conic Curve over Finite Field of size 7
 defined by x^2 - y^2 + 3*x*z + y*z + 2*z^2
sage: e.has_singular_point(point = True)
(True, (2 : 4 : 1))

sage: Conic([1, 1, -1]).has_singular_point()
False
sage: Conic([1, 1, -1]).has_singular_point(point=True)
(False, None)

>>> from sage.all import *
>>> c = Conic(QQ, [Integer(1),Integer(0),Integer(1)]); c
Projective Conic Curve over Rational Field defined by x^2 + z^2
>>> c.has_singular_point(point = True)
(True, (0 : 1 : 0))

>>> P = GF(Integer(7))['x, y, z']; (x, y, z,) = P._first_ngens(3)
>>> e = Conic((x+y+z)*(x-y+Integer(2)*z)); e
Projective Conic Curve over Finite Field of size 7
 defined by x^2 - y^2 + 3*x*z + y*z + 2*z^2
>>> e.has_singular_point(point = True)
(True, (2 : 4 : 1))

>>> Conic([Integer(1), Integer(1), -Integer(1)]).has_singular_point()
False
>>> Conic([Integer(1), Integer(1), -Integer(1)]).has_singular_point(point=True)
(False, None)

c = Conic(QQ, [1,0,1]); c
c.has_singular_point(point = True)
P.<x,y,z> = GF(7)[]
e = Conic((x+y+z)*(x-y+2*z)); e
e.has_singular_point(point = True)
Conic([1, 1, -1]).has_singular_point()
Conic([1, 1, -1]).has_singular_point(point=True)

has_singular_point is not implemented over all fields of characteristic \(2\). It is implemented over finite fields.

sage: F.<a> = FiniteField(8)                                                # needs sage.rings.finite_rings
sage: Conic([a, a + 1, 1]).has_singular_point(point=True)                   # needs sage.rings.finite_rings
(True, (a + 1 : 0 : 1))

sage: P.<t> = GF(2)[]
sage: C = Conic(P, [t,t,1]); C
Projective Conic Curve over Fraction Field of Univariate Polynomial Ring
 in t over Finite Field of size 2... defined by t*x^2 + t*y^2 + z^2
sage: C.has_singular_point(point=False)
Traceback (most recent call last):
...
NotImplementedError: Sorry, find singular point on conics not implemented
over all fields of characteristic 2.

>>> from sage.all import *
>>> F = FiniteField(Integer(8), names=('a',)); (a,) = F._first_ngens(1)# needs sage.rings.finite_rings
>>> Conic([a, a + Integer(1), Integer(1)]).has_singular_point(point=True)                   # needs sage.rings.finite_rings
(True, (a + 1 : 0 : 1))

>>> P = GF(Integer(2))['t']; (t,) = P._first_ngens(1)
>>> C = Conic(P, [t,t,Integer(1)]); C
Projective Conic Curve over Fraction Field of Univariate Polynomial Ring
 in t over Finite Field of size 2... defined by t*x^2 + t*y^2 + z^2
>>> C.has_singular_point(point=False)
Traceback (most recent call last):
...
NotImplementedError: Sorry, find singular point on conics not implemented
over all fields of characteristic 2.

F.<a> = FiniteField(8)                                                # needs sage.rings.finite_rings
Conic([a, a + 1, 1]).has_singular_point(point=True)                   # needs sage.rings.finite_rings
P.<t> = GF(2)[]
C = Conic(P, [t,t,1]); C
C.has_singular_point(point=False)
hom(x, Y=None)[source]

Return the scheme morphism from self to Y defined by x. Here x can be a matrix or a sequence of polynomials. If Y is omitted, then a natural image is found if possible.

EXAMPLES:

Here are a few morphisms given by matrices. In the first example, Y is omitted, in the second example, Y is specified.

sage: c = Conic([-1, 1, 1])
sage: h = c.hom(Matrix([[1,1,0],[0,1,0],[0,0,1]])); h
Scheme morphism:
  From: Projective Conic Curve over Rational Field defined by -x^2 + y^2 + z^2
  To:   Projective Conic Curve over Rational Field defined by -x^2 + 2*x*y + z^2
  Defn: Defined on coordinates by sending (x : y : z) to (x + y : y : z)
sage: h([-1, 1, 0])                                                         # needs sage.libs.singular
(0 : 1 : 0)

sage: c = Conic([-1, 1, 1])
sage: d = Conic([4, 1, -1])
sage: c.hom(Matrix([[0, 0, 1/2], [0, 1, 0], [1, 0, 0]]), d)
Scheme morphism:
  From: Projective Conic Curve over Rational Field defined by -x^2 + y^2 + z^2
  To:   Projective Conic Curve over Rational Field defined by 4*x^2 + y^2 - z^2
  Defn: Defined on coordinates by sending (x : y : z) to (1/2*z : y : x)

>>> from sage.all import *
>>> c = Conic([-Integer(1), Integer(1), Integer(1)])
>>> h = c.hom(Matrix([[Integer(1),Integer(1),Integer(0)],[Integer(0),Integer(1),Integer(0)],[Integer(0),Integer(0),Integer(1)]])); h
Scheme morphism:
  From: Projective Conic Curve over Rational Field defined by -x^2 + y^2 + z^2
  To:   Projective Conic Curve over Rational Field defined by -x^2 + 2*x*y + z^2
  Defn: Defined on coordinates by sending (x : y : z) to (x + y : y : z)
>>> h([-Integer(1), Integer(1), Integer(0)])                                                         # needs sage.libs.singular
(0 : 1 : 0)

>>> c = Conic([-Integer(1), Integer(1), Integer(1)])
>>> d = Conic([Integer(4), Integer(1), -Integer(1)])
>>> c.hom(Matrix([[Integer(0), Integer(0), Integer(1)/Integer(2)], [Integer(0), Integer(1), Integer(0)], [Integer(1), Integer(0), Integer(0)]]), d)
Scheme morphism:
  From: Projective Conic Curve over Rational Field defined by -x^2 + y^2 + z^2
  To:   Projective Conic Curve over Rational Field defined by 4*x^2 + y^2 - z^2
  Defn: Defined on coordinates by sending (x : y : z) to (1/2*z : y : x)

c = Conic([-1, 1, 1])
h = c.hom(Matrix([[1,1,0],[0,1,0],[0,0,1]])); h
h([-1, 1, 0])                                                         # needs sage.libs.singular
c = Conic([-1, 1, 1])
d = Conic([4, 1, -1])
c.hom(Matrix([[0, 0, 1/2], [0, 1, 0], [1, 0, 0]]), d)

ValueError is raised if the wrong codomain Y is specified:

sage: c = Conic([-1, 1, 1])
sage: c.hom(Matrix([[0, 0, 1/2], [0, 1, 0], [1, 0, 0]]), c)
Traceback (most recent call last):
...
ValueError: The matrix x (= [  0   0 1/2]
                            [  0   1   0]
                            [  1   0   0]) does not define a map
from self (= Projective Conic Curve over Rational Field defined by -x^2 + y^2 + z^2)
to Y (= Projective Conic Curve over Rational Field defined by -x^2 + y^2 + z^2)

>>> from sage.all import *
>>> c = Conic([-Integer(1), Integer(1), Integer(1)])
>>> c.hom(Matrix([[Integer(0), Integer(0), Integer(1)/Integer(2)], [Integer(0), Integer(1), Integer(0)], [Integer(1), Integer(0), Integer(0)]]), c)
Traceback (most recent call last):
...
ValueError: The matrix x (= [  0   0 1/2]
                            [  0   1   0]
                            [  1   0   0]) does not define a map
from self (= Projective Conic Curve over Rational Field defined by -x^2 + y^2 + z^2)
to Y (= Projective Conic Curve over Rational Field defined by -x^2 + y^2 + z^2)

c = Conic([-1, 1, 1])
c.hom(Matrix([[0, 0, 1/2], [0, 1, 0], [1, 0, 0]]), c)

The identity map between two representations of the same conic:

sage: C = Conic([1,2,3,4,5,6])
sage: D = Conic([2,4,6,8,10,12])
sage: C.hom(identity_matrix(3), D)
Scheme morphism:
  From: Projective Conic Curve over Rational Field
        defined by x^2 + 2*x*y + 4*y^2 + 3*x*z + 5*y*z + 6*z^2
  To:   Projective Conic Curve over Rational Field
        defined by 2*x^2 + 4*x*y + 8*y^2 + 6*x*z + 10*y*z + 12*z^2
  Defn: Defined on coordinates by sending (x : y : z) to (x : y : z)

>>> from sage.all import *
>>> C = Conic([Integer(1),Integer(2),Integer(3),Integer(4),Integer(5),Integer(6)])
>>> D = Conic([Integer(2),Integer(4),Integer(6),Integer(8),Integer(10),Integer(12)])
>>> C.hom(identity_matrix(Integer(3)), D)
Scheme morphism:
  From: Projective Conic Curve over Rational Field
        defined by x^2 + 2*x*y + 4*y^2 + 3*x*z + 5*y*z + 6*z^2
  To:   Projective Conic Curve over Rational Field
        defined by 2*x^2 + 4*x*y + 8*y^2 + 6*x*z + 10*y*z + 12*z^2
  Defn: Defined on coordinates by sending (x : y : z) to (x : y : z)

C = Conic([1,2,3,4,5,6])
D = Conic([2,4,6,8,10,12])
C.hom(identity_matrix(3), D)

An example not over the rational numbers:

sage: P.<t> = QQ[]
sage: C = Conic([1,0,0,t,0,1/t])
sage: D = Conic([1/t^2, 0, -2/t^2, t, 0, (t + 1)/t^2])
sage: T = Matrix([[t,0,1], [0,1,0], [0,0,1]])
sage: C.hom(T, D)
Scheme morphism:
  From: Projective Conic Curve over Fraction Field of Univariate
        Polynomial Ring in t over Rational Field defined by x^2 + t*y^2 + 1/t*z^2
  To:   Projective Conic Curve over Fraction Field of Univariate
        Polynomial Ring in t over Rational Field defined by
        1/(t^2)*x^2 + t*y^2 - 2/(t^2)*x*z + (t + 1)/(t^2)*z^2
  Defn: Defined on coordinates by sending (x : y : z) to (t*x + z : y : z)

>>> from sage.all import *
>>> P = QQ['t']; (t,) = P._first_ngens(1)
>>> C = Conic([Integer(1),Integer(0),Integer(0),t,Integer(0),Integer(1)/t])
>>> D = Conic([Integer(1)/t**Integer(2), Integer(0), -Integer(2)/t**Integer(2), t, Integer(0), (t + Integer(1))/t**Integer(2)])
>>> T = Matrix([[t,Integer(0),Integer(1)], [Integer(0),Integer(1),Integer(0)], [Integer(0),Integer(0),Integer(1)]])
>>> C.hom(T, D)
Scheme morphism:
  From: Projective Conic Curve over Fraction Field of Univariate
        Polynomial Ring in t over Rational Field defined by x^2 + t*y^2 + 1/t*z^2
  To:   Projective Conic Curve over Fraction Field of Univariate
        Polynomial Ring in t over Rational Field defined by
        1/(t^2)*x^2 + t*y^2 - 2/(t^2)*x*z + (t + 1)/(t^2)*z^2
  Defn: Defined on coordinates by sending (x : y : z) to (t*x + z : y : z)

P.<t> = QQ[]
C = Conic([1,0,0,t,0,1/t])
D = Conic([1/t^2, 0, -2/t^2, t, 0, (t + 1)/t^2])
T = Matrix([[t,0,1], [0,1,0], [0,0,1]])
C.hom(T, D)
is_diagonal()[source]

Return True if and only if the conic has the form \(a x^2 + b y^2 + c z^2\).

EXAMPLES:

sage: c = Conic([1,1,0,1,0,1]); c
Projective Conic Curve over Rational Field defined by x^2 + x*y + y^2 + z^2
sage: d, t = c.diagonal_matrix()
sage: c.is_diagonal()
False
sage: c.diagonalization()[0].is_diagonal()
True

>>> from sage.all import *
>>> c = Conic([Integer(1),Integer(1),Integer(0),Integer(1),Integer(0),Integer(1)]); c
Projective Conic Curve over Rational Field defined by x^2 + x*y + y^2 + z^2
>>> d, t = c.diagonal_matrix()
>>> c.is_diagonal()
False
>>> c.diagonalization()[Integer(0)].is_diagonal()
True

c = Conic([1,1,0,1,0,1]); c
d, t = c.diagonal_matrix()
c.is_diagonal()
c.diagonalization()[0].is_diagonal()
is_smooth()[source]

Return True if and only if self is smooth.

EXAMPLES:

sage: Conic([1,-1,0]).is_smooth()
False
sage: Conic(GF(2),[1,1,1,1,1,0]).is_smooth()
True

>>> from sage.all import *
>>> Conic([Integer(1),-Integer(1),Integer(0)]).is_smooth()
False
>>> Conic(GF(Integer(2)),[Integer(1),Integer(1),Integer(1),Integer(1),Integer(1),Integer(0)]).is_smooth()
True

Conic([1,-1,0]).is_smooth()
Conic(GF(2),[1,1,1,1,1,0]).is_smooth()
matrix()[source]

Return a matrix \(M\) such that \((x, y, z) M (x, y, z)^t\) is the defining equation of self.

The matrix \(M\) is upper triangular if the base field has characteristic \(2\) and symmetric otherwise.

EXAMPLES:

sage: R.<x, y, z> = QQ[]
sage: C = Conic(x^2 + x*y + y^2 + z^2)
sage: C.matrix()
[  1 1/2   0]
[1/2   1   0]
[  0   0   1]

sage: R.<x, y, z> = GF(2)[]
sage: C = Conic(x^2 + x*y + y^2 + x*z + z^2)
sage: C.matrix()
[1 1 1]
[0 1 0]
[0 0 1]

>>> from sage.all import *
>>> R = QQ['x, y, z']; (x, y, z,) = R._first_ngens(3)
>>> C = Conic(x**Integer(2) + x*y + y**Integer(2) + z**Integer(2))
>>> C.matrix()
[  1 1/2   0]
[1/2   1   0]
[  0   0   1]

>>> R = GF(Integer(2))['x, y, z']; (x, y, z,) = R._first_ngens(3)
>>> C = Conic(x**Integer(2) + x*y + y**Integer(2) + x*z + z**Integer(2))
>>> C.matrix()
[1 1 1]
[0 1 0]
[0 0 1]

R.<x, y, z> = QQ[]
C = Conic(x^2 + x*y + y^2 + z^2)
C.matrix()
R.<x, y, z> = GF(2)[]
C = Conic(x^2 + x*y + y^2 + x*z + z^2)
C.matrix()
parametrization(point=None, morphism=True)[source]

Return a parametrization \(f\) of self together with the inverse of \(f\).

If point is specified, then that point is used for the parametrization. Otherwise, use rational_point() to find a point.

If morphism is True, then \(f\) is returned in the form of a Scheme morphism. Otherwise, it is a tuple of polynomials that gives the parametrization.

EXAMPLES:

An example over a finite field

sage: # needs sage.libs.pari
sage: c = Conic(GF(2), [1,1,1,1,1,0])
sage: f, g = c.parametrization(); f, g
(Scheme morphism:
  From: Projective Space of dimension 1 over Finite Field of size 2
  To:   Projective Conic Curve over Finite Field of size 2
        defined by x^2 + x*y + y^2 + x*z + y*z
  Defn: Defined on coordinates by sending (x : y) to ...,
 Scheme morphism:
  From: Projective Conic Curve over Finite Field of size 2
        defined by x^2 + x*y + y^2 + x*z + y*z
  To:   Projective Space of dimension 1 over Finite Field of size 2
  Defn: Defined on coordinates by sending (x : y : z) to ...)
sage: set(f(p) for p in f.domain())
{(0 : 0 : 1), (0 : 1 : 1), (1 : 0 : 1)}

>>> from sage.all import *
>>> # needs sage.libs.pari
>>> c = Conic(GF(Integer(2)), [Integer(1),Integer(1),Integer(1),Integer(1),Integer(1),Integer(0)])
>>> f, g = c.parametrization(); f, g
(Scheme morphism:
  From: Projective Space of dimension 1 over Finite Field of size 2
  To:   Projective Conic Curve over Finite Field of size 2
        defined by x^2 + x*y + y^2 + x*z + y*z
  Defn: Defined on coordinates by sending (x : y) to ...,
 Scheme morphism:
  From: Projective Conic Curve over Finite Field of size 2
        defined by x^2 + x*y + y^2 + x*z + y*z
  To:   Projective Space of dimension 1 over Finite Field of size 2
  Defn: Defined on coordinates by sending (x : y : z) to ...)
>>> set(f(p) for p in f.domain())
{(0 : 0 : 1), (0 : 1 : 1), (1 : 0 : 1)}

# needs sage.libs.pari
c = Conic(GF(2), [1,1,1,1,1,0])
f, g = c.parametrization(); f, g
set(f(p) for p in f.domain())

Verfication of the example

sage: # needs sage.libs.pari
sage: h = g*f; h
Scheme endomorphism of Projective Space of dimension 1
 over Finite Field of size 2
  Defn: Defined on coordinates by sending (x : y) to ...
sage: h[0]/h[1]
x/y
sage: h.is_one()                    # known bug (see :issue:`31892`)
True
sage: (x,y,z) = c.gens()
sage: x.parent()
Quotient of Multivariate Polynomial Ring in x, y, z
 over Finite Field of size 2 by the ideal (x^2 + x*y + y^2 + x*z + y*z)
sage: k = f*g
sage: k[0]*z-k[2]*x
0
sage: k[1]*z-k[2]*y
0

>>> from sage.all import *
>>> # needs sage.libs.pari
>>> h = g*f; h
Scheme endomorphism of Projective Space of dimension 1
 over Finite Field of size 2
  Defn: Defined on coordinates by sending (x : y) to ...
>>> h[Integer(0)]/h[Integer(1)]
x/y
>>> h.is_one()                    # known bug (see :issue:`31892`)
True
>>> (x,y,z) = c.gens()
>>> x.parent()
Quotient of Multivariate Polynomial Ring in x, y, z
 over Finite Field of size 2 by the ideal (x^2 + x*y + y^2 + x*z + y*z)
>>> k = f*g
>>> k[Integer(0)]*z-k[Integer(2)]*x
0
>>> k[Integer(1)]*z-k[Integer(2)]*y
0

# needs sage.libs.pari
h = g*f; h
h[0]/h[1]
h.is_one()                    # known bug (see :issue:`31892`)
(x,y,z) = c.gens()
x.parent()
k = f*g
k[0]*z-k[2]*x
k[1]*z-k[2]*y

The morphisms are mathematically defined in all points, but don’t work completely in SageMath (see Issue #31892)

sage: # needs sage.libs.pari
sage: f, g = c.parametrization([0,0,1])
sage: g([0,1,1])
(1 : 0)
sage: f([1,0])
(0 : 1 : 1)
sage: f([1,1])
(0 : 0 : 1)
sage: g([0,0,1])
(1 : 1)

>>> from sage.all import *
>>> # needs sage.libs.pari
>>> f, g = c.parametrization([Integer(0),Integer(0),Integer(1)])
>>> g([Integer(0),Integer(1),Integer(1)])
(1 : 0)
>>> f([Integer(1),Integer(0)])
(0 : 1 : 1)
>>> f([Integer(1),Integer(1)])
(0 : 0 : 1)
>>> g([Integer(0),Integer(0),Integer(1)])
(1 : 1)

# needs sage.libs.pari
f, g = c.parametrization([0,0,1])
g([0,1,1])
f([1,0])
f([1,1])
g([0,0,1])

An example with morphism = False

sage: # needs sage.libs.pari
sage: R.<x,y,z> = QQ[]
sage: C = Curve(7*x^2 + 2*y*z + z^2)
sage: (p, i) = C.parametrization(morphism=False); (p, i)
([-2*x*y, x^2 + 7*y^2, -2*x^2], [-1/2*x, 1/7*y + 1/14*z])
sage: C.defining_polynomial()(p)
0
sage: i[0](p) / i[1](p)
x/y

>>> from sage.all import *
>>> # needs sage.libs.pari
>>> R = QQ['x, y, z']; (x, y, z,) = R._first_ngens(3)
>>> C = Curve(Integer(7)*x**Integer(2) + Integer(2)*y*z + z**Integer(2))
>>> (p, i) = C.parametrization(morphism=False); (p, i)
([-2*x*y, x^2 + 7*y^2, -2*x^2], [-1/2*x, 1/7*y + 1/14*z])
>>> C.defining_polynomial()(p)
0
>>> i[Integer(0)](p) / i[Integer(1)](p)
x/y

# needs sage.libs.pari
R.<x,y,z> = QQ[]
C = Curve(7*x^2 + 2*y*z + z^2)
(p, i) = C.parametrization(morphism=False); (p, i)
C.defining_polynomial()(p)
i[0](p) / i[1](p)

A ValueError is raised if self has no rational point

sage: # needs sage.libs.pari
sage: C = Conic(x^2 + y^2 + 7*z^2)
sage: C.parametrization()
Traceback (most recent call last):
...
ValueError: Conic Projective Conic Curve over Rational Field defined by
x^2 + y^2 + 7*z^2 has no rational points over Rational Field!

>>> from sage.all import *
>>> # needs sage.libs.pari
>>> C = Conic(x**Integer(2) + y**Integer(2) + Integer(7)*z**Integer(2))
>>> C.parametrization()
Traceback (most recent call last):
...
ValueError: Conic Projective Conic Curve over Rational Field defined by
x^2 + y^2 + 7*z^2 has no rational points over Rational Field!

# needs sage.libs.pari
C = Conic(x^2 + y^2 + 7*z^2)
C.parametrization()

A ValueError is raised if self is not smooth

sage: # needs sage.libs.pari
sage: C = Conic(x^2 + y^2)
sage: C.parametrization()
Traceback (most recent call last):
...
ValueError: The conic self (=Projective Conic Curve over Rational Field
defined by x^2 + y^2) is not smooth, hence does not have a parametrization.

>>> from sage.all import *
>>> # needs sage.libs.pari
>>> C = Conic(x**Integer(2) + y**Integer(2))
>>> C.parametrization()
Traceback (most recent call last):
...
ValueError: The conic self (=Projective Conic Curve over Rational Field
defined by x^2 + y^2) is not smooth, hence does not have a parametrization.

# needs sage.libs.pari
C = Conic(x^2 + y^2)
C.parametrization()
point(v, check=True)[source]

Construct a point on self corresponding to the input v.

If check is True, then checks if v defines a valid point on self.

If no rational point on self is known yet, then also caches the point for use by rational_point() and parametrization().

EXAMPLES:

sage: c = Conic([1, -1, 1])
sage: c.point([15, 17, 8])
(15/8 : 17/8 : 1)
sage: c.rational_point()
(15/8 : 17/8 : 1)
sage: d = Conic([1, -1, 1])
sage: d.rational_point()                                                    # needs sage.libs.pari
(-1 : 1 : 0)

>>> from sage.all import *
>>> c = Conic([Integer(1), -Integer(1), Integer(1)])
>>> c.point([Integer(15), Integer(17), Integer(8)])
(15/8 : 17/8 : 1)
>>> c.rational_point()
(15/8 : 17/8 : 1)
>>> d = Conic([Integer(1), -Integer(1), Integer(1)])
>>> d.rational_point()                                                    # needs sage.libs.pari
(-1 : 1 : 0)

c = Conic([1, -1, 1])
c.point([15, 17, 8])
c.rational_point()
d = Conic([1, -1, 1])
d.rational_point()                                                    # needs sage.libs.pari
random_rational_point(*args1, **args2)[source]

Return a random rational point of the conic self.

ALGORITHM:

  1. Compute a parametrization \(f\) of self using parametrization().

  2. Computes a random point \((x:y)\) on the projective line.

  3. Output \(f(x:y)\).

The coordinates \(x\) and \(y\) are computed using B.random_element, where B is the base field of self and additional arguments to random_rational_point are passed to random_element.

If the base field is a finite field, then the output is uniformly distributed over the points of self.

EXAMPLES:

sage: # needs sage.libs.pari
sage: c = Conic(GF(2), [1,1,1,1,1,0])
sage: [c.random_rational_point() for i in range(10)]              # random
[(1 : 0 : 1), (1 : 0 : 1), (1 : 0 : 1), (0 : 1 : 1), (1 : 0 : 1),
 (0 : 0 : 1), (1 : 0 : 1), (1 : 0 : 1), (0 : 0 : 1), (1 : 0 : 1)]
sage: d = Conic(QQ, [1, 1, -1])
sage: d.random_rational_point(den_bound=1, num_bound=5)           # random
(-24/25 : 7/25 : 1)
sage: Conic(QQ, [1, 1, 1]).random_rational_point()
Traceback (most recent call last):
...
ValueError: Conic Projective Conic Curve over Rational Field defined by
x^2 + y^2 + z^2 has no rational points over Rational Field!

>>> from sage.all import *
>>> # needs sage.libs.pari
>>> c = Conic(GF(Integer(2)), [Integer(1),Integer(1),Integer(1),Integer(1),Integer(1),Integer(0)])
>>> [c.random_rational_point() for i in range(Integer(10))]              # random
[(1 : 0 : 1), (1 : 0 : 1), (1 : 0 : 1), (0 : 1 : 1), (1 : 0 : 1),
 (0 : 0 : 1), (1 : 0 : 1), (1 : 0 : 1), (0 : 0 : 1), (1 : 0 : 1)]
>>> d = Conic(QQ, [Integer(1), Integer(1), -Integer(1)])
>>> d.random_rational_point(den_bound=Integer(1), num_bound=Integer(5))           # random
(-24/25 : 7/25 : 1)
>>> Conic(QQ, [Integer(1), Integer(1), Integer(1)]).random_rational_point()
Traceback (most recent call last):
...
ValueError: Conic Projective Conic Curve over Rational Field defined by
x^2 + y^2 + z^2 has no rational points over Rational Field!

# needs sage.libs.pari
c = Conic(GF(2), [1,1,1,1,1,0])
[c.random_rational_point() for i in range(10)]              # random
d = Conic(QQ, [1, 1, -1])
d.random_rational_point(den_bound=1, num_bound=5)           # random
Conic(QQ, [1, 1, 1]).random_rational_point()
rational_point(algorithm='default', read_cache=True)[source]

Return a point on self defined over the base field.

This raises a ValueError if no rational point exists.

See self.has_rational_point for the algorithm used and for the use of the parameters algorithm and read_cache.

EXAMPLES:

Examples over \(\QQ\)

sage: R.<x,y,z> = QQ[]

sage: # needs sage.libs.pari
sage: C = Conic(7*x^2 + 2*y*z + z^2)
sage: C.rational_point()
(0 : 1 : 0)
sage: C = Conic(x^2 + 2*y^2 + z^2)
sage: C.rational_point()
Traceback (most recent call last):
...
ValueError: Conic Projective Conic Curve over Rational Field defined by
x^2 + 2*y^2 + z^2 has no rational points over Rational Field!

sage: C = Conic(x^2 + y^2 + 7*z^2)
sage: C.rational_point(algorithm='rnfisnorm')
Traceback (most recent call last):
...
ValueError: Conic Projective Conic Curve over Rational Field defined by
x^2 + y^2 + 7*z^2 has no rational points over Rational Field!

>>> from sage.all import *
>>> R = QQ['x, y, z']; (x, y, z,) = R._first_ngens(3)

>>> # needs sage.libs.pari
>>> C = Conic(Integer(7)*x**Integer(2) + Integer(2)*y*z + z**Integer(2))
>>> C.rational_point()
(0 : 1 : 0)
>>> C = Conic(x**Integer(2) + Integer(2)*y**Integer(2) + z**Integer(2))
>>> C.rational_point()
Traceback (most recent call last):
...
ValueError: Conic Projective Conic Curve over Rational Field defined by
x^2 + 2*y^2 + z^2 has no rational points over Rational Field!

>>> C = Conic(x**Integer(2) + y**Integer(2) + Integer(7)*z**Integer(2))
>>> C.rational_point(algorithm='rnfisnorm')
Traceback (most recent call last):
...
ValueError: Conic Projective Conic Curve over Rational Field defined by
x^2 + y^2 + 7*z^2 has no rational points over Rational Field!

R.<x,y,z> = QQ[]
# needs sage.libs.pari
C = Conic(7*x^2 + 2*y*z + z^2)
C.rational_point()
C = Conic(x^2 + 2*y^2 + z^2)
C.rational_point()
C = Conic(x^2 + y^2 + 7*z^2)
C.rational_point(algorithm='rnfisnorm')

Examples over number fields

sage: # needs sage.rings.number_field
sage: P.<x> = QQ[]
sage: L.<b> = NumberField(x^3 - 5)
sage: C = Conic(L, [3, 2, -b])
sage: p = C.rational_point(algorithm='rnfisnorm')
sage: p                                         # output is random
(1/3*b^2 - 4/3*b + 4/3 : b^2 - 2 : 1)
sage: C.defining_polynomial()(list(p))
0

sage: K.<i> = QuadraticField(-1)                                            # needs sage.rings.number_field
sage: D = Conic(K, [3, 2, 5])                                               # needs sage.rings.number_field
sage: D.rational_point(algorithm='rnfisnorm')   # output is random          # needs sage.rings.number_field
(-3 : 4*i : 1)

sage: # needs sage.libs.pari sage.rings.number_field
sage: L.<s> = QuadraticField(2)
sage: Conic(QQ, [1, 1, -3]).has_rational_point()
False
sage: E = Conic(L, [1, 1, -3])
sage: E.rational_point()                        # output is random
(-1 : -s : 1)

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> P = QQ['x']; (x,) = P._first_ngens(1)
>>> L = NumberField(x**Integer(3) - Integer(5), names=('b',)); (b,) = L._first_ngens(1)
>>> C = Conic(L, [Integer(3), Integer(2), -b])
>>> p = C.rational_point(algorithm='rnfisnorm')
>>> p                                         # output is random
(1/3*b^2 - 4/3*b + 4/3 : b^2 - 2 : 1)
>>> C.defining_polynomial()(list(p))
0

>>> K = QuadraticField(-Integer(1), names=('i',)); (i,) = K._first_ngens(1)# needs sage.rings.number_field
>>> D = Conic(K, [Integer(3), Integer(2), Integer(5)])                                               # needs sage.rings.number_field
>>> D.rational_point(algorithm='rnfisnorm')   # output is random          # needs sage.rings.number_field
(-3 : 4*i : 1)

>>> # needs sage.libs.pari sage.rings.number_field
>>> L = QuadraticField(Integer(2), names=('s',)); (s,) = L._first_ngens(1)
>>> Conic(QQ, [Integer(1), Integer(1), -Integer(3)]).has_rational_point()
False
>>> E = Conic(L, [Integer(1), Integer(1), -Integer(3)])
>>> E.rational_point()                        # output is random
(-1 : -s : 1)

# needs sage.rings.number_field
P.<x> = QQ[]
L.<b> = NumberField(x^3 - 5)
C = Conic(L, [3, 2, -b])
p = C.rational_point(algorithm='rnfisnorm')
p                                         # output is random
C.defining_polynomial()(list(p))
K.<i> = QuadraticField(-1)                                            # needs sage.rings.number_field
D = Conic(K, [3, 2, 5])                                               # needs sage.rings.number_field
D.rational_point(algorithm='rnfisnorm')   # output is random          # needs sage.rings.number_field
# needs sage.libs.pari sage.rings.number_field
L.<s> = QuadraticField(2)
Conic(QQ, [1, 1, -3]).has_rational_point()
E = Conic(L, [1, 1, -3])
E.rational_point()                        # output is random

Currently Magma is better at solving conics over number fields than Sage, so it helps to use the algorithm ‘magma’ if Magma is installed:

sage: # optional - magma, needs sage.rings.number_field
sage: q = C.rational_point(algorithm='magma',
....:                      read_cache=False)
sage: q                                         # output is random
(1/5*b^2 : 1/5*b^2 : 1)
sage: C.defining_polynomial()(list(q))
0
sage: len(str(p)) > 1.5*len(str(q))
True
sage: D.rational_point(algorithm='magma',       # random
....:                  read_cache=False)
(1 : 2*i : 1)
sage: E.rational_point(algorithm='magma',       # random
....:                  read_cache=False)
(-s : 1 : 1)

sage: # needs sage.libs.pari sage.rings.number_field
sage: F = Conic([L.gen(), 30, -20])
sage: q = F.rational_point(algorithm='magma')       # optional - magma
sage: q  # random                                   # optional - magma
(-10/7*s + 40/7 : 5/7*s - 6/7 : 1)
sage: p = F.rational_point(read_cache=False)
sage: p  # random
(788210*s - 1114700 : -171135*s + 242022 : 1)
sage: len(str(p)) > len(str(q))                     # optional - magma
True

sage: # needs sage.rings.number_field
sage: G = Conic([L.gen(), 30, -21])
sage: G.has_rational_point(algorithm='magma')       # optional - magma
False
sage: G.has_rational_point(read_cache=False)        # needs sage.libs.pari
False
sage: G.has_rational_point(algorithm='local',
....:                      read_cache=False)
False
sage: G.rational_point(algorithm='magma')           # optional - magma
Traceback (most recent call last):
...
ValueError: Conic Projective Conic Curve over Number Field in s
with defining polynomial x^2 - 2 with s = 1.414213562373095?
defined by s*x^2 + 30*y^2 - 21*z^2 has no rational points over
Number Field in s with defining polynomial x^2 - 2 with s = 1.414213562373095?!
sage: G.rational_point(algorithm='magma',           # optional - magma
....:                  read_cache=False)
Traceback (most recent call last):
...
ValueError: Conic Projective Conic Curve over Number Field in s
with defining polynomial x^2 - 2 with s = 1.414213562373095?
defined by s*x^2 + 30*y^2 - 21*z^2 has no rational points over
Number Field in s with defining polynomial x^2 - 2 with s = 1.414213562373095?!

>>> from sage.all import *
>>> # optional - magma, needs sage.rings.number_field
>>> q = C.rational_point(algorithm='magma',
...                      read_cache=False)
>>> q                                         # output is random
(1/5*b^2 : 1/5*b^2 : 1)
>>> C.defining_polynomial()(list(q))
0
>>> len(str(p)) > RealNumber('1.5')*len(str(q))
True
>>> D.rational_point(algorithm='magma',       # random
...                  read_cache=False)
(1 : 2*i : 1)
>>> E.rational_point(algorithm='magma',       # random
...                  read_cache=False)
(-s : 1 : 1)

>>> # needs sage.libs.pari sage.rings.number_field
>>> F = Conic([L.gen(), Integer(30), -Integer(20)])
>>> q = F.rational_point(algorithm='magma')       # optional - magma
>>> q  # random                                   # optional - magma
(-10/7*s + 40/7 : 5/7*s - 6/7 : 1)
>>> p = F.rational_point(read_cache=False)
>>> p  # random
(788210*s - 1114700 : -171135*s + 242022 : 1)
>>> len(str(p)) > len(str(q))                     # optional - magma
True

>>> # needs sage.rings.number_field
>>> G = Conic([L.gen(), Integer(30), -Integer(21)])
>>> G.has_rational_point(algorithm='magma')       # optional - magma
False
>>> G.has_rational_point(read_cache=False)        # needs sage.libs.pari
False
>>> G.has_rational_point(algorithm='local',
...                      read_cache=False)
False
>>> G.rational_point(algorithm='magma')           # optional - magma
Traceback (most recent call last):
...
ValueError: Conic Projective Conic Curve over Number Field in s
with defining polynomial x^2 - 2 with s = 1.414213562373095?
defined by s*x^2 + 30*y^2 - 21*z^2 has no rational points over
Number Field in s with defining polynomial x^2 - 2 with s = 1.414213562373095?!
>>> G.rational_point(algorithm='magma',           # optional - magma
...                  read_cache=False)
Traceback (most recent call last):
...
ValueError: Conic Projective Conic Curve over Number Field in s
with defining polynomial x^2 - 2 with s = 1.414213562373095?
defined by s*x^2 + 30*y^2 - 21*z^2 has no rational points over
Number Field in s with defining polynomial x^2 - 2 with s = 1.414213562373095?!

# optional - magma, needs sage.rings.number_field
q = C.rational_point(algorithm='magma',
                     read_cache=False)
q                                         # output is random
C.defining_polynomial()(list(q))
len(str(p)) > 1.5*len(str(q))
D.rational_point(algorithm='magma',       # random
                 read_cache=False)
E.rational_point(algorithm='magma',       # random
                 read_cache=False)
# needs sage.libs.pari sage.rings.number_field
F = Conic([L.gen(), 30, -20])
q = F.rational_point(algorithm='magma')       # optional - magma
q  # random                                   # optional - magma
p = F.rational_point(read_cache=False)
p  # random
len(str(p)) > len(str(q))                     # optional - magma
# needs sage.rings.number_field
G = Conic([L.gen(), 30, -21])
G.has_rational_point(algorithm='magma')       # optional - magma
G.has_rational_point(read_cache=False)        # needs sage.libs.pari
G.has_rational_point(algorithm='local',
                     read_cache=False)
G.rational_point(algorithm='magma')           # optional - magma
G.rational_point(algorithm='magma',           # optional - magma
                 read_cache=False)

Examples over finite fields

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

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

F.<a> = FiniteField(7^20)                                             # needs sage.rings.finite_rings
C = Conic([1, a, -5]); C                                              # needs sage.rings.finite_rings
C.rational_point()                        # output is random          # needs sage.rings.finite_rings

Examples over \(\RR\) and \(\CC\)

sage: Conic(CC, [1, 2, 3]).rational_point()
(0 : 1.22474487139159*I : 1)

sage: Conic(RR, [1, 1, 1]).rational_point()
Traceback (most recent call last):
...
ValueError: Conic Projective Conic Curve over Real Field
with 53 bits of precision defined by x^2 + y^2 + z^2 has
no rational points over Real Field with 53 bits of precision!

>>> from sage.all import *
>>> Conic(CC, [Integer(1), Integer(2), Integer(3)]).rational_point()
(0 : 1.22474487139159*I : 1)

>>> Conic(RR, [Integer(1), Integer(1), Integer(1)]).rational_point()
Traceback (most recent call last):
...
ValueError: Conic Projective Conic Curve over Real Field
with 53 bits of precision defined by x^2 + y^2 + z^2 has
no rational points over Real Field with 53 bits of precision!

Conic(CC, [1, 2, 3]).rational_point()
Conic(RR, [1, 1, 1]).rational_point()
singular_point()[source]

Return a singular rational point of self.

EXAMPLES:

sage: Conic(GF(2), [1,1,1,1,1,1]).singular_point()
(1 : 1 : 1)

>>> from sage.all import *
>>> Conic(GF(Integer(2)), [Integer(1),Integer(1),Integer(1),Integer(1),Integer(1),Integer(1)]).singular_point()
(1 : 1 : 1)

Conic(GF(2), [1,1,1,1,1,1]).singular_point()

ValueError is raised if the conic has no rational singular point

sage: Conic(QQ, [1,1,1,1,1,1]).singular_point()
Traceback (most recent call last):
...
ValueError: The conic self (= Projective Conic Curve over Rational Field
defined by x^2 + x*y + y^2 + x*z + y*z + z^2) has no rational singular point

>>> from sage.all import *
>>> Conic(QQ, [Integer(1),Integer(1),Integer(1),Integer(1),Integer(1),Integer(1)]).singular_point()
Traceback (most recent call last):
...
ValueError: The conic self (= Projective Conic Curve over Rational Field
defined by x^2 + x*y + y^2 + x*z + y*z + z^2) has no rational singular point

Conic(QQ, [1,1,1,1,1,1]).singular_point()
symmetric_matrix()[source]

The symmetric matrix \(M\) such that \((x y z) M (x y z)^t\) is the defining equation of self.

EXAMPLES:

sage: R.<x, y, z> = QQ[]
sage: C = Conic(x^2 + x*y/2 + y^2 + z^2)
sage: C.symmetric_matrix()
[  1 1/4   0]
[1/4   1   0]
[  0   0   1]

sage: C = Conic(x^2 + 2*x*y + y^2 + 3*x*z + z^2)
sage: v = vector([x, y, z])
sage: v * C.symmetric_matrix() * v
x^2 + 2*x*y + y^2 + 3*x*z + z^2

>>> from sage.all import *
>>> R = QQ['x, y, z']; (x, y, z,) = R._first_ngens(3)
>>> C = Conic(x**Integer(2) + x*y/Integer(2) + y**Integer(2) + z**Integer(2))
>>> C.symmetric_matrix()
[  1 1/4   0]
[1/4   1   0]
[  0   0   1]

>>> C = Conic(x**Integer(2) + Integer(2)*x*y + y**Integer(2) + Integer(3)*x*z + z**Integer(2))
>>> v = vector([x, y, z])
>>> v * C.symmetric_matrix() * v
x^2 + 2*x*y + y^2 + 3*x*z + z^2

R.<x, y, z> = QQ[]
C = Conic(x^2 + x*y/2 + y^2 + z^2)
C.symmetric_matrix()
C = Conic(x^2 + 2*x*y + y^2 + 3*x*z + z^2)
v = vector([x, y, z])
v * C.symmetric_matrix() * v
upper_triangular_matrix()[source]

The upper-triangular matrix \(M\) such that \((x y z) M (x y z)^t\) is the defining equation of self.

EXAMPLES:

sage: R.<x, y, z> = QQ[]
sage: C = Conic(x^2 + x*y + y^2 + z^2)
sage: C.upper_triangular_matrix()
[1 1 0]
[0 1 0]
[0 0 1]

sage: C = Conic(x^2 + 2*x*y + y^2 + 3*x*z + z^2)
sage: v = vector([x, y, z])
sage: v * C.upper_triangular_matrix() * v
x^2 + 2*x*y + y^2 + 3*x*z + z^2

>>> from sage.all import *
>>> R = QQ['x, y, z']; (x, y, z,) = R._first_ngens(3)
>>> C = Conic(x**Integer(2) + x*y + y**Integer(2) + z**Integer(2))
>>> C.upper_triangular_matrix()
[1 1 0]
[0 1 0]
[0 0 1]

>>> C = Conic(x**Integer(2) + Integer(2)*x*y + y**Integer(2) + Integer(3)*x*z + z**Integer(2))
>>> v = vector([x, y, z])
>>> v * C.upper_triangular_matrix() * v
x^2 + 2*x*y + y^2 + 3*x*z + z^2

R.<x, y, z> = QQ[]
C = Conic(x^2 + x*y + y^2 + z^2)
C.upper_triangular_matrix()
C = Conic(x^2 + 2*x*y + y^2 + 3*x*z + z^2)
v = vector([x, y, z])
v * C.upper_triangular_matrix() * v
variable_names()[source]

Return the variable names of the defining polynomial of self.

EXAMPLES:

sage: c = Conic([1,1,0,1,0,1], 'x,y,z')
sage: c.variable_names()
('x', 'y', 'z')
sage: c.variable_name()
'x'

>>> from sage.all import *
>>> c = Conic([Integer(1),Integer(1),Integer(0),Integer(1),Integer(0),Integer(1)], 'x,y,z')
>>> c.variable_names()
('x', 'y', 'z')
>>> c.variable_name()
'x'

c = Conic([1,1,0,1,0,1], 'x,y,z')
c.variable_names()
c.variable_name()

The function variable_names() is required for the following construction:

sage: C.<p,q,r> = Conic(QQ, [1, 1, 1]); C                                   # needs sage.libs.singular
Projective Conic Curve over Rational Field defined by p^2 + q^2 + r^2

>>> from sage.all import *
>>> C = Conic(QQ, [Integer(1), Integer(1), Integer(1)], names=('p', 'q', 'r',)); (p, q, r,) = C._first_ngens(3); C                                   # needs sage.libs.singular
Projective Conic Curve over Rational Field defined by p^2 + q^2 + r^2

C.<p,q,r> = Conic(QQ, [1, 1, 1]); C                                   # needs sage.libs.singular