Ternary quadratic form with integer coefficients¶

AUTHOR:

Gustavo Rama

Based in code of Gonzalo Tornaria

The form \(a\cdot x^2 + b\cdot y^2 + c\cdot z^2 + r\cdot yz + s\cdot xz + t\cdot xy\) is stored as a tuple (a, b, c, r, s, t) of integers.

class sage.quadratic_forms.ternary_qf.TernaryQF(v)[source]¶

Bases: SageObject

The TernaryQF class represents a quadratic form in 3 variables with coefficients in \(\ZZ\).

INPUT:

v – list or tuple of 6 entries: [a,b,c,r,s,t]

OUTPUT: the ternary quadratic form \(a\cdot x^2 + b\cdot y^2 + c\cdot z^2 + r\cdot y\cdot z + s\cdot x\cdot z + t\cdot x\cdot y\)

EXAMPLES:

Sage

sage: Q = TernaryQF([1, 2, 3, 4, 5, 6]); Q
Ternary quadratic form with integer coefficients:
[1 2 3]
[4 5 6]
sage: A = matrix(ZZ, 3, [1, -7, 1, 0, -2, 1, 0, -1, 0])
sage: Q(A)
Ternary quadratic form with integer coefficients:
[1 187 9]
[-85 8 -31]
sage: TestSuite(TernaryQF).run()

Python

>>> from sage.all import *
>>> Q = TernaryQF([Integer(1), Integer(2), Integer(3), Integer(4), Integer(5), Integer(6)]); Q
Ternary quadratic form with integer coefficients:
[1 2 3]
[4 5 6]
>>> A = matrix(ZZ, Integer(3), [Integer(1), -Integer(7), Integer(1), Integer(0), -Integer(2), Integer(1), Integer(0), -Integer(1), Integer(0)])
>>> Q(A)
Ternary quadratic form with integer coefficients:
[1 187 9]
[-85 8 -31]
>>> TestSuite(TernaryQF).run()

adjoint()[source]¶

Return the adjoint form associated to the given ternary quadratic form.

That is, the Hessian matrix of the adjoint form is twice the classical adjoint matrix of the Hessian matrix of the given form.

EXAMPLES:

Sage

sage: Q = TernaryQF([1, 1, 17, 0, 0, 1])
sage: Q.adjoint()
Ternary quadratic form with integer coefficients:
[68 68 3]
[0 0 -68]
sage: Q.adjoint().matrix() == 2*Q.matrix().adjoint_classical()
True

Python

>>> from sage.all import *
>>> Q = TernaryQF([Integer(1), Integer(1), Integer(17), Integer(0), Integer(0), Integer(1)])
>>> Q.adjoint()
Ternary quadratic form with integer coefficients:
[68 68 3]
[0 0 -68]
>>> Q.adjoint().matrix() == Integer(2)*Q.matrix().adjoint_classical()
True

automorphism_spin_norm(A)[source]¶

Return the spin norm of the automorphism \(A\).

EXAMPLES:

Sage

sage: Q = TernaryQF([9, 12, 30, -26, -28, 20])
sage: A = matrix(ZZ, 3, [9, 10, -10, -6, -7, 6, 2, 2, -3])
sage: A.det()
1
sage: Q(A) == Q
True
sage: Q.automorphism_spin_norm(A)
7

Python

>>> from sage.all import *
>>> Q = TernaryQF([Integer(9), Integer(12), Integer(30), -Integer(26), -Integer(28), Integer(20)])
>>> A = matrix(ZZ, Integer(3), [Integer(9), Integer(10), -Integer(10), -Integer(6), -Integer(7), Integer(6), Integer(2), Integer(2), -Integer(3)])
>>> A.det()
1
>>> Q(A) == Q
True
>>> Q.automorphism_spin_norm(A)
7

automorphism_symmetries(A)[source]¶

Given the automorphism \(A\), if \(A\) is the identity, return the empty list. Otherwise, return a list of two vectors \(v_1\), \(v_2\) such that the product of the symmetries of the ternary quadratic form given by the two vectors is \(A\).

EXAMPLES:

Sage

sage: Q = TernaryQF([9, 12, 30, -26, -28, 20])
sage: A = matrix(ZZ, 3, [9, 10, -10, -6, -7, 6, 2, 2, -3])
sage: Q(A) == Q
True
sage: v1, v2 = Q.automorphism_symmetries(A)
sage: v1, v2
((8, -6, 2), (1, -5/4, -1/4))
sage: A1 = Q.symmetry(v1)
sage: A1
[    9     9   -13]
[   -6 -23/4  39/4]
[    2   9/4  -9/4]
sage: A2 = Q.symmetry(v2)
sage: A2
[    1     1     3]
[    0  -1/4 -15/4]
[    0  -1/4   1/4]
sage: A1*A2 == A
True
sage: Q.automorphism_symmetries(identity_matrix(ZZ,3))
[]

Python

>>> from sage.all import *
>>> Q = TernaryQF([Integer(9), Integer(12), Integer(30), -Integer(26), -Integer(28), Integer(20)])
>>> A = matrix(ZZ, Integer(3), [Integer(9), Integer(10), -Integer(10), -Integer(6), -Integer(7), Integer(6), Integer(2), Integer(2), -Integer(3)])
>>> Q(A) == Q
True
>>> v1, v2 = Q.automorphism_symmetries(A)
>>> v1, v2
((8, -6, 2), (1, -5/4, -1/4))
>>> A1 = Q.symmetry(v1)
>>> A1
[    9     9   -13]
[   -6 -23/4  39/4]
[    2   9/4  -9/4]
>>> A2 = Q.symmetry(v2)
>>> A2
[    1     1     3]
[    0  -1/4 -15/4]
[    0  -1/4   1/4]
>>> A1*A2 == A
True
>>> Q.automorphism_symmetries(identity_matrix(ZZ,Integer(3)))
[]

automorphisms(slow=True)[source]¶

Return a list with the automorphisms of the definite ternary quadratic form.

EXAMPLES:

Sage

sage: Q = TernaryQF([1, 1, 7, 0, 0, 0])
sage: auts = Q.automorphisms(); auts
[
[-1  0  0]  [-1  0  0]  [ 0 -1  0]  [ 0 -1  0]  [ 0  1  0]  [ 0  1  0]
[ 0 -1  0]  [ 0  1  0]  [-1  0  0]  [ 1  0  0]  [-1  0  0]  [ 1  0  0]
[ 0  0  1], [ 0  0 -1], [ 0  0 -1], [ 0  0  1], [ 0  0  1], [ 0  0 -1],
[ 1  0  0]  [1 0 0]
[ 0 -1  0]  [0 1 0]
[ 0  0 -1], [0 0 1]
]
sage: all(Q == Q(A) for A in auts)
True
sage: Q = TernaryQF([3, 4, 5, 3, 3, 2])
sage: Q.automorphisms(slow=False)
[
[1 0 0]
[0 1 0]
[0 0 1]
]
sage: Q = TernaryQF([4, 2, 4, 3, -4, -5])
sage: auts = Q.automorphisms(slow=False)
sage: auts
[
[1 0 0]  [ 2 -1 -1]
[0 1 0]  [ 3 -2 -1]
[0 0 1], [ 0  0 -1]
]
sage: A = auts[1]
sage: Q(A) == Q
True
sage: Qr, M_red = Q.reduced_form_eisenstein()
sage: Qr
Ternary quadratic form with integer coefficients:
[1 2 3]
[-1 0 -1]
sage: Q(A*M_red) == Qr
True

Python

>>> from sage.all import *
>>> Q = TernaryQF([Integer(1), Integer(1), Integer(7), Integer(0), Integer(0), Integer(0)])
>>> auts = Q.automorphisms(); auts
[
[-1  0  0]  [-1  0  0]  [ 0 -1  0]  [ 0 -1  0]  [ 0  1  0]  [ 0  1  0]
[ 0 -1  0]  [ 0  1  0]  [-1  0  0]  [ 1  0  0]  [-1  0  0]  [ 1  0  0]
[ 0  0  1], [ 0  0 -1], [ 0  0 -1], [ 0  0  1], [ 0  0  1], [ 0  0 -1],
[ 1  0  0]  [1 0 0]
[ 0 -1  0]  [0 1 0]
[ 0  0 -1], [0 0 1]
]
>>> all(Q == Q(A) for A in auts)
True
>>> Q = TernaryQF([Integer(3), Integer(4), Integer(5), Integer(3), Integer(3), Integer(2)])
>>> Q.automorphisms(slow=False)
[
[1 0 0]
[0 1 0]
[0 0 1]
]
>>> Q = TernaryQF([Integer(4), Integer(2), Integer(4), Integer(3), -Integer(4), -Integer(5)])
>>> auts = Q.automorphisms(slow=False)
>>> auts
[
[1 0 0]  [ 2 -1 -1]
[0 1 0]  [ 3 -2 -1]
[0 0 1], [ 0  0 -1]
]
>>> A = auts[Integer(1)]
>>> Q(A) == Q
True
>>> Qr, M_red = Q.reduced_form_eisenstein()
>>> Qr
Ternary quadratic form with integer coefficients:
[1 2 3]
[-1 0 -1]
>>> Q(A*M_red) == Qr
True

basic_lemma(p)[source]¶

Find a number represented by self and coprime to the prime \(p\).

EXAMPLES:

Sage

sage: Q = TernaryQF([3, 3, 3, -2, 0, -1])
sage: Q.basic_lemma(3)
4

Python

>>> from sage.all import *
>>> Q = TernaryQF([Integer(3), Integer(3), Integer(3), -Integer(2), Integer(0), -Integer(1)])
>>> Q.basic_lemma(Integer(3))
4

coefficient(n)[source]¶

Return the \(n\)-th coefficient of the ternary quadratic form.

INPUT:

n – integer with \(0 \leq n \leq 5\)

EXAMPLES:

Sage

sage: Q = TernaryQF([1, 2, 3, 4, 5, 6]); Q
Ternary quadratic form with integer coefficients:
[1 2 3]
[4 5 6]
sage: Q.coefficient(2)
3
sage: Q.coefficient(5)
6

Python

>>> from sage.all import *
>>> Q = TernaryQF([Integer(1), Integer(2), Integer(3), Integer(4), Integer(5), Integer(6)]); Q
Ternary quadratic form with integer coefficients:
[1 2 3]
[4 5 6]
>>> Q.coefficient(Integer(2))
3
>>> Q.coefficient(Integer(5))
6

coefficients()[source]¶

Return the list of coefficients of the ternary quadratic form.

EXAMPLES:

Sage

sage: Q = TernaryQF([1, 2, 3, 4, 5, 6]); Q
Ternary quadratic form with integer coefficients:
[1 2 3]
[4 5 6]
sage: Q.coefficients()
(1, 2, 3, 4, 5, 6)

Python

>>> from sage.all import *
>>> Q = TernaryQF([Integer(1), Integer(2), Integer(3), Integer(4), Integer(5), Integer(6)]); Q
Ternary quadratic form with integer coefficients:
[1 2 3]
[4 5 6]
>>> Q.coefficients()
(1, 2, 3, 4, 5, 6)

content()[source]¶

Return the greatest common divisor of the coefficients of the given ternary quadratic form.

EXAMPLES:

Sage

sage: Q = TernaryQF([1, 1, 2, 0, 0, 0])
sage: Q.content()
1
sage: Q = TernaryQF([2, 4, 6, 0, 0, 0])
sage: Q.content()
2
sage: Q.scale_by_factor(100).content()
200

Python

>>> from sage.all import *
>>> Q = TernaryQF([Integer(1), Integer(1), Integer(2), Integer(0), Integer(0), Integer(0)])
>>> Q.content()
1
>>> Q = TernaryQF([Integer(2), Integer(4), Integer(6), Integer(0), Integer(0), Integer(0)])
>>> Q.content()
2
>>> Q.scale_by_factor(Integer(100)).content()
200

delta()[source]¶

Return the omega of the adjoint of the given ternary quadratic form, which is the same as the omega of the reciprocal form.

EXAMPLES:

Sage

sage: Q = TernaryQF([1, 2, 2, -1, 0, -1])
sage: Q.delta()
208
sage: Q.adjoint().omega()
208
sage: Q = TernaryQF([1, -1, 1, 0, 0, 0])
sage: Q.delta()
4
sage: Q.omega()
4

Python

>>> from sage.all import *
>>> Q = TernaryQF([Integer(1), Integer(2), Integer(2), -Integer(1), Integer(0), -Integer(1)])
>>> Q.delta()
208
>>> Q.adjoint().omega()
208
>>> Q = TernaryQF([Integer(1), -Integer(1), Integer(1), Integer(0), Integer(0), Integer(0)])
>>> Q.delta()
4
>>> Q.omega()
4

disc()[source]¶

Return the discriminant of the ternary quadratic form, this is the determinant of the matrix divided by 2.

EXAMPLES:

Sage

sage: Q = TernaryQF([1, 1, 2, 0, -1, 4])
sage: Q.disc()
-25
sage: Q.matrix().det()
-50

Python

>>> from sage.all import *
>>> Q = TernaryQF([Integer(1), Integer(1), Integer(2), Integer(0), -Integer(1), Integer(4)])
>>> Q.disc()
-25
>>> Q.matrix().det()
-50

divisor()[source]¶

Return the content of the adjoint form associated to the given form.

EXAMPLES:

Sage

sage: Q = TernaryQF([1, 1, 17, 0, 0, 0])
sage: Q.divisor()
4

Python

>>> from sage.all import *
>>> Q = TernaryQF([Integer(1), Integer(1), Integer(17), Integer(0), Integer(0), Integer(0)])
>>> Q.divisor()
4

find_p_neighbor_from_vec(p, v, mat=False)[source]¶

Finds the reduced equivalent of the \(p\)-neighbor of this ternary quadratic form associated to a given vector \(v\) satisfying:

\(Q(v) = 0\) (mod \(p\))
\(v\) is a non-singular point of the conic \(Q(v) = 0\) (mod \(p\)).

REFERENCES:

Gonzalo Tornaria’s Thesis, Thrm 3.5, p34.

EXAMPLES:

Sage

sage: # needs sage.libs.pari
sage: Q = TernaryQF([1, 3, 3, -2, 0, -1]); Q
Ternary quadratic form with integer coefficients:
[1 3 3]
[-2 0 -1]
sage: Q.disc()
29
sage: v = (9, 7, 1)
sage: v in Q.find_zeros_mod_p(11)
True
sage: Q11, M = Q.find_p_neighbor_from_vec(11, v, mat=True)
sage: Q11
Ternary quadratic form with integer coefficients:
[1 2 4]
[-1 -1 0]
sage: M
[    -1  -5/11   7/11]
[     0 -10/11   3/11]
[     0  -3/11  13/11]
sage: Q(M) == Q11
True

Python

>>> from sage.all import *
>>> # needs sage.libs.pari
>>> Q = TernaryQF([Integer(1), Integer(3), Integer(3), -Integer(2), Integer(0), -Integer(1)]); Q
Ternary quadratic form with integer coefficients:
[1 3 3]
[-2 0 -1]
>>> Q.disc()
29
>>> v = (Integer(9), Integer(7), Integer(1))
>>> v in Q.find_zeros_mod_p(Integer(11))
True
>>> Q11, M = Q.find_p_neighbor_from_vec(Integer(11), v, mat=True)
>>> Q11
Ternary quadratic form with integer coefficients:
[1 2 4]
[-1 -1 0]
>>> M
[    -1  -5/11   7/11]
[     0 -10/11   3/11]
[     0  -3/11  13/11]
>>> Q(M) == Q11
True

Test that it works with (0, 0, 1):

Sage

sage: Q.find_p_neighbor_from_vec(3, (0,0,1))                                # needs sage.libs.pari
Ternary quadratic form with integer coefficients:
[1 3 3]
[-2 0 -1]

Python

>>> from sage.all import *
>>> Q.find_p_neighbor_from_vec(Integer(3), (Integer(0),Integer(0),Integer(1)))                                # needs sage.libs.pari
Ternary quadratic form with integer coefficients:
[1 3 3]
[-2 0 -1]

find_p_neighbors(p, mat=False)[source]¶

Find a list with all the reduced equivalent of the \(p\)-neighbors of this ternary quadratic form, given by the zeros mod \(p\) of the form. See find_p_neighbor_from_vec() for more information.

EXAMPLES:

Sage

sage: # needs sage.libs.pari
sage: Q0 = TernaryQF([1, 3, 3, -2, 0, -1]); Q0
Ternary quadratic form with integer coefficients:
[1 3 3]
[-2 0 -1]
sage: neig = Q0.find_p_neighbors(5)
sage: len(neig)
6
sage: Q1 = TernaryQF([1, 1, 10, 1, 1, 1])
sage: Q2 = TernaryQF([1, 2, 4, -1, -1, 0])
sage: neig.count(Q0)
2
sage: neig.count(Q1)
1
sage: neig.count(Q2)
3

Python

>>> from sage.all import *
>>> # needs sage.libs.pari
>>> Q0 = TernaryQF([Integer(1), Integer(3), Integer(3), -Integer(2), Integer(0), -Integer(1)]); Q0
Ternary quadratic form with integer coefficients:
[1 3 3]
[-2 0 -1]
>>> neig = Q0.find_p_neighbors(Integer(5))
>>> len(neig)
6
>>> Q1 = TernaryQF([Integer(1), Integer(1), Integer(10), Integer(1), Integer(1), Integer(1)])
>>> Q2 = TernaryQF([Integer(1), Integer(2), Integer(4), -Integer(1), -Integer(1), Integer(0)])
>>> neig.count(Q0)
2
>>> neig.count(Q1)
1
>>> neig.count(Q2)
3

find_zeros_mod_p(p)[source]¶

Find the zeros of the given ternary quadratic positive definite form modulo a prime \(p\), where \(p\) doesn’t divide the discriminant of the form.

EXAMPLES:

Sage

sage: Q = TernaryQF([4, 7, 8, -4, -1, -3])
sage: Q.is_positive_definite()
True
sage: Q.disc().factor()
3 * 13 * 19
sage: Q.find_zeros_mod_p(2)
[(1, 0, 0), (1, 1, 0), (0, 0, 1)]
sage: zeros_17 = Q.find_zeros_mod_p(17)                                     # needs sage.libs.pari
sage: len(zeros_17)                                                         # needs sage.libs.pari
18
sage: [Q(v)%17 for v in zeros_17]                                           # needs sage.libs.pari
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

Python

>>> from sage.all import *
>>> Q = TernaryQF([Integer(4), Integer(7), Integer(8), -Integer(4), -Integer(1), -Integer(3)])
>>> Q.is_positive_definite()
True
>>> Q.disc().factor()
3 * 13 * 19
>>> Q.find_zeros_mod_p(Integer(2))
[(1, 0, 0), (1, 1, 0), (0, 0, 1)]
>>> zeros_17 = Q.find_zeros_mod_p(Integer(17))                                     # needs sage.libs.pari
>>> len(zeros_17)                                                         # needs sage.libs.pari
18
>>> [Q(v)%Integer(17) for v in zeros_17]                                           # needs sage.libs.pari
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

is_definite()[source]¶

Determine if the ternary quadratic form is definite.

EXAMPLES:

Sage

sage: Q = TernaryQF([10, 10, 1, -1, 2, 3])
sage: Q.is_definite()
True
sage: (-Q).is_definite()
True
sage: Q = TernaryQF([1, 1, 2, -3, 0, -1])
sage: Q.is_definite()
False

Python

>>> from sage.all import *
>>> Q = TernaryQF([Integer(10), Integer(10), Integer(1), -Integer(1), Integer(2), Integer(3)])
>>> Q.is_definite()
True
>>> (-Q).is_definite()
True
>>> Q = TernaryQF([Integer(1), Integer(1), Integer(2), -Integer(3), Integer(0), -Integer(1)])
>>> Q.is_definite()
False

is_eisenstein_reduced()[source]¶

Determine if the ternary quadratic form is Eisenstein reduced.

That is, if we have a ternary quadratic form:

[a b c]
[r s t]

then

\(a \leq b \leq c\);
\(r\), \(s\), and \(t\) are all positive or all nonpositive;
\(a \geq |t|\); \(a \geq |s|\); \(b \geq |r|\);
\(a+b+r+s+t \geq 0\);
\(a=t\) implies \(s \leq 2\cdot r\); \(a=s\) implies \(t \leq 2\cdot r\); \(b=r\) implies \(t \leq 2\cdot s\);
\(a=-t\) implies \(s=0\); \(a=-s\) implies \(t=0\); \(b=-r\) implies \(t=0\);
\(a+b+r+s+t = 0\) implies \(2\cdot a+2\cdot s+t \leq 0\);
\(a=b\) implies \(|r| \leq |s|\); \(b=c\) implies \(|s| \leq |t|\).

EXAMPLES:

Sage

sage: Q = TernaryQF([1, 1, 1, 0, 0, 0])
sage: Q.is_eisenstein_reduced()
True
sage: Q = TernaryQF([34, 14, 44, 12, 25, -22])
sage: Q.is_eisenstein_reduced()
False

Python

>>> from sage.all import *
>>> Q = TernaryQF([Integer(1), Integer(1), Integer(1), Integer(0), Integer(0), Integer(0)])
>>> Q.is_eisenstein_reduced()
True
>>> Q = TernaryQF([Integer(34), Integer(14), Integer(44), Integer(12), Integer(25), -Integer(22)])
>>> Q.is_eisenstein_reduced()
False

is_negative_definite()[source]¶

Determine if the ternary quadratic form is negative definite.

EXAMPLES:

Sage

sage: Q = TernaryQF([-8, -9, -10, 1, 9, -3])
sage: Q.is_negative_definite()
True
sage: Q = TernaryQF([-4, -1, 6, -5, 1, -5])
sage: Q((0, 0, 1))
6
sage: Q.is_negative_definite()
False

Python

>>> from sage.all import *
>>> Q = TernaryQF([-Integer(8), -Integer(9), -Integer(10), Integer(1), Integer(9), -Integer(3)])
>>> Q.is_negative_definite()
True
>>> Q = TernaryQF([-Integer(4), -Integer(1), Integer(6), -Integer(5), Integer(1), -Integer(5)])
>>> Q((Integer(0), Integer(0), Integer(1)))
6
>>> Q.is_negative_definite()
False

is_positive_definite()[source]¶

Determine if the ternary quadratic form is positive definite.

EXAMPLES:

Sage

sage: Q = TernaryQF([10, 10, 1, -1, 2, 3])
sage: Q.is_positive_definite()
True
sage: (-Q).is_positive_definite()
False
sage: Q = TernaryQF([1, 1, 0, 0, 0, 0])
sage: Q.is_positive_definite()
False
sage: Q = TernaryQF([1, 1, 1, -1, -2, -3])
sage: Q((1,1,1))
-3
sage: Q.is_positive_definite()
False

Python

>>> from sage.all import *
>>> Q = TernaryQF([Integer(10), Integer(10), Integer(1), -Integer(1), Integer(2), Integer(3)])
>>> Q.is_positive_definite()
True
>>> (-Q).is_positive_definite()
False
>>> Q = TernaryQF([Integer(1), Integer(1), Integer(0), Integer(0), Integer(0), Integer(0)])
>>> Q.is_positive_definite()
False
>>> Q = TernaryQF([Integer(1), Integer(1), Integer(1), -Integer(1), -Integer(2), -Integer(3)])
>>> Q((Integer(1),Integer(1),Integer(1)))
-3
>>> Q.is_positive_definite()
False

is_primitive()[source]¶

Determine if the ternary quadratic form is primitive.

This means that the greatest common divisor of the coefficients of the form is 1.

EXAMPLES:

Sage

sage: Q = TernaryQF([1, 2, 3, 4, 5, 6])
sage: Q.is_primitive()
True
sage: Q.content()
1
sage: Q = TernaryQF([10, 10, 10, 5, 5, 5])
sage: Q.content()
5
sage: Q.is_primitive()
False

Python

>>> from sage.all import *
>>> Q = TernaryQF([Integer(1), Integer(2), Integer(3), Integer(4), Integer(5), Integer(6)])
>>> Q.is_primitive()
True
>>> Q.content()
1
>>> Q = TernaryQF([Integer(10), Integer(10), Integer(10), Integer(5), Integer(5), Integer(5)])
>>> Q.content()
5
>>> Q.is_primitive()
False

level()[source]¶

Return the level of the ternary quadratic form, which is 4 times the discriminant divided by the divisor.

EXAMPLES:

Sage

sage: Q = TernaryQF([1, 2, 2, -1, 0, -1])
sage: Q.level()
52
sage: 4*Q.disc()/Q.divisor()
52

Python

>>> from sage.all import *
>>> Q = TernaryQF([Integer(1), Integer(2), Integer(2), -Integer(1), Integer(0), -Integer(1)])
>>> Q.level()
52
>>> Integer(4)*Q.disc()/Q.divisor()
52

matrix()[source]¶

Return the Hessian matrix associated to the ternary quadratic form. That is, if \(Q\) is a ternary quadratic form, \(Q(x,y,z) = a\cdot x^2 + b\cdot y^2 + c\cdot z^2 + r\cdot y\cdot z + s\cdot x\cdot z + t\cdot x\cdot y\), then the Hessian matrix associated to \(Q\) is

[2\cdot a t s]
[t 2\cdot b r]
[s r 2\cdot c]

EXAMPLES:

Sage

sage: Q = TernaryQF([1,1,2,0,-1,4]); Q
Ternary quadratic form with integer coefficients:
[1 1 2]
[0 -1 4]
sage: M = Q.matrix(); M
[ 2  4 -1]
[ 4  2  0]
[-1  0  4]
sage: v = vector((1, 2, 3))
sage: Q(v)
28
sage: (v*M*v.column())[0]//2
28

Python

>>> from sage.all import *
>>> Q = TernaryQF([Integer(1),Integer(1),Integer(2),Integer(0),-Integer(1),Integer(4)]); Q
Ternary quadratic form with integer coefficients:
[1 1 2]
[0 -1 4]
>>> M = Q.matrix(); M
[ 2  4 -1]
[ 4  2  0]
[-1  0  4]
>>> v = vector((Integer(1), Integer(2), Integer(3)))
>>> Q(v)
28
>>> (v*M*v.column())[Integer(0)]//Integer(2)
28

number_of_automorphisms(slow=True)[source]¶

Return the number of automorphisms of the definite ternary quadratic form.

EXAMPLES:

Sage

sage: Q = TernaryQF([1, 1, 7, 0, 0, 0])
sage: A = matrix(ZZ, 3, [0, 1, 0, -1, 5, 0, -8, -1, 1])
sage: A.det()
1
sage: Q1 = Q(A); Q1
Ternary quadratic form with integer coefficients:
[449 33 7]
[-14 -112 102]
sage: Q1.number_of_automorphisms()
8
sage: Q = TernaryQF([-19, -7, -6, -12, 20, 23])
sage: Q.is_negative_definite()
True
sage: Q.number_of_automorphisms(slow=False)
24

Python

>>> from sage.all import *
>>> Q = TernaryQF([Integer(1), Integer(1), Integer(7), Integer(0), Integer(0), Integer(0)])
>>> A = matrix(ZZ, Integer(3), [Integer(0), Integer(1), Integer(0), -Integer(1), Integer(5), Integer(0), -Integer(8), -Integer(1), Integer(1)])
>>> A.det()
1
>>> Q1 = Q(A); Q1
Ternary quadratic form with integer coefficients:
[449 33 7]
[-14 -112 102]
>>> Q1.number_of_automorphisms()
8
>>> Q = TernaryQF([-Integer(19), -Integer(7), -Integer(6), -Integer(12), Integer(20), Integer(23)])
>>> Q.is_negative_definite()
True
>>> Q.number_of_automorphisms(slow=False)
24

omega()[source]¶

Return the content of the adjoint of the primitive associated ternary quadratic form.

EXAMPLES:

Sage

sage: Q = TernaryQF([4, 11, 12, 0, -4, 0])
sage: Q.omega()
176
sage: Q.primitive().adjoint().content()
176

Python

>>> from sage.all import *
>>> Q = TernaryQF([Integer(4), Integer(11), Integer(12), Integer(0), -Integer(4), Integer(0)])
>>> Q.omega()
176
>>> Q.primitive().adjoint().content()
176

polynomial(names='x,y,z')[source]¶

Return the polynomial associated to the ternary quadratic form.

EXAMPLES:

Sage

sage: Q = TernaryQF([1, 1, 0, 2, -3, -1]); Q
Ternary quadratic form with integer coefficients:
[1 1 0]
[2 -3 -1]
sage: p = Q.polynomial(); p
x^2 - x*y + y^2 - 3*x*z + 2*y*z
sage: p.parent()
Multivariate Polynomial Ring in x, y, z over Integer Ring

Python

>>> from sage.all import *
>>> Q = TernaryQF([Integer(1), Integer(1), Integer(0), Integer(2), -Integer(3), -Integer(1)]); Q
Ternary quadratic form with integer coefficients:
[1 1 0]
[2 -3 -1]
>>> p = Q.polynomial(); p
x^2 - x*y + y^2 - 3*x*z + 2*y*z
>>> p.parent()
Multivariate Polynomial Ring in x, y, z over Integer Ring

possible_automorphisms = None¶

primitive()[source]¶

Return the primitive version of the ternary quadratic form.

EXAMPLES:

Sage

sage: Q = TernaryQF([2, 2, 2, 1, 1, 1])
sage: Q.is_primitive()
True
sage: Q.primitive()
Ternary quadratic form with integer coefficients:
[2 2 2]
[1 1 1]
sage: Q.primitive() == Q
True
sage: Q = TernaryQF([10, 10, 10, 5, 5, 5])
sage: Q.primitive()
Ternary quadratic form with integer coefficients:
[2 2 2]
[1 1 1]

Python

>>> from sage.all import *
>>> Q = TernaryQF([Integer(2), Integer(2), Integer(2), Integer(1), Integer(1), Integer(1)])
>>> Q.is_primitive()
True
>>> Q.primitive()
Ternary quadratic form with integer coefficients:
[2 2 2]
[1 1 1]
>>> Q.primitive() == Q
True
>>> Q = TernaryQF([Integer(10), Integer(10), Integer(10), Integer(5), Integer(5), Integer(5)])
>>> Q.primitive()
Ternary quadratic form with integer coefficients:
[2 2 2]
[1 1 1]

pseudorandom_primitive_zero_mod_p(p)[source]¶

Return a tuple of the form \(v = (a, b, 1)\) such that is a zero of the given ternary quadratic positive definite form modulo an odd prime \(p\), where \(p\) doesn’t divides the discriminant of the form.

EXAMPLES:

Sage

sage: Q = TernaryQF([1, 1, 11, 0, -1, 0])
sage: Q.disc()
43
sage: Q.pseudorandom_primitive_zero_mod_p(3)  # random
(1, 2, 1)
sage: Q((1, 2, 1))
15
sage: v = Q.pseudorandom_primitive_zero_mod_p(1009)                        # needs sage.libs.pari
sage: Q(v) % 1009                                                          # needs sage.libs.pari
0
sage: v[2]                                                                 # needs sage.libs.pari
1

Python

>>> from sage.all import *
>>> Q = TernaryQF([Integer(1), Integer(1), Integer(11), Integer(0), -Integer(1), Integer(0)])
>>> Q.disc()
43
>>> Q.pseudorandom_primitive_zero_mod_p(Integer(3))  # random
(1, 2, 1)
>>> Q((Integer(1), Integer(2), Integer(1)))
15
>>> v = Q.pseudorandom_primitive_zero_mod_p(Integer(1009))                        # needs sage.libs.pari
>>> Q(v) % Integer(1009)                                                          # needs sage.libs.pari
0
>>> v[Integer(2)]                                                                 # needs sage.libs.pari
1

quadratic_form()[source]¶

Return a QuadraticForm with the same coefficients as self over \(\ZZ\).

EXAMPLES:

Sage

sage: Q = TernaryQF([1, 2, 3, 1, 1, 1])
sage: QF1 = Q.quadratic_form(); QF1
Quadratic form in 3 variables over Integer Ring with coefficients:
[ 1 1 1 ]
[ * 2 1 ]
[ * * 3 ]
sage: QF2 = QuadraticForm(ZZ, 3, [1, 1, 1, 2, 1, 3])
sage: bool(QF1 == QF2)
True

Python

>>> from sage.all import *
>>> Q = TernaryQF([Integer(1), Integer(2), Integer(3), Integer(1), Integer(1), Integer(1)])
>>> QF1 = Q.quadratic_form(); QF1
Quadratic form in 3 variables over Integer Ring with coefficients:
[ 1 1 1 ]
[ * 2 1 ]
[ * * 3 ]
>>> QF2 = QuadraticForm(ZZ, Integer(3), [Integer(1), Integer(1), Integer(1), Integer(2), Integer(1), Integer(3)])
>>> bool(QF1 == QF2)
True

reciprocal()[source]¶

Return the reciprocal quadratic form associated to the given form.

This is defined as the multiple of the primitive adjoint with the same content as the given form.

EXAMPLES:

Sage

sage: Q = TernaryQF([2, 2, 14, 0, 0, 0])
sage: Q.reciprocal()
Ternary quadratic form with integer coefficients:
[14 14 2]
[0 0 0]
sage: Q.content()
2
sage: Q.reciprocal().content()
2
sage: Q.adjoint().content()
16

Python

>>> from sage.all import *
>>> Q = TernaryQF([Integer(2), Integer(2), Integer(14), Integer(0), Integer(0), Integer(0)])
>>> Q.reciprocal()
Ternary quadratic form with integer coefficients:
[14 14 2]
[0 0 0]
>>> Q.content()
2
>>> Q.reciprocal().content()
2
>>> Q.adjoint().content()
16

reciprocal_reduced()[source]¶

Return the reduced form of the reciprocal form of the given ternary quadratic form.

EXAMPLES:

Sage

sage: Q = TernaryQF([1, 1, 3, 0, -1, 0])
sage: Qrr = Q.reciprocal_reduced(); Qrr
Ternary quadratic form with integer coefficients:
[4 11 12]
[0 -4 0]
sage: Q.is_eisenstein_reduced()
True
sage: Qr = Q.reciprocal()
sage: Qr.reduced_form_eisenstein(matrix=False) == Qrr
True

Python

>>> from sage.all import *
>>> Q = TernaryQF([Integer(1), Integer(1), Integer(3), Integer(0), -Integer(1), Integer(0)])
>>> Qrr = Q.reciprocal_reduced(); Qrr
Ternary quadratic form with integer coefficients:
[4 11 12]
[0 -4 0]
>>> Q.is_eisenstein_reduced()
True
>>> Qr = Q.reciprocal()
>>> Qr.reduced_form_eisenstein(matrix=False) == Qrr
True

reduced_form_eisenstein(matrix=True)[source]¶

Return the Eisenstein reduced form equivalent to the given positive ternary quadratic form, which is unique.

EXAMPLES:

Sage

sage: Q = TernaryQF([293, 315, 756, 908, 929, 522])
sage: Qr, m = Q.reduced_form_eisenstein()
sage: Qr
Ternary quadratic form with integer coefficients:
[1 2 2]
[-1 0 -1]
sage: Qr.is_eisenstein_reduced()
True
sage: m
[ -54  137  -38]
[ -23   58  -16]
[  47 -119   33]
sage: m.det()
1
sage: Q(m) == Qr
True
sage: Q = TernaryQF([12,36,3,14,-7,-19])
sage: Q.reduced_form_eisenstein(matrix = False)
Ternary quadratic form with integer coefficients:
[3 8 20]
[3 2 1]

Python

>>> from sage.all import *
>>> Q = TernaryQF([Integer(293), Integer(315), Integer(756), Integer(908), Integer(929), Integer(522)])
>>> Qr, m = Q.reduced_form_eisenstein()
>>> Qr
Ternary quadratic form with integer coefficients:
[1 2 2]
[-1 0 -1]
>>> Qr.is_eisenstein_reduced()
True
>>> m
[ -54  137  -38]
[ -23   58  -16]
[  47 -119   33]
>>> m.det()
1
>>> Q(m) == Qr
True
>>> Q = TernaryQF([Integer(12),Integer(36),Integer(3),Integer(14),-Integer(7),-Integer(19)])
>>> Q.reduced_form_eisenstein(matrix = False)
Ternary quadratic form with integer coefficients:
[3 8 20]
[3 2 1]

scale_by_factor(k)[source]¶

Scale the values of the ternary quadratic form by the number k.

OUTPUT:

If k times the content of the ternary quadratic form is an integer, return a ternary quadratic form; otherwise, return a quadratic form of dimension 3.

EXAMPLES:

Sage

sage: Q = TernaryQF([2, 2, 4, 0, -2, 8])
sage: Q
Ternary quadratic form with integer coefficients:
[2 2 4]
[0 -2 8]
sage: Q.scale_by_factor(5)
Ternary quadratic form with integer coefficients:
[10 10 20]
[0 -10 40]
sage: Q.scale_by_factor(1/2)
Ternary quadratic form with integer coefficients:
[1 1 2]
[0 -1 4]
sage: Q.scale_by_factor(1/3)
Quadratic form in 3 variables over Rational Field with coefficients:
[ 2/3 8/3 -2/3 ]
[ * 2/3 0 ]
[ * * 4/3 ]

Python

>>> from sage.all import *
>>> Q = TernaryQF([Integer(2), Integer(2), Integer(4), Integer(0), -Integer(2), Integer(8)])
>>> Q
Ternary quadratic form with integer coefficients:
[2 2 4]
[0 -2 8]
>>> Q.scale_by_factor(Integer(5))
Ternary quadratic form with integer coefficients:
[10 10 20]
[0 -10 40]
>>> Q.scale_by_factor(Integer(1)/Integer(2))
Ternary quadratic form with integer coefficients:
[1 1 2]
[0 -1 4]
>>> Q.scale_by_factor(Integer(1)/Integer(3))
Quadratic form in 3 variables over Rational Field with coefficients:
[ 2/3 8/3 -2/3 ]
[ * 2/3 0 ]
[ * * 4/3 ]

symmetry(v)[source]¶

Return \(A\), the automorphism of the ternary quadratic form such that:

\(Av = -v\),
\(Au = 0\), if \(u\) is orthogonal to \(v\),

where \(v\) is a given vector.

EXAMPLES:

Sage

sage: Q = TernaryQF([4, 5, 8, 5, 2, 2])
sage: v = vector((1,1,1))
sage: M = Q.symmetry(v)
sage: M
[  7/13 -17/26 -23/26]
[ -6/13   9/26 -23/26]
[ -6/13 -17/26   3/26]
sage: M.det()
-1
sage: M*v
(-1, -1, -1)
sage: v1 = vector((23, 0, -12))
sage: v2 = vector((0, 23, -17))
sage: v1*Q.matrix()*v
0
sage: v2*Q.matrix()*v
0
sage: M*v1 == v1
True
sage: M*v2 == v2
True

Python

>>> from sage.all import *
>>> Q = TernaryQF([Integer(4), Integer(5), Integer(8), Integer(5), Integer(2), Integer(2)])
>>> v = vector((Integer(1),Integer(1),Integer(1)))
>>> M = Q.symmetry(v)
>>> M
[  7/13 -17/26 -23/26]
[ -6/13   9/26 -23/26]
[ -6/13 -17/26   3/26]
>>> M.det()
-1
>>> M*v
(-1, -1, -1)
>>> v1 = vector((Integer(23), Integer(0), -Integer(12)))
>>> v2 = vector((Integer(0), Integer(23), -Integer(17)))
>>> v1*Q.matrix()*v
0
>>> v2*Q.matrix()*v
0
>>> M*v1 == v1
True
>>> M*v2 == v2
True

xi(p)[source]¶

Return the value of the genus characters Xi_p… which may be missing one character. We allow \(-1\) as a prime.

REFERENCES:

Dickson’s “Studies in the Theory of Numbers”

EXAMPLES:

Sage

sage: Q1 = TernaryQF([26, 42, 53, -36, -17, -3])
sage: Q2 = Q1.find_p_neighbors(2)[1]
sage: Q1.omega()
3
sage: Q1.xi(3), Q2.xi(3)
(-1, -1)

Python

>>> from sage.all import *
>>> Q1 = TernaryQF([Integer(26), Integer(42), Integer(53), -Integer(36), -Integer(17), -Integer(3)])
>>> Q2 = Q1.find_p_neighbors(Integer(2))[Integer(1)]
>>> Q1.omega()
3
>>> Q1.xi(Integer(3)), Q2.xi(Integer(3))
(-1, -1)

xi_rec(p)[source]¶

Return Xi(p) for the reciprocal form.

EXAMPLES:

Sage

sage: Q1 = TernaryQF([1, 1, 7, 0, 0, 0])
sage: Q2 = Q1.find_p_neighbors(3)[0]
sage: Q1.delta()
28
sage: Q1.xi_rec(7), Q2.xi_rec(7)
(1, 1)

Python

>>> from sage.all import *
>>> Q1 = TernaryQF([Integer(1), Integer(1), Integer(7), Integer(0), Integer(0), Integer(0)])
>>> Q2 = Q1.find_p_neighbors(Integer(3))[Integer(0)]
>>> Q1.delta()
28
>>> Q1.xi_rec(Integer(7)), Q2.xi_rec(Integer(7))
(1, 1)

sage.quadratic_forms.ternary_qf.find_a_ternary_qf_by_level_disc(N, d)[source]¶

Find a reduced ternary quadratic form given its discriminant \(d\) and level \(N\). If \(N|4d\) and \(d|N^2\), then it may be a form with that discriminant and level.

EXAMPLES:

Sage

sage: Q1 = find_a_ternary_qf_by_level_disc(44, 11); Q1
Ternary quadratic form with integer coefficients:
[1 1 3]
[0 -1 0]
sage: Q2 = find_a_ternary_qf_by_level_disc(44, 11^2 * 16)
sage: Q2
Ternary quadratic form with integer coefficients:
[3 15 15]
[-14 -2 -2]
sage: Q1.is_eisenstein_reduced()
True
sage: Q1.level()
44
sage: Q1.disc()
11
sage: find_a_ternary_qf_by_level_disc(44, 22)
sage: find_a_ternary_qf_by_level_disc(44, 33)
Traceback (most recent call last):
...
ValueError: There are no ternary forms of this level and discriminant

Python

>>> from sage.all import *
>>> Q1 = find_a_ternary_qf_by_level_disc(Integer(44), Integer(11)); Q1
Ternary quadratic form with integer coefficients:
[1 1 3]
[0 -1 0]
>>> Q2 = find_a_ternary_qf_by_level_disc(Integer(44), Integer(11)**Integer(2) * Integer(16))
>>> Q2
Ternary quadratic form with integer coefficients:
[3 15 15]
[-14 -2 -2]
>>> Q1.is_eisenstein_reduced()
True
>>> Q1.level()
44
>>> Q1.disc()
11
>>> find_a_ternary_qf_by_level_disc(Integer(44), Integer(22))
>>> find_a_ternary_qf_by_level_disc(Integer(44), Integer(33))
Traceback (most recent call last):
...
ValueError: There are no ternary forms of this level and discriminant

sage.quadratic_forms.ternary_qf.find_all_ternary_qf_by_level_disc(N, d)[source]¶

Find the coefficients of all the reduced ternary quadratic forms given its discriminant \(d\) and level \(N\).

If \(N|4d\) and \(d|N^2\), then it may be some forms with that discriminant and level.

EXAMPLES:

Sage

sage: find_all_ternary_qf_by_level_disc(44, 11)
[Ternary quadratic form with integer coefficients:
[1 1 3]
[0 -1 0], Ternary quadratic form with integer coefficients:
[1 1 4]
[1 1 1]]
sage: find_all_ternary_qf_by_level_disc(44, 11^2 * 16)
[Ternary quadratic form with integer coefficients:
[3 15 15]
[-14 -2 -2], Ternary quadratic form with integer coefficients:
[4 11 12]
[0 -4 0]]
sage: Q = TernaryQF([1, 1, 3, 0, -1, 0])
sage: Q.is_eisenstein_reduced()
True
sage: Q.reciprocal_reduced()
Ternary quadratic form with integer coefficients:
[4 11 12]
[0 -4 0]
sage: find_all_ternary_qf_by_level_disc(44, 22)
[]
sage: find_all_ternary_qf_by_level_disc(44, 33)
Traceback (most recent call last):
...
ValueError: There are no ternary forms of this level and discriminant

Python

>>> from sage.all import *
>>> find_all_ternary_qf_by_level_disc(Integer(44), Integer(11))
[Ternary quadratic form with integer coefficients:
[1 1 3]
[0 -1 0], Ternary quadratic form with integer coefficients:
[1 1 4]
[1 1 1]]
>>> find_all_ternary_qf_by_level_disc(Integer(44), Integer(11)**Integer(2) * Integer(16))
[Ternary quadratic form with integer coefficients:
[3 15 15]
[-14 -2 -2], Ternary quadratic form with integer coefficients:
[4 11 12]
[0 -4 0]]
>>> Q = TernaryQF([Integer(1), Integer(1), Integer(3), Integer(0), -Integer(1), Integer(0)])
>>> Q.is_eisenstein_reduced()
True
>>> Q.reciprocal_reduced()
Ternary quadratic form with integer coefficients:
[4 11 12]
[0 -4 0]
>>> find_all_ternary_qf_by_level_disc(Integer(44), Integer(22))
[]
>>> find_all_ternary_qf_by_level_disc(Integer(44), Integer(33))
Traceback (most recent call last):
...
ValueError: There are no ternary forms of this level and discriminant