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)

Bases: SageObject

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

INPUT:

• v – a 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: 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()

adjoint()

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: 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

automorphism_spin_norm(A)

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

EXAMPLES:

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

automorphism_symmetries(A)

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: 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))
[]

automorphisms(slow=True)

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

EXAMPLES:

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

basic_lemma(p)

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

EXAMPLES:

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

coefficient(n)

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

INPUT:

• n – integer with \(0 \leq n \leq 5\).

EXAMPLES:

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

coefficients()

Return the list of coefficients of the ternary quadratic form.

EXAMPLES:

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)

content()

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

EXAMPLES:

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

delta()

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: 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

disc()

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

EXAMPLES:

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

divisor()

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

EXAMPLES:

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

find_p_neighbor_from_vec(p, v, mat=False)

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

1. \(Q(v) = 0\) (mod \(p\))

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

REFERENCES:

Gonzalo Tornaria’s Thesis, Thrm 3.5, p34.

EXAMPLES:

sage: # 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

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

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]

find_p_neighbors(p, mat=False)

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: # 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

find_zeros_mod_p(p)

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: 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]

is_definite()

Determine if the ternary quadratic form is definite.

EXAMPLES:

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

is_eisenstein_reduced()

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

1. \(a \leq b \leq c\);

2. \(r\), \(s\), and \(t\) are all positive or all nonpositive;

3. \(a \geq |t|\); \(a \geq |s|\); \(b \geq |r|\);

4. \(a+b+r+s+t \geq 0\);

5. \(a=t\) implies \(s \leq 2\cdot r\); \(a=s\) implies \(t \leq 2\cdot r\); \(b=r\) implies \(t \leq 2\cdot s\);

6. \(a=-t\) implies \(s=0\); \(a=-s\) implies \(t=0\); \(b=-r\) implies \(t=0\);

7. \(a+b+r+s+t = 0\) implies \(2\cdot a+2\cdot s+t \leq 0\);

8. \(a=b\) implies \(|r| \leq |s|\); \(b=c\) implies \(|s| \leq |t|\).

EXAMPLES:

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

is_negative_definite()

Determine if the ternary quadratic form is negative definite.

EXAMPLES:

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

is_positive_definite()

Determine if the ternary quadratic form is positive definite.

EXAMPLES:

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

is_primitive()

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: 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

level()

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

EXAMPLES:

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

matrix()

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: 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

number_of_automorphisms(slow=True)

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

EXAMPLES:

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

omega()

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

EXAMPLES:

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

polynomial(names='x,y,z')

Return the polynomial associated to the ternary quadratic form.

EXAMPLES:

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

possible_automorphisms = None

primitive()

Return the primitive version of the ternary quadratic form.

EXAMPLES:

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]

pseudorandom_primitive_zero_mod_p(p)

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: 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

quadratic_form()

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

EXAMPLES:

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

reciprocal()

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: 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

reciprocal_reduced()

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

EXAMPLES:

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

reduced_form_eisenstein(matrix=True)

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

EXAMPLES:

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]

scale_by_factor(k)

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: 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 ]

symmetry(v)

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: 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

xi(p)

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: 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)

xi_rec(p)

Return Xi(p) for the reciprocal form.

EXAMPLES:

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)

sage.quadratic_forms.ternary_qf.find_a_ternary_qf_by_level_disc(N, d)

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: 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

sage.quadratic_forms.ternary_qf.find_all_ternary_qf_by_level_disc(N, d)

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: 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