def _multiply_theta_char(l, f):
r"""
Return the coefficient at ``f`` of the theta series `\prod_t \theta_t`
where `t` runs through the list ``l`` of theta characteristics.
INPUT:
- ``l`` -- a list of quadruples `t` in `{0,1}^4`
- ``f`` -- a triple `(a,b,c)` of rational numbers such that the
quadratic form `[a,b,c]` is semi positive definite
(i.e. `a,c \geq 0` and `b^2-4ac \leq 0`).
EXAMPLES::
sage: from sage.modular.siegel.theta_constant import _multiply_theta_char
sage: from sage.modular.siegel.theta_constant import _compute_theta_char_poly
sage: tc = [(1, 1, 0, 0), (0, 0, 1, 1), (1, 1, 0, 0), (0, 0, 1, 1)]
sage: _multiply_theta_char(tc, [2, 0, 6])
-32
sage: _multiply_theta_char(tc, [2, 0, 10])
32
sage: _multiply_theta_char(tc, [0, 0, 0])
0
"""
if 0 == len(l):
return (1 if (0, 0, 0) == f else 0)
a, b, c = f
m1, m2, m3, m4 = l[0]
# if the characteristic is not even:
if 1 == (m1 * m3 + m2 * m4) % 2:
return 0
coeff = 0
from sage.misc.all import isqrt, xsrange
for u in xsrange(m1, isqrt(a) + 1, 2):
for v in xsrange(m2, isqrt(c) + 1, 2):
if 0 == u and 0 == v:
coeff += _multiply_theta_char(l[1:], (a, b, c))
continue
ap, bp, cp = (a - u * u, b - 2 * u * v, c - v * v)
if bp * bp - 4 * ap * cp <= 0:
val = (2 if 0 == (u * m3 + v * m4) % 4 else -2)
coeff += val * _multiply_theta_char(l[1:], (ap, bp, cp))
if u != 0 and v != 0:
ap, bp, cp = (a - u * u, b + 2 * u * v, c - v * v)
if bp * bp - 4 * ap * cp <= 0:
val = (2 if 0 == (u * m3 - v * m4) % 4 else -2)
coeff += val * _multiply_theta_char(l[1:], (ap, bp, cp))
return coeff
def _multiply_theta_char(l, f):
r"""
Return the coefficient at ``f`` of the theta series `\prod_t \theta_t`
where `t` runs through the list ``l`` of theta characteristics.
INPUT:
- ``l`` -- a list of quadruples `t` in `{0,1}^4`
- ``f`` -- a triple `(a,b,c)` of rational numbers such that the
quadratic form `[a,b,c]` is semi positive definite
(i.e. `a,c \geq 0` and `b^2-4ac \leq 0`).
EXAMPLES::
sage: from sage.modular.siegel.theta_constant import _multiply_theta_char
sage: from sage.modular.siegel.theta_constant import _compute_theta_char_poly
sage: tc = [(1, 1, 0, 0), (0, 0, 1, 1), (1, 1, 0, 0), (0, 0, 1, 1)]
sage: _multiply_theta_char(tc, [2, 0, 6])
-32
sage: _multiply_theta_char(tc, [2, 0, 10])
32
sage: _multiply_theta_char(tc, [0, 0, 0])
0
"""
if 0 == len(l):
return (1 if (0, 0, 0) == f else 0)
a, b, c = f
m1, m2, m3, m4 = l[0]
# if the characteristic is not even:
if 1 == (m1*m3 + m2*m4)%2:
return 0
coeff = 0
from sage.misc.all import isqrt, xsrange
for u in xsrange(m1, isqrt(a)+1, 2):
for v in xsrange(m2, isqrt(c)+1, 2):
if 0 == u and 0 == v:
coeff += _multiply_theta_char(l[1:], (a, b, c))
continue
ap, bp, cp = (a-u*u, b-2*u*v, c-v*v)
if bp*bp-4*ap*cp <= 0:
val = (2 if 0 == (u*m3 + v*m4)%4 else -2)
coeff += val * _multiply_theta_char(l[1:], (ap, bp, cp))
if u != 0 and v != 0:
ap, bp, cp = (a-u*u, b+2*u*v, c-v*v)
if bp*bp-4*ap*cp <= 0:
val = (2 if 0 == (u*m3 - v*m4)%4 else -2)
coeff += val * _multiply_theta_char(l[1:], (ap, bp, cp))
return coeff
def BinaryQF_reduced_representatives(D, primitive_only=False):
r"""
Returns a list of inequivalent reduced representatives for the
equivalence classes of positive definite binary forms of
discriminant D.
INPUT:
- `D` -- (integer) A negative discriminant.
- ``primitive_only`` -- (bool, default False) flag controlling whether only
primitive forms are included.
OUTPUT:
(list) A lexicographically-ordered list of inequivalent reduced
representatives for the equivalence classes of positive definite binary
forms of discriminant `D`. If ``primitive_only`` is ``True`` then
imprimitive forms (which only exist when `D` is not fundamental) are
omitted; otherwise they are included.
EXAMPLES::
sage: BinaryQF_reduced_representatives(-4)
[x^2 + y^2]
sage: BinaryQF_reduced_representatives(-163)
[x^2 + x*y + 41*y^2]
sage: BinaryQF_reduced_representatives(-12)
[x^2 + 3*y^2, 2*x^2 + 2*x*y + 2*y^2]
sage: BinaryQF_reduced_representatives(-16)
[x^2 + 4*y^2, 2*x^2 + 2*y^2]
sage: BinaryQF_reduced_representatives(-63)
[x^2 + x*y + 16*y^2, 2*x^2 - x*y + 8*y^2, 2*x^2 + x*y + 8*y^2, 3*x^2 + 3*x*y + 6*y^2, 4*x^2 + x*y + 4*y^2]
The number of inequivalent reduced binary forms with a fixed negative
fundamental discriminant D is the class number of the quadratic field
`Q(\sqrt{D})`::
sage: len(BinaryQF_reduced_representatives(-13*4))
2
sage: QuadraticField(-13*4, 'a').class_number()
2
sage: p=next_prime(2^20); p
1048583
sage: len(BinaryQF_reduced_representatives(-p))
689
sage: QuadraticField(-p, 'a').class_number()
689
sage: BinaryQF_reduced_representatives(-23*9)
[x^2 + x*y + 52*y^2,
2*x^2 - x*y + 26*y^2,
2*x^2 + x*y + 26*y^2,
3*x^2 + 3*x*y + 18*y^2,
4*x^2 - x*y + 13*y^2,
4*x^2 + x*y + 13*y^2,
6*x^2 - 3*x*y + 9*y^2,
6*x^2 + 3*x*y + 9*y^2,
8*x^2 + 7*x*y + 8*y^2]
sage: BinaryQF_reduced_representatives(-23*9, primitive_only=True)
[x^2 + x*y + 52*y^2,
2*x^2 - x*y + 26*y^2,
2*x^2 + x*y + 26*y^2,
4*x^2 - x*y + 13*y^2,
4*x^2 + x*y + 13*y^2,
8*x^2 + 7*x*y + 8*y^2]
TESTS::
sage: BinaryQF_reduced_representatives(5)
Traceback (most recent call last):
...
ValueError: discriminant must be negative and congruent to 0 or 1 modulo 4
"""
D = ZZ(D)
if not ( D < 0 and (D % 4 in [0,1])):
raise ValueError, "discriminant must be negative and congruent to 0 or 1 modulo 4"
# For a fundamental discriminant all forms are primitive so we need not check:
if primitive_only:
primitive_only = not is_fundamental_discriminant(D)
form_list = []
from sage.misc.all import xsrange
from sage.rings.arith import gcd
# Only iterate over positive a and over b of the same
# parity as D such that 4a^2 + D <= b^2 <= a^2
for a in xsrange(1,1+((-D)//3).isqrt()):
a4 = 4*a
s = D + a*a4
w = 1+(s-1).isqrt() if s > 0 else 0
if w%2 != D%2: w += 1
for b in xsrange(w,a+1,2):
t = b*b-D
if t % a4 == 0:
c = t // a4
if (not primitive_only) or gcd([a,b,c])==1:
if b>0 and a>b and c>a:
form_list.append(BinaryQF([a,-b,c]))
form_list.append(BinaryQF([a,b,c]))
form_list.sort()
return form_list
def BinaryQF_reduced_representatives(D, primitive_only=False):
r"""
Returns a list of inequivalent reduced representatives for the
equivalence classes of positive definite binary forms of
discriminant D.
INPUT:
- `D` -- (integer) A negative discriminant.
- ``primitive_only`` -- (bool, default False) flag controlling whether only
primitive forms are included.
OUTPUT:
(list) A lexicographically-ordered list of inequivalent reduced
representatives for the equivalence classes of positive definite binary
forms of discriminant `D`. If ``primitive_only`` is ``True`` then
imprimitive forms (which only exist when `D` is not fundamental) are
omitted; otherwise they are included.
EXAMPLES::
sage: BinaryQF_reduced_representatives(-4)
[x^2 + y^2]
sage: BinaryQF_reduced_representatives(-163)
[x^2 + x*y + 41*y^2]
sage: BinaryQF_reduced_representatives(-12)
[x^2 + 3*y^2, 2*x^2 + 2*x*y + 2*y^2]
sage: BinaryQF_reduced_representatives(-16)
[x^2 + 4*y^2, 2*x^2 + 2*y^2]
sage: BinaryQF_reduced_representatives(-63)
[x^2 + x*y + 16*y^2, 2*x^2 - x*y + 8*y^2, 2*x^2 + x*y + 8*y^2, 3*x^2 + 3*x*y + 6*y^2, 4*x^2 + x*y + 4*y^2]
The number of inequivalent reduced binary forms with a fixed negative
fundamental discriminant D is the class number of the quadratic field
`Q(\sqrt{D})`::
sage: len(BinaryQF_reduced_representatives(-13*4))
2
sage: QuadraticField(-13*4, 'a').class_number()
2
sage: p=next_prime(2^20); p
1048583
sage: len(BinaryQF_reduced_representatives(-p))
689
sage: QuadraticField(-p, 'a').class_number()
689
sage: BinaryQF_reduced_representatives(-23*9)
[x^2 + x*y + 52*y^2,
2*x^2 - x*y + 26*y^2,
2*x^2 + x*y + 26*y^2,
3*x^2 + 3*x*y + 18*y^2,
4*x^2 - x*y + 13*y^2,
4*x^2 + x*y + 13*y^2,
6*x^2 - 3*x*y + 9*y^2,
6*x^2 + 3*x*y + 9*y^2,
8*x^2 + 7*x*y + 8*y^2]
sage: BinaryQF_reduced_representatives(-23*9, primitive_only=True)
[x^2 + x*y + 52*y^2,
2*x^2 - x*y + 26*y^2,
2*x^2 + x*y + 26*y^2,
4*x^2 - x*y + 13*y^2,
4*x^2 + x*y + 13*y^2,
8*x^2 + 7*x*y + 8*y^2]
TESTS::
sage: BinaryQF_reduced_representatives(5)
Traceback (most recent call last):
...
ValueError: discriminant must be negative and congruent to 0 or 1 modulo 4
"""
D = ZZ(D)
if not (D < 0 and (D % 4 in [0, 1])):
raise ValueError, "discriminant must be negative and congruent to 0 or 1 modulo 4"
# For a fundamental discriminant all forms are primitive so we need not check:
if primitive_only:
primitive_only = not is_fundamental_discriminant(D)
form_list = []
from sage.misc.all import xsrange
from sage.rings.arith import gcd
# Only iterate over positive a and over b of the same
# parity as D such that 4a^2 + D <= b^2 <= a^2
for a in xsrange(1, 1 + ((-D) // 3).isqrt()):
a4 = 4 * a
s = D + a * a4
w = 1 + (s - 1).isqrt() if s > 0 else 0
if w % 2 != D % 2: w += 1
for b in xsrange(w, a + 1, 2):
t = b * b - D
if t % a4 == 0:
c = t // a4
if (not primitive_only) or gcd([a, b, c]) == 1:
if b > 0 and a > b and c > a:
form_list.append(BinaryQF([a, -b, c]))
form_list.append(BinaryQF([a, b, c]))
form_list.sort()
return form_list