def has_fundamental_discriminant(self):
"""
Checks if the discriminant D of this form is a fundamental
discriminant (i.e. D is the smallest element of its
squareclass with D = 0 or 1 mod 4).
EXAMPLES::
sage: Q = BinaryQF([1,0,1])
sage: Q.discriminant()
-4
sage: Q.has_fundamental_discriminant()
True
sage: Q = BinaryQF([2,0,2])
sage: Q.discriminant()
-16
sage: Q.has_fundamental_discriminant()
False
"""
return is_fundamental_discriminant(self.discriminant())
def has_fundamental_discriminant(self):
"""
Return if the discriminant `D` of this form is a fundamental
discriminant (i.e. `D` is the smallest element of its
squareclass with `D = 0` or `1` modulo `4`).
EXAMPLES::
sage: Q = BinaryQF([1, 0, 1])
sage: Q.discriminant()
-4
sage: Q.has_fundamental_discriminant()
True
sage: Q = BinaryQF([2, 0, 2])
sage: Q.discriminant()
-16
sage: Q.has_fundamental_discriminant()
False
"""
return is_fundamental_discriminant(self.discriminant())
def has_fundamental_discriminant(self):
"""
Checks if the discriminant D of this form is a fundamental
discriminant (i.e. D is the smallest element of its
squareclass with D = 0 or 1 mod 4).
EXAMPLES::
sage: Q = BinaryQF([1,0,1])
sage: Q.discriminant()
-4
sage: Q.has_fundamental_discriminant()
True
sage: Q = BinaryQF([2,0,2])
sage: Q.discriminant()
-16
sage: Q.has_fundamental_discriminant()
False
"""
return is_fundamental_discriminant(self.discriminant())
def discriminants_with_bounded_class_number(hmax, B=None, proof=None):
r"""
Return dictionary with keys class numbers `h\le hmax` and values the
list of all pairs `(D, f)`, with `D<0` a fundamental discriminant such
that `Df^2` has class number `h`. If the optional bound `B` is given,
return only those pairs with fundamental `|D| \le B`, though `f` can
still be arbitrarily large.
INPUT:
- ``hmax`` -- integer
- `B` -- integer or None; if None returns all pairs
- ``proof`` -- this code calls the PARI function ``qfbclassno``, so it
could give wrong answers when ``proof``==``False``. The default is
whatever ``proof.number_field()`` is. If ``proof==False`` and `B` is
``None``, at least the number of discriminants is correct, since it
is double checked with Watkins's table.
OUTPUT:
- dictionary
In case `B` is not given, we use Mark Watkins's: "Class numbers of
imaginary quadratic fields" to compute a `B` that captures all `h`
up to `hmax` (only available for `hmax\le100`).
EXAMPLES::
sage: v = sage.schemes.elliptic_curves.cm.discriminants_with_bounded_class_number(3)
sage: sorted(v)
[1, 2, 3]
sage: v[1]
[(-3, 3), (-3, 2), (-3, 1), (-4, 2), (-4, 1), (-7, 2), (-7, 1), (-8, 1), (-11, 1), (-19, 1), (-43, 1), (-67, 1), (-163, 1)]
sage: v[2]
[(-3, 7), (-3, 5), (-3, 4), (-4, 5), (-4, 4), (-4, 3), (-7, 4), (-8, 3), (-8, 2), (-11, 3), (-15, 2), (-15, 1), (-20, 1), (-24, 1), (-35, 1), (-40, 1), (-51, 1), (-52, 1), (-88, 1), (-91, 1), (-115, 1), (-123, 1), (-148, 1), (-187, 1), (-232, 1), (-235, 1), (-267, 1), (-403, 1), (-427, 1)]
sage: v[3]
[(-3, 9), (-3, 6), (-11, 2), (-19, 2), (-23, 2), (-23, 1), (-31, 2), (-31, 1), (-43, 2), (-59, 1), (-67, 2), (-83, 1), (-107, 1), (-139, 1), (-163, 2), (-211, 1), (-283, 1), (-307, 1), (-331, 1), (-379, 1), (-499, 1), (-547, 1), (-643, 1), (-883, 1), (-907, 1)]
sage: v = sage.schemes.elliptic_curves.cm.discriminants_with_bounded_class_number(8, proof=False)
sage: sorted(len(v[h]) for h in v)
[13, 25, 29, 29, 38, 84, 101, 208]
Find all class numbers for discriminant up to 50::
sage: sage.schemes.elliptic_curves.cm.discriminants_with_bounded_class_number(hmax=5, B=50)
{1: [(-3, 3), (-3, 2), (-3, 1), (-4, 2), (-4, 1), (-7, 2), (-7, 1), (-8, 1), (-11, 1), (-19, 1), (-43, 1)], 2: [(-3, 7), (-3, 5), (-3, 4), (-4, 5), (-4, 4), (-4, 3), (-7, 4), (-8, 3), (-8, 2), (-11, 3), (-15, 2), (-15, 1), (-20, 1), (-24, 1), (-35, 1), (-40, 1)], 3: [(-3, 9), (-3, 6), (-11, 2), (-19, 2), (-23, 2), (-23, 1), (-31, 2), (-31, 1), (-43, 2)], 4: [(-3, 13), (-3, 11), (-3, 8), (-4, 10), (-4, 8), (-4, 7), (-4, 6), (-7, 8), (-7, 6), (-7, 3), (-8, 6), (-8, 4), (-11, 5), (-15, 4), (-19, 5), (-19, 3), (-20, 3), (-20, 2), (-24, 2), (-35, 3), (-39, 2), (-39, 1), (-40, 2), (-43, 3)], 5: [(-47, 2), (-47, 1)]}
"""
# imports that are needed only for this function
from sage.structure.proof.proof import get_flag
# deal with input defaults and type checking
proof = get_flag(proof, 'number_field')
hmax = Integer(hmax)
# T stores the output
T = {}
# Easy case -- instead of giving error, give meaningful output
if hmax < 1:
return T
if B is None:
# Determine how far we have to go by applying Watkins's theorem.
v = [
largest_fundamental_disc_with_class_number(h)
for h in range(1, hmax + 1)
]
B = max([b for b, _ in v])
fund_count = [0] + [cnt for _, cnt in v]
else:
# Nothing to do -- set to None so we can use this later to know not
# to do a double check about how many we find.
fund_count = None
B = Integer(B)
if B <= 2:
# This is an easy special case, since there are no fundamental discriminants
# this small.
return T
# This lower bound gets used in an inner loop below.
from math import log
def lb(f):
"""Lower bound on euler_phi."""
# 1.79 > e^gamma = 1.7810724...
if f <= 1: return 0 # don't do log(log(1)) = log(0)
return f / (1.79 * log(log(f)) + 3.0 / log(log(f)))
for D in range(-B, -2):
D = Integer(D)
if is_fundamental_discriminant(D):
h_D = D.class_number(proof)
# For each fundamental discriminant D, loop through the f's such
# that h(D*f^2) could possibly be <= hmax. As explained to me by Cremona,
# we have h(D*f^2) >= (1/c)*h(D)*phi_D(f) >= (1/c)*h(D)*euler_phi(f), where
# phi_D(f) is like euler_phi(f) but the factor (1-1/p) is replaced
# by a factor of (1-kr(D,p)*p), where kr(D/p) is the Kronecker symbol.
# The factor c is 1 unless D=-4 and f>1 (when c=2) or D=-3 and f>1 (when c=3).
# Since (1-1/p) <= 1 and (1-1/p) <= (1+1/p), we see that
# euler_phi(f) <= phi_D(f).
#
# We have the following analytic lower bound on euler_phi:
#
# euler_phi(f) >= lb(f) = f / (exp(euler_gamma)*log(log(n)) + 3/log(log(n))).
#
# See Theorem 8 of Peter Clark's
# http://math.uga.edu/~pete/4400arithmeticorders.pdf
# which is a consequence of Theorem 15 of
# [Rosser and Schoenfeld, 1962].
#
# By Calculus, we see that the lb(f) is an increasing function of f >= 2.
#
# NOTE: You can visibly "see" that it is a lower bound in Sage with
# lb(n) = n/(exp(euler_gamma)*log(log(n)) + 3/log(log(n)))
# plot(lb, (n, 1, 10^4), color='red') + plot(lambda x: euler_phi(int(x)), 1, 10^4).show()
#
# So we consider f=1,2,..., until the first f with lb(f)*h_D > c*h_max.
# (Note that lb(f) is <= 0 for f=1,2, so nothing special is needed there.)
#
# TODO: Maybe we could do better using a bound for phi_D(f).
#
f = Integer(1)
chmax = hmax
if D == -3:
chmax *= 3
else:
if D == -4:
chmax *= 2
while lb(f) * h_D <= chmax:
if f == 1:
h = h_D
else:
h = (D * f * f).class_number(proof)
# If the class number of this order is within the range, then
# use it. (NOTE: In some cases there is a simple relation between
# the class number for D and D*f^2, and this could be used to
# optimize this inner loop a little.)
if h <= hmax:
z = (D, f)
if h in T:
T[h].append(z)
else:
T[h] = [z]
f += 1
for h in T:
T[h] = list(reversed(T[h]))
if fund_count is not None:
# Double check that we found the right number of fundamental
# discriminants; we might as well, since Watkins provides this
# data.
for h in T:
if len([DD for DD, ff in T[h] if ff == 1]) != fund_count[h]:
raise RuntimeError(
"number of discriminants inconsistent with Watkins's table"
)
return T
def discriminants_with_bounded_class_number(hmax, B=None, proof=None):
"""
Return dictionary with keys class numbers `h\le hmax` and values the
list of all pairs `(D, f)`, with `D<0` a fundamental discriminant such
that `Df^2` has class number `h`. If the optional bound `B` is given,
return only those pairs with fundamental `|D| \le B`, though `f` can
still be arbitrarily large.
INPUT:
- ``hmax`` -- integer
- `B` -- integer or None; if None returns all pairs
- ``proof`` -- this code calls the PARI function ``qfbclassno``, so it
could give wrong answers when ``proof``==``False``. The default is
whatever ``proof.number_field()`` is. If ``proof==False`` and `B` is
``None``, at least the number of discriminants is correct, since it
is double checked with Watkins's table.
OUTPUT:
- dictionary
In case `B` is not given, we use Mark Watkins's: "Class numbers of
imaginary quadratic fields" to compute a `B` that captures all `h`
up to `hmax` (only available for `hmax\le100`).
EXAMPLES::
sage: v = sage.schemes.elliptic_curves.cm.discriminants_with_bounded_class_number(3)
sage: list(v)
[1, 2, 3]
sage: v[1]
[(-3, 3), (-3, 2), (-3, 1), (-4, 2), (-4, 1), (-7, 2), (-7, 1), (-8, 1), (-11, 1), (-19, 1), (-43, 1), (-67, 1), (-163, 1)]
sage: v[2]
[(-3, 7), (-3, 5), (-3, 4), (-4, 5), (-4, 4), (-4, 3), (-7, 4), (-8, 3), (-8, 2), (-11, 3), (-15, 2), (-15, 1), (-20, 1), (-24, 1), (-35, 1), (-40, 1), (-51, 1), (-52, 1), (-88, 1), (-91, 1), (-115, 1), (-123, 1), (-148, 1), (-187, 1), (-232, 1), (-235, 1), (-267, 1), (-403, 1), (-427, 1)]
sage: v[3]
[(-3, 9), (-3, 6), (-11, 2), (-19, 2), (-23, 2), (-23, 1), (-31, 2), (-31, 1), (-43, 2), (-59, 1), (-67, 2), (-83, 1), (-107, 1), (-139, 1), (-163, 2), (-211, 1), (-283, 1), (-307, 1), (-331, 1), (-379, 1), (-499, 1), (-547, 1), (-643, 1), (-883, 1), (-907, 1)]
sage: v = sage.schemes.elliptic_curves.cm.discriminants_with_bounded_class_number(8, proof=False)
sage: [len(v[h]) for h in v]
[13, 29, 25, 84, 29, 101, 38, 208]
Find all class numbers for discriminant up to 50::
sage: sage.schemes.elliptic_curves.cm.discriminants_with_bounded_class_number(hmax=5, B=50)
{1: [(-3, 3), (-3, 2), (-3, 1), (-4, 2), (-4, 1), (-7, 2), (-7, 1), (-8, 1), (-11, 1), (-19, 1), (-43, 1)], 2: [(-3, 7), (-3, 5), (-3, 4), (-4, 5), (-4, 4), (-4, 3), (-7, 4), (-8, 3), (-8, 2), (-11, 3), (-15, 2), (-15, 1), (-20, 1), (-24, 1), (-35, 1), (-40, 1)], 3: [(-3, 9), (-3, 6), (-11, 2), (-19, 2), (-23, 2), (-23, 1), (-31, 2), (-31, 1), (-43, 2)], 4: [(-3, 13), (-3, 11), (-3, 8), (-4, 10), (-4, 8), (-4, 7), (-4, 6), (-7, 8), (-7, 6), (-7, 3), (-8, 6), (-8, 4), (-11, 5), (-15, 4), (-19, 5), (-19, 3), (-20, 3), (-20, 2), (-24, 2), (-35, 3), (-39, 2), (-39, 1), (-40, 2), (-43, 3)], 5: [(-47, 2), (-47, 1)]}
"""
# imports that are needed only for this function
from sage.structure.proof.proof import get_flag
import math
from sage.misc.functional import round
# deal with input defaults and type checking
proof = get_flag(proof, 'number_field')
hmax = Integer(hmax)
# T stores the output
T = {}
# Easy case -- instead of giving error, give meaningful output
if hmax < 1:
return T
if B is None:
# Determine how far we have to go by applying Watkins's theorem.
v = [largest_fundamental_disc_with_class_number(h) for h in range(1, hmax+1)]
B = max([b for b,_ in v])
fund_count = [0] + [cnt for _,cnt in v]
else:
# Nothing to do -- set to None so we can use this later to know not
# to do a double check about how many we find.
fund_count = None
B = Integer(B)
if B <= 2:
# This is an easy special case, since there are no fundamental discriminants
# this small.
return T
# This lower bound gets used in an inner loop below.
from math import log
def lb(f):
"""Lower bound on euler_phi."""
# 1.79 > e^gamma = 1.7810724...
if f <= 1: return 0 # don't do log(log(1)) = log(0)
return f/(1.79*log(log(f)) + 3.0/log(log(f)))
for D in range(-B, -2):
D = Integer(D)
if is_fundamental_discriminant(D):
h_D = D.class_number(proof)
# For each fundamental discriminant D, loop through the f's such
# that h(D*f^2) could possibly be <= hmax. As explained to me by Cremona,
# we have h(D*f^2) >= (1/c)*h(D)*phi_D(f) >= (1/c)*h(D)*euler_phi(f), where
# phi_D(f) is like euler_phi(f) but the factor (1-1/p) is replaced
# by a factor of (1-kr(D,p)*p), where kr(D/p) is the Kronecker symbol.
# The factor c is 1 unless D=-4 and f>1 (when c=2) or D=-3 and f>1 (when c=3).
# Since (1-1/p) <= 1 and (1-1/p) <= (1+1/p), we see that
# euler_phi(f) <= phi_D(f).
#
# We have the following analytic lower bound on euler_phi:
#
# euler_phi(f) >= lb(f) = f / (exp(euler_gamma)*log(log(n)) + 3/log(log(n))).
#
# See Theorem 8 of Peter Clark's
# http://math.uga.edu/~pete/4400arithmeticorders.pdf
# which is a consequence of Theorem 15 of
# [Rosser and Schoenfeld, 1962].
#
# By Calculus, we see that the lb(f) is an increasing function of f >= 2.
#
# NOTE: You can visibly "see" that it is a lower bound in Sage with
# lb(n) = n/(exp(euler_gamma)*log(log(n)) + 3/log(log(n)))
# plot(lb, (n, 1, 10^4), color='red') + plot(lambda x: euler_phi(int(x)), 1, 10^4).show()
#
# So we consider f=1,2,..., until the first f with lb(f)*h_D > c*h_max.
# (Note that lb(f) is <= 0 for f=1,2, so nothing special is needed there.)
#
# TODO: Maybe we could do better using a bound for for phi_D(f).
#
f = Integer(1)
chmax=hmax
if D==-3:
chmax*=3
else:
if D==-4:
chmax*=2
while lb(f)*h_D <= chmax:
if f == 1:
h = h_D
else:
h = (D*f*f).class_number(proof)
# If the class number of this order is within the range, then
# use it. (NOTE: In some cases there is a simple relation between
# the class number for D and D*f^2, and this could be used to
# optimize this inner loop a little.)
if h <= hmax:
z = (D, f)
if h in T:
T[h].append(z)
else:
T[h] = [z]
f += 1
for h in T:
T[h] = list(reversed(T[h]))
if fund_count is not None:
# Double check that we found the right number of fundamental
# discriminants; we might as well, since Watkins provides this
# data.
for h in T:
if len([D for D,f in T[h] if f==1]) != fund_count[h]:
raise RuntimeError("number of discriminants inconsistent with Watkins's table")
return T
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"""
Return representatives for the classes of binary quadratic forms
of discriminant `D`.
INPUT:
- ``D`` -- (integer) a discriminant
- ``primitive_only`` -- (boolean, default True): if True, only
return primitive forms.
OUTPUT:
(list) A lexicographically-ordered list of inequivalent reduced
representatives for the equivalence classes of binary quadratic
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
`\QQ(\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(73)
[-6*x^2 + 5*x*y + 2*y^2,
-6*x^2 + 7*x*y + y^2,
-4*x^2 + 3*x*y + 4*y^2,
-4*x^2 + 3*x*y + 4*y^2,
-4*x^2 + 5*x*y + 3*y^2,
-3*x^2 + 5*x*y + 4*y^2,
-3*x^2 + 7*x*y + 2*y^2,
-2*x^2 + 5*x*y + 6*y^2,
-2*x^2 + 7*x*y + 3*y^2,
-x^2 + 7*x*y + 6*y^2,
x^2 + 7*x*y - 6*y^2,
2*x^2 + 5*x*y - 6*y^2,
2*x^2 + 7*x*y - 3*y^2,
3*x^2 + 5*x*y - 4*y^2,
3*x^2 + 7*x*y - 2*y^2,
4*x^2 + 3*x*y - 4*y^2,
4*x^2 + 3*x*y - 4*y^2,
4*x^2 + 5*x*y - 3*y^2,
6*x^2 + 5*x*y - 2*y^2,
6*x^2 + 7*x*y - y^2]
sage: BinaryQF_reduced_representatives(76, primitive_only=True)
[-5*x^2 + 4*x*y + 3*y^2,
-5*x^2 + 6*x*y + 2*y^2,
-3*x^2 + 4*x*y + 5*y^2,
-3*x^2 + 8*x*y + y^2,
-2*x^2 + 6*x*y + 5*y^2,
-x^2 + 8*x*y + 3*y^2,
x^2 + 8*x*y - 3*y^2,
2*x^2 + 6*x*y - 5*y^2,
3*x^2 + 4*x*y - 5*y^2,
3*x^2 + 8*x*y - y^2,
5*x^2 + 4*x*y - 3*y^2,
5*x^2 + 6*x*y - 2*y^2]
Check that the primitive_only keyword does something::
sage: BinaryQF_reduced_representatives(4*5, primitive_only=True)
[-x^2 + 4*x*y + y^2,
-x^2 + 4*x*y + y^2,
x^2 + 4*x*y - y^2,
x^2 + 4*x*y - y^2]
sage: BinaryQF_reduced_representatives(4*5, primitive_only=False)
[-2*x^2 + 2*x*y + 2*y^2,
-2*x^2 + 2*x*y + 2*y^2,
-x^2 + 4*x*y + y^2,
-x^2 + 4*x*y + y^2,
x^2 + 4*x*y - y^2,
x^2 + 4*x*y - y^2,
2*x^2 + 2*x*y - 2*y^2,
2*x^2 + 2*x*y - 2*y^2]
"""
D = ZZ(D)
# 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.arith.srange import xsrange
D4 = D % 4
if D4 == 2 or D4 == 3:
raise ValueError("%s is not a discriminant" % D)
if D > 0: # Indefinite
# We follow the description of Buchmann/Vollmer 6.7.1
if D.is_square():
# Buchmann/Vollmer 6.7.1. require D squarefree.
raise ValueError("%s is a square" % D)
sqrt_d = D.sqrt(prec=53)
for b in xsrange(1, sqrt_d.floor() + 1):
if (D - b) % 2 != 0:
continue
A = (D - b**2) / 4
Low_a = ((sqrt_d - b) / 2).ceil()
High_a = (A.sqrt(prec=53)).floor()
for a in xsrange(Low_a, High_a + 1):
if a == 0:
continue
c = -A / a
if c in ZZ:
if (not primitive_only) or gcd([a, b, c]) == 1:
Q = BinaryQF(a, b, c)
Q1 = BinaryQF(-a, b, -c)
Q2 = BinaryQF(c, b, a)
Q3 = BinaryQF(-c, b, -a)
form_list.append(Q)
form_list.append(Q1)
form_list.append(Q2)
form_list.append(Q3)
else: # Definite
# 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