def crt(inlist):
"""
Chinese Remainder Theorem, Algo. 1.3.11 of Cohen's Book.
"""
k = len(inlist)
if k < 2:
raise ValueError("nothing to be done for one element")
ccj = [None] # gabage to simplify loop index
ellist = list(inlist)
ellist.sort()
modulus = 1
for j in range(1, k):
modulus *= ellist[j - 1][1]
d, tpl = gcd.gcd_of_list([modulus % ellist[j][1], ellist[j][1]])
if d > 1:
raise ValueError("moduli are not pairwise coprime")
ccj.append(tpl[0])
yj = [ellist[0][0] % ellist[0][1]]
for j in range(1, k):
ctp = yj[-1]
for i in range(j - 2, -1, -1):
ctp = yj[i] + ellist[i][1] * ctp
yj.append(((ellist[j][0] - ctp) * ccj[j]) % ellist[j][1])
outp = yj.pop()
for j in range(k - 2, -1, -1):
outp = yj.pop() + (ellist[j][1]) * outp
return outp
def class_number(disc, limit_of_disc=100000000):
"""
Return class number with the given discriminant by counting reduced forms.
Not only fundamental discriminant.
"""
if disc & 3 not in (0, 1):
raise ValueError("a discriminant must be 0 or 1 mod 4")
if disc >= 0:
raise ValueError("a discriminant must be negative")
if -disc >= limit_of_disc:
warnings.warn("the discriminant seems to have too big absolute value")
h = 1
b = disc & 1
c_b = int(math.sqrt(-disc / 3))
while b <= c_b:
q = (b**2 - disc) >> 2
a = b
if a <= 1:
a = 2
while a**2 <= q:
if q % a == 0 and gcd.gcd_of_list([a, b, q // a])[0] == 1:
if a == b or a**2 == q or b == 0:
h += 1
else:
h += 2
a += 1
b += 2
return h
def class_number(disc, limit_of_disc=100000000):
"""
Return class number with the given discriminant by counting reduced forms.
Not only fundamental discriminant.
"""
if disc % 4 not in (0, 1):
raise ValueError("a discriminant must be 0 or 1 mod 4")
if disc >= 0:
raise ValueError("a discriminant must be negative")
if -disc >= limit_of_disc:
warnings.warn("the discriminant seems to have too big absolute value")
h = 1
b = disc % 2
c_b = int(math.sqrt(-disc / 3))
while b <= c_b:
q = (b**2 - disc) // 4
a = b
if a <= 1:
a = 2
while a**2 <= q:
if q % a == 0 and gcd.gcd_of_list([a, b, q//a])[0] == 1:
if a == b or a**2 == q or b == 0:
h += 1
else:
h += 2
a += 1
b += 2
return h
def crt(inlist):
"""
Chinese Remainder Theorem, Algo. 1.3.11 of Cohen's Book.
"""
k = len(inlist)
if k < 2:
raise ValueError("nothing to be done for one element")
ccj = [None] # gabage to simplify loop index
ellist = list(inlist)
ellist.sort()
modulus = 1
for j in range(1, k):
modulus *= ellist[j - 1][1]
d, tpl = gcd.gcd_of_list([modulus % ellist[j][1], ellist[j][1]])
if d > 1:
raise ValueError("moduli are not pairwise coprime")
ccj.append(tpl[0])
yj = [ellist[0][0] % ellist[0][1]]
for j in range(1, k):
ctp = yj[-1]
for i in range(j - 2, -1, -1):
ctp = yj[i] + ellist[i][1] * ctp
yj.append(((ellist[j][0] - ctp) * ccj[j]) % ellist[j][1])
outp = yj.pop()
for j in range(k - 2, -1, -1):
outp = yj.pop() + (ellist[j][1]) * outp
return outp
def compositePDF(f_1, f_2):
"""
Return the reduced form of composition of the given forms 'f_1' and 'f_2'.
'f_1' and 'f_2' are quadratic forms with same disc.
"""
if gcd.gcd_of_list(f_1)[0] != 1:
raise ValueError(
"coefficients of a quadratic form must be relatively prime")
if gcd.gcd_of_list(f_2)[0] != 1:
raise ValueError(
"coefficients of a quadratic form must be relatively prime")
if disc(f_1) != disc(f_2):
raise ValueError("two quadratic forms must have same discriminant")
if f_1[0] > f_2[0]:
f_1, f_2 = f_2, f_1
s = (f_1[1] + f_2[1]) >> 1
n = f_2[1] - s
if f_2[0] % f_1[0] == 0:
y_1 = 0
d = f_1[0]
else:
u, v, d = euclid_exd(f_2[0], f_1[0])
y_1 = u
if s % d == 0:
y_2 = -1
x_2 = 0
d_1 = d
else:
u, v, d_1 = euclid_exd(s, d)
x_2 = u
y_2 = -v
v_1 = f_1[0] // d_1
v_2 = f_2[0] // d_1
r = (y_1 * y_2 * n - x_2 * f_2[2]) % v_1
b_3 = f_2[1] + 2 * v_2 * r
a_3 = v_1 * v_2
c_3 = (f_2[2] * d_1 + r * (f_2[1] + v_2 * r)) // v_1
return reducePDF((a_3, b_3, c_3))
def compositePDF(f_1, f_2):
"""
Return the reduced form of composition of the given forms 'f_1' and 'f_2'.
'f_1' and 'f_2' are quadratic forms with same disc.
"""
if gcd.gcd_of_list(f_1)[0] != 1:
raise ValueError("coefficients of a quadratic form must be relatively prime")
if gcd.gcd_of_list(f_2)[0] != 1:
raise ValueError("coefficients of a quadratic form must be relatively prime")
if disc(f_1) != disc(f_2):
raise ValueError("two quadratic forms must have same discriminant")
if f_1[0] > f_2[0]:
f_1, f_2 = f_2, f_1
s = (f_1[1] + f_2[1]) // 2
n = f_2[1] - s
if f_2[0] % f_1[0] == 0:
y_1 = 0
d = f_1[0]
else:
u, v, d = euclid_exd(f_2[0], f_1[0])
y_1 = u
if s % d == 0:
y_2 = -1
x_2 = 0
d_1 = d
else:
u, v, d_1 = euclid_exd(s, d)
x_2 = u
y_2 = -v
v_1 = f_1[0] // d_1
v_2 = f_2[0] // d_1
r = (y_1*y_2*n - x_2*f_2[2]) % v_1
b_3 = f_2[1] + 2*v_2*r
a_3 = v_1*v_2
c_3 = (f_2[2]*d_1 + r*(f_2[1] + v_2*r)) // v_1
return reducePDF((a_3, b_3, c_3))
def class_group(disc, limit_of_disc=100000000):
"""
Return the class number and the class group with the given discriminant
by counting reduced forms. Not only fundamental discriminant.
"""
if disc % 4 not in (0, 1):
raise ValueError("a discriminant must be 0 or 1 mod 4")
if disc >= 0:
raise ValueError("a discriminant must be negative")
if -disc >= limit_of_disc:
warnings.warn("the discriminant seems to have too big absolute value")
h = 1
b = disc % 2
c_b = int(math.sqrt(-disc / 3))
ret_list = []
f_a = 1
if disc % 4 == 0:
f_b = 0
f_c = -(disc // 4)
else:
f_b = 1
f_c = -((disc - 1) // 4)
unit = (f_a, f_b, f_c)
ret_list.append(ReducedQuadraticForm(unit, unit))
while b <= c_b:
q = (b**2 - disc) // 4
a = b
if a <= 1:
a = 2
while a**2 <= q:
if q % a == 0 and gcd.gcd_of_list([a, b, q//a])[0] == 1:
f_c = (b**2 - disc) // (4 * a)
if a == b or a**2 == q or b == 0:
h += 1
ret_list.append(ReducedQuadraticForm((a, b, f_c), unit))
else:
h += 2
ret_list.append(ReducedQuadraticForm((a, b, f_c), unit))
ret_list.append(ReducedQuadraticForm((a, -b, f_c), unit))
a += 1
b += 2
return h, ret_list
def class_group(disc, limit_of_disc=100000000):
"""
Return the class number and the class group with the given discriminant
by counting reduced forms. Not only fundamental discriminant.
"""
if disc & 3 not in (0, 1):
raise ValueError("a discriminant must be 0 or 1 mod 4")
if disc >= 0:
raise ValueError("a discriminant must be negative")
if -disc >= limit_of_disc:
warnings.warn("the discriminant seems to have too big absolute value")
h = 1
b = disc & 1
c_b = int(math.sqrt(-disc / 3))
ret_list = []
f_a = 1
if disc & 3 == 0:
f_b = 0
f_c = -(disc >> 2)
else:
f_b = 1
f_c = -((disc - 1) >> 2)
unit = (f_a, f_b, f_c)
ret_list.append(ReducedQuadraticForm(unit, unit))
while b <= c_b:
q = (b**2 - disc) >> 2
a = b
if a <= 1:
a = 2
while a**2 <= q:
if q % a == 0 and gcd.gcd_of_list([a, b, q // a])[0] == 1:
f_c = (b**2 - disc) // (4 * a)
if a == b or a**2 == q or b == 0:
h += 1
ret_list.append(ReducedQuadraticForm((a, b, f_c), unit))
else:
h += 2
ret_list.append(ReducedQuadraticForm((a, b, f_c), unit))
ret_list.append(ReducedQuadraticForm((a, -b, f_c), unit))
a += 1
b += 2
return h, ret_list
def __init__(self, valuelist, polynomial, precompute=False):
if len(polynomial) != len(valuelist[0])+1:
raise ValueError
self.value = valuelist
self.coeff = valuelist[0]
self.denom = valuelist[1]
self.degree = len(polynomial) - 1
self.polynomial = polynomial
self.field = NumberField(self.polynomial)
Gcd = gcd.gcd_of_list(self.coeff)
GCD = gcd.gcd(Gcd[0], self.denom)
if GCD != 1:
self.coeff = [i//GCD for i in self.coeff]
self.denom = self.denom//GCD
if precompute:
self.getConj()
self.getApprox()
self.getCharPoly()
def __init__(self, valuelist, polynomial, precompute=False):
if len(polynomial) != len(valuelist[0]) + 1:
raise ValueError
self.value = valuelist
self.coeff = valuelist[0]
self.denom = valuelist[1]
self.degree = len(polynomial) - 1
self.polynomial = polynomial
self.field = NumberField(self.polynomial)
Gcd = gcd.gcd_of_list(self.coeff)
GCD = gcd.gcd(Gcd[0], self.denom)
if GCD != 1:
self.coeff = [i // GCD for i in self.coeff]
self.denom = self.denom // GCD
if precompute:
self.getConj()
self.getApprox()
self.getCharPoly()
def randomele(disc, unit):
"""
Return a reduced random form with the given discriminant and the given unit.
Also random element is not unit.
"""
limit = int(math.sqrt(-disc / 3))
while True:
a = bigrandom.randrange(1, limit + 1)
ind = 0
while ind < 2*a:
b = bigrandom.randrange(a)
if bigrandom.randrange(2):
b = -b
tp = disc - b**2
if tp % (-4 * a) == 0:
c = tp // (-4 * a)
if gcd.gcd_of_list([a, b, c])[0] != 1:
continue
red = reducePDF((a, b, c))
if red != unit:
return ReducedQuadraticForm(red, unit)
ind += 1
def randomele(disc, unit):
"""
Return a reduced random form with the given discriminant and the given unit.
Also random element is not unit.
"""
limit = int(math.sqrt(-disc / 3))
while True:
a = bigrandom.randrange(1, limit + 1)
ind = 0
while ind < 2 * a:
b = bigrandom.randrange(a)
if bigrandom.randrange(2):
b = -b
tp = disc - b**2
if tp % (-4 * a) == 0:
c = tp // (-4 * a)
if gcd.gcd_of_list([a, b, c])[0] != 1:
continue
red = reducePDF((a, b, c))
if red != unit:
return ReducedQuadraticForm(red, unit)
ind += 1
def testGcdOfList(self):
self.assertEqual([8, [1]], gcd.gcd_of_list([8]))
self.assertEqual([1, [-4, 3]], gcd.gcd_of_list([8, 11]))
self.assertEqual([1, [-4, 3, 0, 0, 0, 0, 0]],
gcd.gcd_of_list([8, 11, 10, 9, 6, 5, 4]))