def spectrum(n, field):
q = field.order
p = field.char
# (1)
eps = 1 if n % 2 == 0 else e
a1 = [(q ** n - eps) // (q - e)]
# (2)
a2 = SemisimpleElements(q, n, min_length=2, sign=e)
# (3)
a3 = []
k = 1
d = gcd(n, q - e)
while True:
n1 = n - p ** (k - 1) - 1
if n1 < 1: break
eps = 1 if n1 % 2 == 0 else e
a3.append(p ** k * (q ** n1 - eps) / gcd(d, n1))
k += 1
# (4)
a4 = MixedElements(q, n, lambda k: p ** (k - 1) + 1,
lambda k: p ** k, min_length=2, sign=e)
# (5)
k = numeric.get_exponent(n - 1, p)
a5 = [] if k is None else [p * (n - 1) * d]
# (6)
a6 = [p * gcd(2, q - 1) * (q + e)] if n == 4 else []
return itertools.chain(a1, a2, a3, a4, a5, a6)
def _omega_pm_spectrum_odd_c(n: int, field: 'Field', sign: int) -> Iterable[int]:
"""Spectra of Omega^e_{2n}(q) for odd q.
Based on [1, Corollary 8] and [3, Lemma 2.3]
Point 3 of [1, Corollary 8] contains an error
that was corrected in [3, Lemma 2.3]
"""
n //= 2
q = field.order
p = field.char
def nk(k):
return (p ** (k - 1) + 3) // 2
# (1)
a1 = [(q ** n - sign) // 2]
# (2)
a2 = SemisimpleElements(q, n, min_length=2, parity=sign)
# (3)
a3 = []
k = 1
while True:
n_k = nk(k)
if n_k >= n:
break
for delta in [1, -1]:
dk = gcd(4, q ** n_k - sign * delta) // 2
a3.append(
p ** k *
lcm(
dk,
(q**(n - n_k) - delta) // dk
)
)
k += 1
# (4)
a4 = MixedElements(q, n, nk, lambda k: p ** k, min_length=2)
# (5)
a5 = []
for elem in SemisimpleElements(q, n - 2, min_length=2, parity=sign):
a5.append(elem.lcm(SpectraElement(p, q, [1], [-1])))
a5.append(elem.lcm(SpectraElement(p, q, [1], [1])))
# (6)
t = (q ** (n - 2) - sign) // 2
a6 = [p * lcm(q - 1, t), p * lcm(q + 1, t)]
# (7)
k = numeric.get_exponent(2 * n - 3, p)
a7 = [] if k is None else [p * (2 * n - 3) * gcd(4, q ** n - sign) // 2]
# (8)
a8 = [p * (q * q - 1), p * (q * q + 1)] if n == 4 and sign == 1 else []
# (9)
a9 = [9 * (q - 1), 9 * (q + 1)] if n == 4 and p == 3 and sign == 1 else []
return itertools.chain(a1, a2, a3, a4, a5, a6, a7, a8, a9)
def spectrum(n: int, field: 'Field') -> Iterable[int]:
q = field.order
p = field.char
# (1)
eps = 1 if n % 2 == 0 else e
a1 = [(q ** n - eps) // (q - e)]
# (2)
a2 = SemisimpleElements(q, n, min_length=2, sign=e)
# (3)
a3 = []
k = 1
d = gcd(n, q - e)
while True:
n1 = n - p ** (k - 1) - 1
if n1 < 1:
break
eps = 1 if n1 % 2 == 0 else e
a3.append(p ** k * (q ** n1 - eps) // gcd(d, n1))
k += 1
# (4)
a4 = MixedElements(q, n,
lambda k: p ** (k - 1) + 1,
lambda k: p ** k, min_length=2, sign=e)
# (5)
k = numeric.get_exponent(n - 1, p)
a5 = [] if k is None else [p * (n - 1) * d]
# (6)
a6 = [p * gcd(2, q - 1) * (q + e)] if n == 4 else []
return itertools.chain(a1, a2, a3, a4, a5, a6)
def spectrum(n, field):
q = field.order
p = field.char
d = gcd(n, q - e)
# (1)
eps = 1 if n % 2 == 0 else e
a1 = [(q ** n - eps) // ((q - e) * d)]
# (2)
a2 = []
eps = lambda s: 1 if s % 2 == 0 else e
for n1 in xrange(1, (n + 2) // 2):
pair = (n1, n - n1)
signs = (-eps(n1), -eps(n - n1))
a2.append(lcm(q ** pair[0] + signs[0], q ** pair[1] + signs[1]) //
gcd(n // gcd(n1, n - n1), q - e))
# (3)
a3 = SemisimpleElements(q, n, min_length=3, sign=e)
# (4)
a4 = []
k = 1
while True:
n1 = n - p ** (k - 1) - 1
if n1 < 1: break
eps = 1 if n1 % 2 == 0 else e
a4.append(p ** k * (q ** n1 - eps) / d)
k += 1
# (5)
a5 = MixedElements(q, n, lambda k: p ** (k - 1) + 1,
lambda k: p ** k, min_length=2, sign=e)
# (6)
k = numeric.get_exponent(n - 1, p)
a6 = [] if k is None else [p * (n - 1)]
return itertools.chain(a1, a2, a3, a4, a5, a6)
def _omega_pm_spectrum_odd_c(n, field, sign):
"""Spectra of Omega^e_{2n}(q) for odd q.
[1, Corollary 8]
"""
n //= 2
q = field.order
p = field.char
nk = lambda k: (p**(k - 1) + 3) // 2
# (1)
a1 = [(q**n - sign) // 2]
# (2)
a2 = SemisimpleElements(q, n, min_length=2, parity=sign)
# (3)
a3 = []
k = 1
while True:
n_k = nk(k)
if n_k >= n:
break
dk = gcd(4, q**n_k - sign) // 2
a3.append(p**k * lcm(dk, (q**(n - n_k) + 1) // dk))
a3.append(p**k * lcm(dk, (q**(n - n_k) - 1) // dk))
k += 1
# (4)
a4 = MixedElements(q, n, nk, lambda k: p**k, min_length=2)
# (5)
a5 = []
for elem in SemisimpleElements(q, n - 2, min_length=2, parity=sign):
a5.append(elem.lcm(SpectraElement(p, q, [1], [-1])))
a5.append(elem.lcm(SpectraElement(p, q, [1], [1])))
# (6)
t = (q**(n - 2) - sign) // 2
a6 = [p * lcm(q - 1, t), p * lcm(q + 1, t)]
# (7)
k = numeric.get_exponent(2 * n - 3, p)
a7 = [] if k is None else [p * (2 * n - 3) * gcd(4, q**n - sign) // 2]
# (8)
a8 = [p * (q * q - 1), p * (q * q + 1)] if n == 4 and sign == 1 else []
# (9)
a9 = [9 * (q - 1), 9 * (q + 1)] if n == 4 and p == 3 and sign == 1 else []
return itertools.chain(a1, a2, a3, a4, a5, a6, a7, a8, a9)
def _add_element(self, a):
"""Add new spectrum element
"""
# these are the indices of future neighbors of a
neighbors = []
# memorize initial length so that we don't have to process newly added
# vertices
l = len(self._vertices)
for i in range(l):
b = self._vertices[i]
d = numeric.gcd(a, b)
if d == 1:
continue
bd = numeric.prime_part(b, d)
if bd == 1:
self._vertices[i] = d
for neighbor in neighbors:
self._set_adjacency(i, neighbor, True)
neighbors.append(i)
else:
self._vertices[i] = bd
dIndex = self.clone_vertex(i, d)
for neighbor in neighbors:
self._adjacency[dIndex][neighbor] = True
neighbors.append(dIndex)
a = numeric.prime_part(a, d)
if a == 1:
break
if a > 1:
index = self._add_vertex(a)
for neighbor in neighbors:
self._set_adjacency(index, neighbor, True)
def _add_element(self, a):
"""Add new spectrum element
"""
# these are the indices of future neighbors of a
neighbors = []
# memorize initial length so that we don't have to process newly added
# vertices
l = len(self._vertices)
for i in range(l):
b = self._vertices[i]
d = numeric.gcd(a, b)
if d == 1: continue
bd = numeric.prime_part(b, d)
if bd == 1:
self._vertices[i] = d
for neighbor in neighbors:
self._set_adjacency(i, neighbor, True)
neighbors.append(i)
else:
self._vertices[i] = bd
dIndex = self.clone_vertex(i, d)
for neighbor in neighbors:
self._adjacency[dIndex][neighbor] = True
neighbors.append(dIndex)
a = numeric.prime_part(a, d)
if a == 1: break
if a > 1:
index = self._add_vertex(a)
for neighbor in neighbors:
self._set_adjacency(index, neighbor, True)
def _projective_special_unitary_order(n, field):
q = field.order
return (Integer({field.char: field.pow * (n * (n - 1) / 2)}) * prod(
(Integer(q**i - 1) * Integer(q**i + 1)
for i in xrange(2, n // 2 + 1))) * prod(
(Integer(q**(2 * i + 1) + 1))
for i in xrange(1, (n + 1) // 2)) * Integer(q - 1) * Integer(
(q + 1) // gcd(n, q + 1)))
def _projective_special_unitary_order(n, field):
q = field.order
return (Integer({field.char: field.pow * (n * (n - 1) / 2)}) *
prod((Integer(q ** i - 1) *
Integer(q ** i + 1) for i in xrange(2, n // 2 + 1))) *
prod((Integer(q ** (2 * i + 1) + 1)) for i in xrange(1,
(n + 1) // 2)) *
Integer(q - 1) * Integer((q + 1) // gcd(n, q + 1)))
def spectrum(n, field):
q = field.order
p = field.char
d = gcd(n, q - e)
# (1)
eps = 1 if n % 2 == 0 else e
a1 = [(q**n - eps) // ((q - e) * d)]
# (2)
a2 = []
eps = lambda s: 1 if s % 2 == 0 else e
for n1 in xrange(1, (n + 2) // 2):
pair = (n1, n - n1)
signs = (-eps(n1), -eps(n - n1))
a2.append(
lcm(q**pair[0] + signs[0], q**pair[1] + signs[1]) //
gcd(n // gcd(n1, n - n1), q - e))
# (3)
a3 = SemisimpleElements(q, n, min_length=3, sign=e)
# (4)
a4 = []
k = 1
while True:
n1 = n - p**(k - 1) - 1
if n1 < 1:
break
eps = 1 if n1 % 2 == 0 else e
a4.append(p**k * (q**n1 - eps) / d)
k += 1
# (5)
a5 = MixedElements(q,
n,
lambda k: p**(k - 1) + 1,
lambda k: p**k,
min_length=2,
sign=e)
# (6)
k = numeric.get_exponent(n - 1, p)
a6 = [] if k is None else [p * (n - 1)]
return itertools.chain(a1, a2, a3, a4, a5, a6)
def order(n, field):
omega_order = _omega_pm_order(e)(n, field)
q = field.order
n //= 2
# gcd(4, q^n-e)
omega_order.div_by_prime(2)
if e == 1:
divisor = 1 if (n % 2 == 1 and q % 4 == 3) else 2
else:
divisor = 2 if (n % 2 == 1 and q % 4 == 3) else 1
if divisor > 1:
omega_order.div_by_prime(2)
return omega_order * gcd(q - e, 2)
def order(n, field):
omega_order = _omega_pm_order(e)(n, field)
q = field.order
n //= 2
# gcd(4, q^n-e)
omega_order.div_by_prime(2)
if e == 1:
divisor = 1 if (n % 2 == 1 and q % 4 == 3) else 2
else:
divisor = 2 if (n % 2 == 1 and q % 4 == 3) else 1
if divisor > 1:
omega_order.div_by_prime(2)
return omega_order * gcd(q - e, 2)
def _omega_pm_spectrum_odd_c(n, field, sign):
n //= 2
q = field.order
p = field.char
nk = lambda k: (p ** (k - 1) + 3) // 2
# (1)
a1 = [(q ** n - sign) // 2]
# (2)
a2 = SemisimpleElements(q, n, min_length=2, parity=sign)
# (3)
a3 = []
k = 1
while True:
n_k = nk(k)
if n_k >= n: break
dk = gcd(4, q ** n_k - sign) // 2
a3.append(p ** k * lcm(dk, (q ** (n - n_k) + 1) // dk))
a3.append(p ** k * lcm(dk, (q ** (n - n_k) - 1) // dk))
k += 1
# (4)
a4 = MixedElements(q, n, nk, lambda k: p ** k, min_length=2)
# (5)
a5 = []
for elem in SemisimpleElements(q, n - 2, min_length=2, parity=sign):
a5.append(elem.lcm(SpectraElement(p, q, [1], [-1])))
a5.append(elem.lcm(SpectraElement(p, q, [1], [1])))
# (6)
t = (q ** (n - 2) - sign) // 2
a6 = [p * lcm(q - 1, t), p * lcm(q + 1, t)]
# (7)
k = numeric.get_exponent(2 * n - 3, p)
a7 = [] if k is None else [p * (2 * n - 3) * gcd(4, q ** n - sign) // 2]
# (8)
a8 = [p * (q * q - 1), p * (q * q + 1)] if n == 4 and sign == 1 else []
# (9)
a9 = [9 * (q - 1), 9 * (q + 1)] if n == 4 and p == 3 and sign == 1 else []
return itertools.chain(a1, a2, a3, a4, a5, a6, a7, a8, a9)
def spectrum(field):
q = field.order
p = field.char
d = gcd(3, q - e)
# (1)
a1 = [(q**6 - 1) // d, (q**6 + e * q**3 + 1) // d,
(q * q + e * q + 1) * (q**4 - q * q + 1) // d,
(q - e) * (q * q + 1) * (q**3 + e) // d,
(q * q - 1) * (q**4 + 1) // d, (q + e) * (q**5 - e) // d,
q**5 - e]
# (2)
a2 = [
p * x
for x in [(q**6 - 1) // (d * (q - e)), (q**5 - e) // d, q**4 -
1, (q**3 - e) * (q + e), (q - e) * (q**3 + e) // d]
]
# (3)
pA2 = 4 if p == 2 else p
a3 = [
pA2 * x for x in [(q**3 - e) * (q + e) // d, (q**4 + q * q + 1) //
d, (q**4 - 1) // d]
]
# (4)
pA3 = p * p if p in (2, 3) else p
a4 = [pA3 * x for x in [(q * q + 1) * (q - e) // d, q**q - 1]]
# (5)
pD4 = 8 if p == 2 else p * p if p in (3, 5) else p
a5 = [
pD4 * x
for x in [q - e, (q * q - 1) // d, (q * q + e * q + 1) // d]
]
# (6)
pD5 = 8 if p == 2 else p * p if p in (3, 5, 7) else p
a6 = [pD5 * (q - e) // d]
# (7)
pE6 = 16 if p == 2 else 27 if p == 3 else p * p if p in (5, 7,
11) else p
a7 = [pE6]
return itertools.chain(a1, a2, a3, a4, a5, a6, a7)
def _e6_order(field):
q = field.order
return (_order_product(field, 36, [6, 4, 3, 3, 2, 1, 1], [9, 5, 3, 3, 1]) *
Integer((q - 1) // gcd(3, q - 1)))
def test_symplectic_gcd(self, n):
max_elems = max_orders.symplectic_2(n)
expected = numeric.gcd(*max_elems)
self.assertEqual(expected, max_orders.symplectic_2_gcd(n))
def _projective_special_linear_order(n, field):
q = field.order
return (Integer({field.char: field.pow * (n * (n - 1) / 2)}) *
prod((Integer(q ** i - 1) for i in xrange(3, n + 1))) *
Integer(q + 1) * Integer((q - 1) // gcd(n, q - 1)))
def _e6_order(field):
q = field.order
return (_order_product(field, 36, [6, 4, 3, 3, 2, 1, 1], [9, 5, 3, 3, 1]) *
Integer((q - 1) // gcd(3, q - 1)))
def _e7_order(field):
q = field.order
return (_order_product(field, 63, [9, 7, 6, 5, 4, 3, 3, 2, 1, 1],
[9, 7, 5, 3, 3, 1]) * Integer((q - 1) // gcd(2, q - 1)))
def _2e6_order(field):
q = field.order
return (_order_product(field, 36, [9, 6, 5, 4, 3, 3, 2, 1], [3, 3, 1, 1]) *
Integer((q + 1) // gcd(3, q + 1)))
def _projective_special_linear_order(n, field):
q = field.order
return (Integer({field.char: field.pow * (n * (n - 1) / 2)}) * prod(
(Integer(q**i - 1)
for i in xrange(3, n + 1))) * Integer(q + 1) * Integer(
(q - 1) // gcd(n, q - 1)))
def _e7_order(field):
q = field.order
return (_order_product(field, 63, [9, 7, 6, 5, 4, 3, 3, 2, 1, 1],
[9, 7, 5, 3, 3, 1]) * Integer(
(q - 1) // gcd(2, q - 1)))
def _2e6_order(field):
q = field.order
return (_order_product(field, 36, [9, 6, 5, 4, 3, 3, 2, 1], [3, 3, 1, 1]) *
Integer((q + 1) // gcd(3, q + 1)))