def __init__(self, parent, x, bits=None, nterms=None):
"""
EXAMPLES::
sage: sage.rings.contfrac.ContinuedFraction(CFF,[1,2,3,4,1,2])
[1, 2, 3, 4, 1, 2]
sage: sage.rings.contfrac.ContinuedFraction(CFF,[1,2,3,4,-1,2])
Traceback (most recent call last):
...
ValueError: each entry except the first must be positive
"""
FieldElement.__init__(self, parent)
if isinstance(x, ContinuedFraction):
self._x = list(x._x)
elif isinstance(x, (list, tuple)):
x = [ZZ(a) for a in x]
for i in range(1, len(x)):
if x[i] <= 0:
raise ValueError, "each entry except the first must be positive"
self._x = list(x)
else:
self._x = [
ZZ(a)
for a in continued_fraction_list(x, bits=bits, nterms=nterms)
]
def selmer_group(self, S, m, proof=True, orders=False):
r"""
Compute the group `\QQ(S,m)`.
INPUT:
- ``S`` -- a set of primes
- ``m`` -- a positive integer
- ``proof`` -- ignored
- ``orders`` (default False) -- if True, output two lists, the
generators and their orders
OUTPUT:
A list of generators of `\QQ(S,m)` (and, optionally, their
orders in `\QQ^\times/(\QQ^\times)^m`). This is the subgroup
of `\QQ^\times/(\QQ^\times)^m` consisting of elements `a` such
that the valuation of `a` is divisible by `m` at all primes
not in `S`. It is equal to the group of `S`-units modulo
`m`-th powers. The group `\QQ(S,m)` contains the subgroup of
those `a` such that `\QQ(\sqrt[m]{a})/\QQ` is unramified at
all primes of `\QQ` outside of `S`, but may contain it
properly when not all primes dividing `m` are in `S`.
EXAMPLES::
sage: QQ.selmer_group((), 2)
[-1]
sage: QQ.selmer_group((3,), 2)
[-1, 3]
sage: QQ.selmer_group((5,), 2)
[-1, 5]
The previous examples show that the group generated by the
output may be strictly larger than the 'true' Selmer group of
elements giving extensions unramified outside `S`.
When `m` is even, `-1` is a generator of order `2`::
sage: QQ.selmer_group((2,3,5,7,), 2, orders=True)
([-1, 2, 3, 5, 7], [2, 2, 2, 2, 2])
sage: QQ.selmer_group((2,3,5,7,), 3, orders=True)
([2, 3, 5, 7], [3, 3, 3, 3])
"""
gens = list(S)
ords = [ZZ(m)] * len(S)
if m % 2 == 0:
gens = [ZZ(-1)] + gens
ords = [ZZ(2)] + ords
if orders:
return gens, ords
else:
return gens
def characteristic(self):
"""
Return 0, since the continued fraction field has characteristic 0.
EXAMPLES::
sage: c = CFF.characteristic(); c
0
sage: parent(c)
Integer Ring
"""
return ZZ(0)
def random_element(self, num_bound=None, den_bound=None, *args, **kwds):
"""
Return an random element of `\QQ`.
EXAMPLES::
sage: QQ.random_element(10,10)
1/4
Passes extra positional or keyword arguments through::
sage: QQ.random_element(10,10, distribution='1/n')
-1
"""
global ZZ
if ZZ is None:
import integer_ring
ZZ = integer_ring.ZZ
if num_bound == None:
num = ZZ.random_element(*args, **kwds)
den = ZZ.random_element(*args, **kwds)
while den == 0: den = ZZ.random_element(*args, **kwds)
return self((num, den))
else:
if num_bound == 0:
num_bound = 2
if den_bound is None:
den_bound = num_bound
if den_bound < 1:
den_bound = 2
num = ZZ.random_element(-num_bound, num_bound+1, *args, **kwds)
den = ZZ.random_element(1, den_bound+1, *args, **kwds)
while den == 0: den = ZZ.random_element(1, den_bound+1, *args, **kwds)
return self((num,den))
def random_element(self, num_bound=None, den_bound=None, *args, **kwds):
"""
EXAMPLES::
sage: QQ.random_element(10,10)
1/4
Passes extra positional or keyword arguments through::
sage: QQ.random_element(10,10, distribution='1/n')
-1
"""
global ZZ
if ZZ is None:
import integer_ring
ZZ = integer_ring.ZZ
if num_bound == None:
num = ZZ.random_element(*args, **kwds)
den = ZZ.random_element(*args, **kwds)
while den == 0:
den = ZZ.random_element(*args, **kwds)
return self((num, den))
else:
if num_bound == 0:
num_bound = 2
if den_bound is None:
den_bound = num_bound
if den_bound < 1:
den_bound = 2
num = ZZ.random_element(-num_bound, num_bound + 1, *args, **kwds)
den = ZZ.random_element(1, den_bound + 1, *args, **kwds)
while den == 0:
den = ZZ.random_element(1, den_bound + 1, *args, **kwds)
return self((num, den))
def primes_of_bounded_norm_iter(self, B):
r"""
Iterator yielding all primes less than or equal to `B`.
INPUT:
- ``B`` -- a positive integer; upper bound on the primes generated.
OUTPUT:
An iterator over all integer primes less than or equal to `B`.
.. note::
This function exists for compatibility with the related number
field method, though it returns prime integers, not ideals.
EXAMPLES::
sage: it = QQ.primes_of_bounded_norm_iter(10)
sage: list(it)
[2, 3, 5, 7]
sage: list(QQ.primes_of_bounded_norm_iter(1))
[]
"""
try:
B = ZZ(B.ceil())
except (TypeError, AttributeError):
raise TypeError("%s is not valid bound on prime ideals" % B)
if B < 2:
raise StopIteration
from sage.rings.arith import primes
for p in primes(B + 1):
yield p
def primes_of_bounded_norm_iter(self, B):
r"""
Iterator yielding all primes less than or equal to `B`.
INPUT:
- ``B`` -- a positive integer; upper bound on the primes generated.
OUTPUT:
An iterator over all integer primes less than or equal to `B`.
.. note::
This function exists for compatibility with the related number
field method, though it returns prime integers, not ideals.
EXAMPLES::
sage: it = QQ.primes_of_bounded_norm_iter(10)
sage: list(it)
[2, 3, 5, 7]
sage: list(QQ.primes_of_bounded_norm_iter(1))
[]
"""
try:
B = ZZ(B.ceil())
except (TypeError, AttributeError):
raise TypeError("%s is not valid bound on prime ideals" % B)
if B < 2:
raise StopIteration
from sage.rings.arith import primes
for p in primes(B + 1):
yield p
def random_element(self, num_bound=None, den_bound=None, *args, **kwds):
"""
Return an random element of `\QQ`.
Elements are constructed by randomly choosing integers
for the numerator and denominator, not neccessarily coprime.
INPUT:
- ``num_bound`` -- a positive integer, specifying a bound
on the absolute value of the numerator.
If absent, no bound is enforced.
- ``den_bound`` -- a positive integer, specifying a bound
on the value of the denominator.
If absent, the bound for the numerator will be reused.
Any extra positional or keyword arguments are passed through to
:meth:`sage.rings.integer_ring.IntegerRing_class.random_element`.
EXAMPLES::
sage: QQ.random_element()
-4
sage: QQ.random_element()
0
sage: QQ.random_element()
-1/2
In the following example, the resulting numbers range from
-5/1 to 5/1 (both inclusive),
while the smallest possible positive value is 1/10::
sage: QQ.random_element(5, 10)
-2/7
Extra positional or keyword arguments are passed through::
sage: QQ.random_element(distribution='1/n')
0
sage: QQ.random_element(distribution='1/n')
-1
"""
global ZZ
if ZZ is None:
import integer_ring
ZZ = integer_ring.ZZ
if num_bound is None:
num = ZZ.random_element(*args, **kwds)
den = ZZ.random_element(*args, **kwds)
while den == 0: den = ZZ.random_element(*args, **kwds)
return self((num, den))
else:
if num_bound == 0:
num_bound = 2
if den_bound is None:
den_bound = num_bound
if den_bound < 1:
den_bound = 2
num = ZZ.random_element(-num_bound, num_bound+1, *args, **kwds)
den = ZZ.random_element(1, den_bound+1, *args, **kwds)
while den == 0: den = ZZ.random_element(1, den_bound+1, *args, **kwds)
return self((num,den))
def random_element(self, num_bound=None, den_bound=None, *args, **kwds):
"""
Return an random element of `\QQ`.
Elements are constructed by randomly choosing integers
for the numerator and denominator, not neccessarily coprime.
INPUT:
- ``num_bound`` -- a positive integer, specifying a bound
on the absolute value of the numerator.
If absent, no bound is enforced.
- ``den_bound`` -- a positive integer, specifying a bound
on the value of the denominator.
If absent, the bound for the numerator will be reused.
Any extra positional or keyword arguments are passed through to
:meth:`sage.rings.integer_ring.IntegerRing_class.random_element`.
EXAMPLES::
sage: QQ.random_element()
-4
sage: QQ.random_element()
0
sage: QQ.random_element()
-1/2
In the following example, the resulting numbers range from
-5/1 to 5/1 (both inclusive),
while the smallest possible positive value is 1/10::
sage: QQ.random_element(5, 10)
-2/7
Extra positional or keyword arguments are passed through::
sage: QQ.random_element(distribution='1/n')
0
sage: QQ.random_element(distribution='1/n')
-1
"""
global ZZ
if ZZ is None:
import integer_ring
ZZ = integer_ring.ZZ
if num_bound is None:
num = ZZ.random_element(*args, **kwds)
den = ZZ.random_element(*args, **kwds)
while den == 0:
den = ZZ.random_element(*args, **kwds)
return self((num, den))
else:
if num_bound == 0:
num_bound = 2
if den_bound is None:
den_bound = num_bound
if den_bound < 1:
den_bound = 2
num = ZZ.random_element(-num_bound, num_bound + 1, *args, **kwds)
den = ZZ.random_element(1, den_bound + 1, *args, **kwds)
while den == 0:
den = ZZ.random_element(1, den_bound + 1, *args, **kwds)
return self((num, den))
