def _element_constructor_(self, u):
"""
Returns the abstract group element corresponding to the unit u.
INPUT:
- ``u`` -- Any object from which an element of the unit group's number
field `K` may be constructed; an error is raised if an element of `K`
cannot be constructed from u, or if the element constructed is not a
unit.
EXAMPLES::
sage: x = polygen(QQ)
sage: K.<a> = NumberField(x^2-38)
sage: UK = UnitGroup(K)
sage: UK(1)
1
sage: UK(-1)
u0
sage: UK.gens()
(u0, u1)
sage: UK.gens_values()
[-1, 6*a - 37]
sage: UK.ngens()
2
sage: [UK(u) for u in UK.gens()]
[u0, u1]
sage: [UK(u).exponents() for u in UK.gens()]
[(1, 0), (0, 1)]
sage: UK(a)
Traceback (most recent call last):
...
ValueError: a is not a unit
"""
K = self.__number_field
pK = self.__pari_number_field
try:
u = K(u)
except TypeError:
raise ValueError, "%s is not an element of %s" % (u, K)
if self.__S:
m = pK.bnfissunit(self.__S_unit_data, pari(u)).mattranspose()
if m.ncols() == 0:
raise ValueError, "%s is not an S-unit" % u
else:
if not u.is_integral() or u.norm().abs() != 1:
raise ValueError, "%s is not a unit" % u
m = pK.bnfisunit(pari(u)).mattranspose()
# convert column matrix to a list:
m = [ZZ(m[0, i].python()) for i in range(m.ncols())]
# NB pari puts the torsion after the fundamental units, before
# the extra S-units but we have the torsion first:
m = [m[self.__nfu]] + m[:self.__nfu] + m[self.__nfu + 1:]
return self.element_class(self, m)
def _element_constructor_(self, u):
"""
Returns the abstract group element corresponding to the unit u.
INPUT:
- ``u`` -- Any object from which an element of the unit group's number
field `K` may be constructed; an error is raised if an element of `K`
cannot be constructed from u, or if the element constructed is not a
unit.
EXAMPLES::
sage: x = polygen(QQ)
sage: K.<a> = NumberField(x^2-38)
sage: UK = UnitGroup(K)
sage: UK(1)
1
sage: UK(-1)
u0
sage: UK.gens()
(u0, u1)
sage: UK.gens_values()
[-1, 6*a - 37]
sage: UK.ngens()
2
sage: [UK(u) for u in UK.gens()]
[u0, u1]
sage: [UK(u).exponents() for u in UK.gens()]
[(1, 0), (0, 1)]
sage: UK(a)
Traceback (most recent call last):
...
ValueError: a is not a unit
"""
K = self.__number_field
pK = self.__pari_number_field
try:
u = K(u)
except TypeError:
raise ValueError, "%s is not an element of %s"%(u,K)
if self.__S:
m = pK.bnfissunit(self.__S_unit_data, pari(u)).mattranspose()
if m.ncols()==0:
raise ValueError, "%s is not an S-unit"%u
else:
if not u.is_integral() or u.norm().abs() != 1:
raise ValueError, "%s is not a unit"%u
m = pK.bnfisunit(pari(u)).mattranspose()
# convert column matrix to a list:
m = [ZZ(m[0,i].python()) for i in range(m.ncols())]
# NB pari puts the torsion after the fundamental units, before
# the extra S-units but we have the torsion first:
m = [m[self.__nfu]] + m[:self.__nfu] + m[self.__nfu+1:]
return self.element_class(self, m)
def primes_of_bounded_norm(B):
"""
Return the prime ideals of the integers of the field Q(sqrt(5)) of
norm at most B, ordered first by norm, then by the image of the
golden ratio mod the prime in GF(p)={0,1,...,p-1}.
INPUT:
- B -- integer
OUTPUT:
- list of prime ideals
EXAMPLES::
sage: import psage
sage: psage.modform.hilbert.sqrt5.primes_of_bounded_norm(4)
[Fractional ideal (2)]
sage: len(psage.modform.hilbert.sqrt5.primes_of_bounded_norm(10^4))
1233
sage: v = psage.modform.hilbert.sqrt5.primes_of_bounded_norm(11); v
[Fractional ideal (2), Fractional ideal (2*a - 1), Fractional ideal (3), Fractional ideal (3*a - 1), Fractional ideal (3*a - 2)]
Check that the sort order is as claimed::
sage: P0 = v[-2]; P1 = v[-1]
sage: K = P0.number_field(); K
Number Field in a with defining polynomial x^2 - x - 1
sage: P0.residue_field()(K.gen())
4
sage: P1.residue_field()(K.gen()) # yep, 4 < 8
8
"""
v = []
g = F.gen()
for p in primes(B + 1):
if p == 5:
v.append((5, F.ideal(2 * g - 1)))
elif p % 5 in [2, 3]:
Norm = p * p
if Norm <= B:
v.append((Norm, F.ideal(p)))
else:
s = pari(5).Mod(p).sqrt()
a = int(((1 + s) / 2).lift())
b = int(((1 - s) / 2).lift())
v.append((p, a, F.ideal([p, g - a])))
v.append((p, b, F.ideal([p, g - b])))
v.sort()
return [z[-1] for z in v]
def primes_of_bounded_norm(B):
"""
Return the prime ideals of the integers of the field Q(sqrt(5)) of
norm at most B, ordered first by norm, then by the image of the
golden ratio mod the prime in GF(p)={0,1,...,p-1}.
INPUT:
- B -- integer
OUTPUT:
- list of prime ideals
EXAMPLES::
sage: import psage
sage: psage.modform.hilbert.sqrt5.primes_of_bounded_norm(4)
[Fractional ideal (2)]
sage: len(psage.modform.hilbert.sqrt5.primes_of_bounded_norm(10^4))
1233
sage: v = psage.modform.hilbert.sqrt5.primes_of_bounded_norm(11); v
[Fractional ideal (2), Fractional ideal (2*a - 1), Fractional ideal (3), Fractional ideal (3*a - 1), Fractional ideal (3*a - 2)]
Check that the sort order is as claimed::
sage: P0 = v[-2]; P1 = v[-1]
sage: K = P0.number_field(); K
Number Field in a with defining polynomial x^2 - x - 1
sage: P0.residue_field()(K.gen())
4
sage: P1.residue_field()(K.gen()) # yep, 4 < 8
8
"""
v = []
g = F.gen()
for p in primes(B+1):
if p == 5:
v.append((5, F.ideal(2*g-1)))
elif p%5 in [2,3]:
Norm = p*p
if Norm <= B:
v.append((Norm, F.ideal(p)))
else:
s = pari(5).Mod(p).sqrt()
a = int(((1 + s)/2).lift()); b = int(((1 - s)/2).lift())
v.append((p, a, F.ideal([p, g - a])))
v.append((p, b, F.ideal([p, g - b])))
v.sort()
return [z[-1] for z in v]
def elliptic_j(z):
r"""
Returns the elliptic modular `j`-function evaluated at `z`.
INPUT:
- ``z`` (complex) -- a complex number with positive imaginary part.
OUTPUT:
(complex) The value of `j(z)`.
ALGORITHM:
Calls the ``pari`` function ``ellj()``.
AUTHOR:
John Cremona
EXAMPLES::
sage: elliptic_j(CC(i))
1728.00000000000
sage: elliptic_j(sqrt(-2.0))
8000.00000000000
sage: z = ComplexField(100)(1,sqrt(11))/2
sage: elliptic_j(z)
-32768.000...
sage: elliptic_j(z).real().round()
-32768
::
sage: tau = (1 + sqrt(-163))/2
sage: (-elliptic_j(tau.n(100)).real().round())^(1/3)
640320
"""
CC = z.parent()
from sage.rings.complex_field import is_ComplexField
if not is_ComplexField(CC):
CC = ComplexField()
try:
z = CC(z)
except ValueError:
raise ValueError("elliptic_j only defined for complex arguments.")
from sage.libs.all import pari
return CC(pari(z).ellj())
def elliptic_j(z):
r"""
Returns the elliptic modular `j`-function evaluated at `z`.
INPUT:
- ``z`` (complex) -- a complex number with positive imaginary part.
OUTPUT:
(complex) The value of `j(z)`.
ALGORITHM:
Calls the ``pari`` function ``ellj()``.
AUTHOR:
John Cremona
EXAMPLES::
sage: elliptic_j(CC(i))
1728.00000000000
sage: elliptic_j(sqrt(-2.0))
8000.00000000000
sage: z = ComplexField(100)(1,sqrt(11))/2
sage: elliptic_j(z)
-32768.000...
sage: elliptic_j(z).real().round()
-32768
::
sage: tau = (1 + sqrt(-163))/2
sage: (-elliptic_j(tau.n(100)).real().round())^(1/3)
640320
"""
CC = z.parent()
from sage.rings.complex_field import is_ComplexField
if not is_ComplexField(CC):
CC = ComplexField()
try:
z = CC(z)
except ValueError:
raise ValueError("elliptic_j only defined for complex arguments.")
from sage.libs.all import pari
return CC(pari(z).ellj())
def _element_constructor_(self, u):
"""
Returns the abstract group element corresponding to the unit u.
INPUT:
- ``u`` -- Any object from which an element of the unit group's number
field `K` may be constructed; an error is raised if an element of `K`
cannot be constructed from u, or if the element constructed is not a
unit.
EXAMPLES::
sage: x = polygen(QQ)
sage: K.<a> = NumberField(x^2-38)
sage: UK = UnitGroup(K)
sage: UK(1)
1
sage: UK(-1)
u0
sage: UK.gens()
(u0, u1)
sage: UK.gens_values()
[-1, 6*a - 37]
sage: UK.ngens()
2
sage: [UK(u) for u in UK.gens()]
[u0, u1]
sage: [UK(u).exponents() for u in UK.gens()]
[(1, 0), (0, 1)]
sage: UK(a)
Traceback (most recent call last):
...
ValueError: a is not a unit
"""
K = self.__number_field
try:
u = K(u)
except TypeError:
raise ValueError, "%s is not an element of %s" % (u, K)
if not u.is_integral() or u.norm().abs() != 1:
raise ValueError, "%s is not a unit" % u
m = K.pari_bnf().bnfisunit(pari(u)).mattranspose()
# convert column matrix to a list:
m = [ZZ(m[0, i].python()) for i in range(m.ncols())]
# NB pari puts the torsion at the end!
m.insert(0, m.pop())
return self.element_class(m, self)
def _element_constructor_(self, u):
"""
Returns the abstract group element corresponding to the unit u.
INPUT:
- ``u`` -- Any object from which an element of the unit group's number
field `K` may be constructed; an error is raised if an element of `K`
cannot be constructed from u, or if the element constructed is not a
unit.
EXAMPLES::
sage: x = polygen(QQ)
sage: K.<a> = NumberField(x^2-38)
sage: UK = UnitGroup(K)
sage: UK(1)
1
sage: UK(-1)
u0
sage: UK.gens()
(u0, u1)
sage: UK.gens_values()
[-1, 6*a - 37]
sage: UK.ngens()
2
sage: [UK(u) for u in UK.gens()]
[u0, u1]
sage: [UK(u).exponents() for u in UK.gens()]
[(1, 0), (0, 1)]
sage: UK(a)
Traceback (most recent call last):
...
ValueError: a is not a unit
"""
K = self.__number_field
try:
u = K(u)
except TypeError:
raise ValueError, "%s is not an element of %s"%(u,K)
if not u.is_integral() or u.norm().abs() != 1:
raise ValueError, "%s is not a unit"%u
m = K.pari_bnf().bnfisunit(pari(u)).mattranspose()
# convert column matrix to a list:
m = [ZZ(m[0,i].python()) for i in range(m.ncols())]
# NB pari puts the torsion at the end!
m.insert(0,m.pop())
return self.element_class(m, self)
def _pari_(self, variable=None):
if variable is None:
variable = self.parent().variable_name()
return pari(self.list()).Polrev(variable)
def _pari_(self, variable = None):
if variable is None:
variable = self.parent().variable_name()
return pari(self.list()).Polrev(variable)
def elliptic_j(z, prec=53):
r"""
Returns the elliptic modular `j`-function evaluated at `z`.
INPUT:
- ``z`` (complex) -- a complex number with positive imaginary part.
- ``prec`` (default: 53) -- precision in bits for the complex field.
OUTPUT:
(complex) The value of `j(z)`.
ALGORITHM:
Calls the ``pari`` function ``ellj()``.
AUTHOR:
John Cremona
EXAMPLES::
sage: elliptic_j(CC(i))
1728.00000000000
sage: elliptic_j(sqrt(-2.0))
8000.00000000000
sage: z = ComplexField(100)(1,sqrt(11))/2
sage: elliptic_j(z)
-32768.000...
sage: elliptic_j(z).real().round()
-32768
::
sage: tau = (1 + sqrt(-163))/2
sage: (-elliptic_j(tau.n(100)).real().round())^(1/3)
640320
This example shows the need for higher precision than the default one of
the `ComplexField`, see :trac:`28355`::
sage: -elliptic_j(tau) # rel tol 1e-2
2.62537412640767e17 - 732.558854258998*I
sage: -elliptic_j(tau,75) # rel tol 1e-2
2.625374126407680000000e17 - 0.0001309913593909879441262*I
sage: -elliptic_j(tau,100) # rel tol 1e-2
2.6253741264076799999999999999e17 - 1.3012822400356887122945119790e-12*I
sage: (-elliptic_j(tau, 100).real().round())^(1/3)
640320
"""
CC = z.parent()
if not isinstance(CC, sage.rings.abc.ComplexField):
from sage.rings.complex_mpfr import ComplexField
CC = ComplexField(prec)
try:
z = CC(z)
except ValueError:
raise ValueError("elliptic_j only defined for complex arguments.")
from sage.libs.all import pari
return CC(pari(z).ellj())
