def _evalf_(self, x, parent):
"""
EXAMPLES::
sage: erf(2).n()
0.995322265018953
sage: erf(2).n(200)
0.99532226501895273416206925636725292861089179704006007673835
sage: erf(pi - 1/2*I).n(100)
1.0000111669099367825726058952 + 1.6332655417638522934072124548e-6*I
TESTS:
Check that PARI/GP through the GP interface gives the same answer::
sage: gp.set_real_precision(59) # random
38
sage: print gp.eval("1 - erfc(1)"); print erf(1).n(200);
0.84270079294971486934122063508260925929606699796630290845994
0.84270079294971486934122063508260925929606699796630290845994
"""
try:
prec = parent.prec()
except AttributeError: # not a Sage parent
prec = 0
return parent(1) - parent(pari(x).erfc(prec))
def __init__(self, field, num_integer_primes=10000, max_iterations=100):
r"""
Construct a new iterator of small degree one primes.
EXAMPLES::
sage: x = QQ['x'].gen()
sage: K.<a> = NumberField(x^2 - 3)
sage: K.primes_of_degree_one_list(3) # random
[Fractional ideal (2*a + 1), Fractional ideal (-a + 4), Fractional ideal (3*a + 2)]
"""
self._field = field
self._poly = self._field.absolute_field('b').defining_polynomial()
self._poly = ZZ['x'](self._poly.denominator() *
self._poly()) # make integer polynomial
# this uses that [ O_K : Z[a] ]^2 = | disc(f(x)) / disc(O_K) |
self._prod_of_small_primes = ZZ(
pari(
'TEMPn = %s; TEMPps = primes(TEMPn); prod(X = 1, TEMPn, TEMPps[X])'
% num_integer_primes))
self._prod_of_small_primes //= self._prod_of_small_primes.gcd(
self._poly.discriminant())
self._integer_iter = iter(ZZ)
self._queue = []
self._max_iterations = max_iterations
def _shortest_vectors(self, max_count=None, max_length=0):
"""
Compute shortest vectors and their number and length (potentially).
INPUT:
- ``max_count`` -- maximum number of vectors to store in the result
(default: all);
- ``max_length`` -- maximal length of vectors to consider
(default: 0, i.e., consider shortest vectors).
OUTPUT:
A triple consisting of the number of corresponding (shortest) vectors,
their length, and a list of at most ``max_count`` such vectors.
TESTS::
sage: L = Lattice([[1, 0], [0, 1]])
sage: L.shortest_vectors_count()
4
"""
default = max_length == 0 and max_count == 0
if default and self.__shortest_vectors is not None:
return self.__shortest_vectors
qf = self.reduced_inner_product_matrix()
if max_count is None:
# determine the number of shortest vectors
max_count = self._shortest_vectors(max_length=max_length, max_count=0)[0] #self.degree()
count, length, vectors = pari(qf).qfminim(0, max_count)
result = count, length, vectors.python().columns()
if default:
self.__shortest_vectors = result
return result
def _evalf_(self, x, parent):
"""
EXAMPLES::
sage: erf(2).n()
0.995322265018953
sage: erf(2).n(200)
0.99532226501895273416206925636725292861089179704006007673835
sage: erf(pi - 1/2*I).n(100)
1.0000111669099367825726058952 + 1.6332655417638522934072124548e-6*I
TESTS:
Check that PARI/GP through the GP interface gives the same answer::
sage: gp.set_real_precision(59) # random
38
sage: print gp.eval("1 - erfc(1)"); print erf(1).n(200);
0.84270079294971486934122063508260925929606699796630290845994
0.84270079294971486934122063508260925929606699796630290845994
"""
try:
prec = parent.prec()
except AttributeError: # not a Sage parent
prec = 0
return parent(1) - parent(pari(x).erfc(prec))
def qfparam(G, sol):
r"""
Parametrizes the conic defined by the matrix ``G``.
INPUT:
- ``G`` -- a `3 \times 3`-matrix over `\QQ`.
- ``sol`` -- a triple of rational numbers providing a solution
to sol*G*sol^t = 0.
OUTPUT:
A triple of polynomials that parametrizes all solutions of
x*G*x^t = 0 up to scaling.
ALGORITHM:
Uses Denis Simon's pari script Qfparam.
EXAMPLES::
sage: from sage.quadratic_forms.qfsolve import qfsolve, qfparam
sage: M = Matrix(QQ, [[0, 0, -12], [0, -12, 0], [-12, 0, -1]]); M
[ 0 0 -12]
[ 0 -12 0]
[-12 0 -1]
sage: sol = qfsolve(M);
sage: ret = qfparam(M, sol); ret
(-t^2 - 12, 24*t, 24*t^2)
sage: ret[0].parent() is QQ['t']
True
"""
gp = _gp_for_simon()
R = QQ['t']
t = R.gen()
s = 'Qfparam(%s, (%s)~)*[t^2,t,1]~' % (G._pari_(), pari(gp(sol))._pari_())
ret = pari(gp(s))
return tuple([R(r) for r in ret])
def qfparam(G, sol):
r"""
Parametrizes the conic defined by the matrix ``G``.
INPUT:
- ``G`` -- a `3 \times 3`-matrix over `\QQ`.
- ``sol`` -- a triple of rational numbers providing a solution
to sol*G*sol^t = 0.
OUTPUT:
A triple of polynomials that parametrizes all solutions of
x*G*x^t = 0 up to scaling.
ALGORITHM:
Uses Denis Simon's pari script Qfparam.
EXAMPLES::
sage: from sage.quadratic_forms.qfsolve import qfsolve, qfparam
sage: M = Matrix(QQ, [[0, 0, -12], [0, -12, 0], [-12, 0, -1]]); M
[ 0 0 -12]
[ 0 -12 0]
[-12 0 -1]
sage: sol = qfsolve(M);
sage: ret = qfparam(M, sol); ret
(-t^2 - 12, 24*t, 24*t^2)
sage: ret[0].parent() is QQ['t']
True
"""
gp = _gp_for_simon()
R = QQ['t']
t = R.gen()
s = 'Qfparam(%s, (%s)~)*[t^2,t,1]~' % (G._pari_(), pari(gp(sol))._pari_())
ret = pari(gp(s))
return tuple([R(r) for r in ret])
def as_hom(self):
r"""
Return the homomorphism L -> L corresponding to self, where L is the
Galois closure of the ambient number field.
EXAMPLE::
sage: G = QuadraticField(-7,'w').galois_group()
sage: G[1].as_hom()
Ring endomorphism of Number Field in w with defining polynomial x^2 + 7
Defn: w |--> -w
"""
L = self.parent().splitting_field()
a = L(self.parent()._pari_data.galoispermtopol(pari(self.list()).Vecsmall()))
return L.hom(a, L)
def as_hom(self):
r"""
Return the homomorphism L -> L corresponding to self, where L is the
Galois closure of the ambient number field.
EXAMPLE::
sage: G = QuadraticField(-7,'w').galois_group()
sage: G[1].as_hom()
Ring endomorphism of Number Field in w with defining polynomial x^2 + 7
Defn: w |--> -w
"""
L = self.parent().splitting_field()
a = L(self.parent()._pari_data.galoispermtopol(
pari(self.list()).Vecsmall()))
return L.hom(a, L)
def resultant(self, normalize=False):
r"""
Computes the resultant of the defining polynomials of ``self`` if ``self`` is a map on the projective line.
If ``normalize`` is ``True``, then first normalize the coordinate
functions with :meth:`normalize_coordinates`.
INPUT:
- ``normalize`` -- Boolean (optional - default: ``False``)
OUTPUT:
- an element of ``self.codomain().base_ring()``
EXAMPLES::
sage: P.<x,y> = ProjectiveSpace(QQ,1)
sage: H = Hom(P,P)
sage: f = H([x^2+y^2,6*y^2])
sage: f.resultant()
36
::
sage: R.<t> = PolynomialRing(GF(17))
sage: P.<x,y> = ProjectiveSpace(R,1)
sage: H = Hom(P,P)
sage: f = H([t*x^2+t*y^2,6*y^2])
sage: f.resultant()
2*t^2
"""
if self.domain().dimension_relative() > 1:
raise TypeError("Only for dimension 1, use self.primes_of_bad_reduction() to get bad primes")
if normalize is True:
F=copy(self)
F.normalize_coordinates()
else:
F=self
x=self.domain().gen(0)
y=self.domain().gen(1)
d=self.degree()
f=F[0].substitute({y:1})
g=F[1].substitute({y:1})
res=(f.lc()**(d-g.degree())*g.lc()**(d-f.degree())*pari(f).polresultant(g,x))
return(self.codomain().base_ring()(res))
def resultant(self,normalize=False):
r"""
Computes the resultant of the defining polynomials of ``self`` if ``self`` is a map on the projective line.
If ``normalize==True``, then first normalize the coordinate functions with ``normalize_coordinates()``.
INPUT:
- ``normalize`` -- Boolean (optional - default: ``False``)
OUTPUT:
- an element of ``self.codomain().base_ring()``
EXAMPLES::
sage: P.<x,y>=ProjectiveSpace(QQ,1)
sage: H=Hom(P,P)
sage: f=H([x^2+y^2,6*y^2])
sage: f.resultant()
36
::
sage: R.<t>=PolynomialRing(GF(17))
sage: P.<x,y>=ProjectiveSpace(R,1)
sage: H=Hom(P,P)
sage: f=H([t*x^2+t*y^2,6*y^2])
sage: f.resultant()
2*t^2
"""
if self.domain().dimension_relative() > 1:
raise TypeError("Only for dimension 1, use self.primes_of_bad_reduction() to get bad primes")
if normalize==True:
F=copy(self)
F.normalize_coordinates()
else:
F=self
x=self.domain().gen(0)
y=self.domain().gen(1)
d=self.degree()
f=F[0].substitute({y:1})
g=F[1].substitute({y:1})
res=(f.lc()**(d-g.degree())*g.lc()**(d-f.degree())*pari(f).polresultant(g,x))
return(self.codomain().base_ring()(res))
def qfsolve(G, factD=None):
r"""
Find a solution `x = (x_0,...,x_n)` to `x G x^t = 0` for an
`n \times n`-matrix ``G`` over `\QQ`.
If a solution exists, returns a tuple of rational numbers `x`.
Otherwise, returns `-1` if no solutions exists over the reals or a
prime `p` if no solution exists over the `p`-adic field `\QQ_p`.
EXAMPLES::
sage: from sage.quadratic_forms.qfsolve import qfsolve
sage: M = Matrix(QQ, [[0, 0, -12], [0, -12, 0], [-12, 0, -1]]); M
[ 0 0 -12]
[ 0 -12 0]
[-12 0 -1]
sage: sol = qfsolve(M); sol
(1, 0, 0)
sage: sol[0].parent() is QQ
True
sage: M = Matrix(QQ, [[1, 0, 0], [0, 1, 0], [0, 0, 1]])
sage: ret = qfsolve(M); ret
-1
sage: ret.parent() is ZZ
True
sage: M = Matrix(QQ, [[1, 0, 0], [0, 1, 0], [0, 0, -7]])
sage: qfsolve(M)
7
sage: M = Matrix(QQ, [[3, 0, 0, 0], [0, 5, 0, 0], [0, 0, -7, 0], [0, 0, 0, -11]])
sage: qfsolve(M)
(-3, 4, 3, 2)
"""
gp = _gp_for_simon()
if factD is not None:
raise NotImplementedError, "qfsolve not implemented with parameter factD"
ret = pari(gp('Qfsolve(%s)' % G._pari_()))
if ret.type() == 't_COL':
return tuple([QQ(r) for r in ret])
return ZZ(ret)
def qfsolve(G, factD=None):
r"""
Find a solution `x = (x_0,...,x_n)` to `x G x^t = 0` for an
`n \times n`-matrix ``G`` over `\QQ`.
If a solution exists, returns a tuple of rational numbers `x`.
Otherwise, returns `-1` if no solutions exists over the reals or a
prime `p` if no solution exists over the `p`-adic field `\QQ_p`.
EXAMPLES::
sage: from sage.quadratic_forms.qfsolve import qfsolve
sage: M = Matrix(QQ, [[0, 0, -12], [0, -12, 0], [-12, 0, -1]]); M
[ 0 0 -12]
[ 0 -12 0]
[-12 0 -1]
sage: sol = qfsolve(M); sol
(1, 0, 0)
sage: sol[0].parent() is QQ
True
sage: M = Matrix(QQ, [[1, 0, 0], [0, 1, 0], [0, 0, 1]])
sage: ret = qfsolve(M); ret
-1
sage: ret.parent() is ZZ
True
sage: M = Matrix(QQ, [[1, 0, 0], [0, 1, 0], [0, 0, -7]])
sage: qfsolve(M)
7
sage: M = Matrix(QQ, [[3, 0, 0, 0], [0, 5, 0, 0], [0, 0, -7, 0], [0, 0, 0, -11]])
sage: qfsolve(M)
(-3, 4, 3, 2)
"""
gp = _gp_for_simon()
if factD is not None:
raise NotImplementedError, "qfsolve not implemented with parameter factD"
ret = pari(gp('Qfsolve(%s)' % G._pari_()))
if ret.type() == 't_COL':
return tuple([QQ(r) for r in ret])
return ZZ(ret)
def __init__(self, field, num_integer_primes=10000, max_iterations=100):
r"""
Construct a new iterator of small degree one primes.
EXAMPLES::
sage: x = QQ['x'].gen()
sage: K.<a> = NumberField(x^2 - 3)
sage: K.primes_of_degree_one_list(3) # random
[Fractional ideal (2*a + 1), Fractional ideal (-a + 4), Fractional ideal (3*a + 2)]
"""
self._field = field
self._poly = self._field.absolute_field('b').defining_polynomial()
self._poly = ZZ['x'](self._poly.denominator() * self._poly()) # make integer polynomial
# this uses that [ O_K : Z[a] ]^2 = | disc(f(x)) / disc(O_K) |
self._prod_of_small_primes = ZZ(pari('TEMPn = %s; TEMPps = primes(TEMPn); prod(X = 1, TEMPn, TEMPps[X])' % num_integer_primes))
self._prod_of_small_primes //= self._prod_of_small_primes.gcd(self._poly.discriminant())
self._integer_iter = iter(ZZ)
self._queue = []
self._max_iterations = max_iterations
def fixed_field(self):
r"""
Return the fixed field of this subgroup (as a subfield of the Galois
closure of the number field associated to the ambient Galois group).
EXAMPLE::
sage: L.<a> = NumberField(x^4 + 1)
sage: G = L.galois_group()
sage: H = G.decomposition_group(L.primes_above(3)[0])
sage: H.fixed_field()
(Number Field in a0 with defining polynomial x^2 + 2, Ring morphism:
From: Number Field in a0 with defining polynomial x^2 + 2
To: Number Field in a with defining polynomial x^4 + 1
Defn: a0 |--> a^3 + a)
"""
if self.order() == 1:
return self._galois_closure # work around a silly error
vecs = [pari(g.list()).Vecsmall() for g in self._elts]
v = self._ambient._pari_data.galoisfixedfield(vecs)
x = self._galois_closure(v[1])
return self._galois_closure.subfield(x)
def fixed_field(self):
r"""
Return the fixed field of this subgroup (as a subfield of the Galois
closure of the number field associated to the ambient Galois group).
EXAMPLE::
sage: L.<a> = NumberField(x^4 + 1)
sage: G = L.galois_group()
sage: H = G.decomposition_group(L.primes_above(3)[0])
sage: H.fixed_field()
(Number Field in a0 with defining polynomial x^2 + 2, Ring morphism:
From: Number Field in a0 with defining polynomial x^2 + 2
To: Number Field in a with defining polynomial x^4 + 1
Defn: a0 |--> a^3 + a)
"""
if self.order() == 1:
return self._galois_closure # work around a silly error
vecs = [pari(g.list()).Vecsmall() for g in self._elts]
v = self._ambient._pari_data.galoisfixedfield(vecs)
x = self._galois_closure(v[1])
return self._galois_closure.subfield(x)
def enumerate_totallyreal_fields_all(n, B, verbose=0, return_seqs=False):
r"""
Enumerates *all* totally real fields of degree `n` with discriminant `\le B`,
primitive or otherwise.
EXAMPLES::
sage: enumerate_totallyreal_fields_all(4, 2000)
[[725, x^4 - x^3 - 3*x^2 + x + 1],
[1125, x^4 - x^3 - 4*x^2 + 4*x + 1],
[1600, x^4 - 6*x^2 + 4],
[1957, x^4 - 4*x^2 - x + 1],
[2000, x^4 - 5*x^2 + 5]]
In practice most of these will be found by :func:`~sage.rings.number_field.totallyreal.enumerate_totallyreal_fields_prim`, which is guaranteed to return all primitive fields but often returns many non-primitive ones as well. For instance, only one of the five fields in the example above is primitive, but :func:`~sage.rings.number_field.totallyreal.enumerate_totallyreal_fields_prim` finds four out of the five (the exception being `x^4 - 6x^2 + 4`).
"""
S = []
counts = [0, 0, 0]
if len(divisors(n)) > 4:
raise ValueError, "Only implemented for n = p*q with p,q prime"
for d in divisors(n):
if d > 1 and d < n:
Sds = enumerate_totallyreal_fields_prim(
d, int(math.floor((1. * B)**(1. * d / n))), verbose=verbose)
for i in range(len(Sds)):
if verbose:
print "=" * 80
print "Taking F =", Sds[i][1]
F = NumberField(ZZx(Sds[i][1]), 't')
T = enumerate_totallyreal_fields_rel(F,
n / d,
B,
verbose=verbose,
return_seqs=return_seqs)
if return_seqs:
for i in range(3):
counts[i] += T[0][i]
S += [[t[0], pari(t[1]).Polrev()] for t in T[1]]
else:
S += [[t[0], t[1]] for t in T]
j = i + 1
for E in enumerate_totallyreal_fields_prim(
n / d,
int(
math.floor((1. * B)**(1. / d) /
(1. * Sds[i][0])**(n * 1. / d**2)))):
for EF in F.composite_fields(NumberField(ZZx(E[1]), 'u')):
if EF.degree() == n and EF.disc() <= B:
S.append(
[EF.disc(),
pari(EF.absolute_polynomial())])
S += enumerate_totallyreal_fields_prim(n, B, verbose=verbose)
S.sort()
weed_fields(S)
# Output.
if verbose:
saveout = sys.stdout
if type(verbose) == str:
fsock = open(verbose, 'w')
sys.stdout = fsock
# Else, print to screen
print "=" * 80
print "Polynomials tested:", counts[0]
print "Irreducible polynomials:", counts[1]
print "Polynomials with nfdisc <= B:", counts[2]
for i in range(len(S)):
print S[i]
if type(verbose) == str:
fsock.close()
sys.stdout = saveout
if return_seqs:
return [counts, [[s[0], s[1].reverse().Vec()] for s in S]]
else:
return S
def enumerate_totallyreal_fields_rel(F,
m,
B,
a=[],
verbose=0,
return_seqs=False):
r"""
This function enumerates (primitive) totally real field extensions of
degree `m>1` of the totally real field F with discriminant `d \leq B`;
optionally one can specify the first few coefficients, where the sequence ``a``
corresponds to a polynomial by
::
a[d]*x^n + ... + a[0]*x^(n-d)
if ``length(a) = d+1``, so in particular always ``a[d] = 1``.
If verbose == 1 (or 2), then print to the screen (really) verbosely; if
verbose is a string, then print verbosely to the file specified by verbose.
If return_seqs, then return the polynomials as sequences (for easier
exporting to a file).
NOTE:
This is guaranteed to give all primitive such fields, and
seems in practice to give many imprimitive ones.
INPUT:
- ``F`` -- number field, the base field
- ``m`` -- integer, the degree
- ``B`` -- integer, the discriminant bound
- ``a`` -- list (default: []), the coefficient list to begin with
- ``verbose`` -- boolean or string (default: 0)
- ``return_seqs`` -- boolean (default: False)
OUTPUT:
the list of fields with entries [d,fabs,f], where
d is the discriminant, fabs is an absolute defining polynomial,
and f is a defining polynomial relative to F,
sorted by discriminant.
EXAMPLES::
sage: ZZx = ZZ['x']
sage: F.<t> = NumberField(x^2-2)
sage: enumerate_totallyreal_fields_rel(F, 2, 2000)
[[1600, x^4 - 6*x^2 + 4, xF^2 + xF - 1]]
AUTHORS:
- John Voight (2007-11-01)
"""
if not isinstance(m, Integer):
try:
m = Integer(m)
except:
raise TypeError, "cannot coerce m (= %s) to an integer" % m
if (m < 1):
raise ValueError, "m must be at least 1."
n = F.degree() * m
# Initialize
T = tr_data_rel(F, m, B, a)
S = []
Srel = []
dB_odlyzko = odlyzko_bound_totallyreal(n)
dB = math.ceil(40000 * dB_odlyzko**n)
counts = [0, 0, 0, 0]
# Trivial case
if m == 1:
g = pari(F.defining_polynomial()).reverse().Vec()
if return_seqs:
return [[0, 0, 0, 0], [1, g, [-1, 1]]]
else:
return [[1, pari('x-1'), g]]
if verbose:
saveout = sys.stdout
if type(verbose) == str:
fsock = open(verbose, 'w')
sys.stdout = fsock
# Else, print to screen
f_out = [0] * m + [1]
if verbose == 2:
T.incr(f_out, verbose)
else:
T.incr(f_out)
Fx = PolynomialRing(F, 'xF')
nfF = pari(
str(F.defining_polynomial()).replace('x', str(F.primitive_element())))
parit = pari(str(F.primitive_element()))
while f_out[m] <> 0:
counts[0] += 1
if verbose:
print "==>", f_out,
f_str = ''
for i in range(len(f_out)):
f_str += '(' + str(f_out[i]) + ')*x^' + str(i)
if i < len(f_out) - 1:
f_str += '+'
nf = pari(f_str)
if nf.poldegree('t') == 0:
nf = nf.subst('x', 'x-t')
nf = nf.polresultant(nfF, parit)
d = nf.poldisc()
counts[0] += 1
if d > 0 and nf.polsturm_full() == n:
da = int_has_small_square_divisor(Integer(d))
if d > dB or d <= B * da:
counts[1] += 1
if nf.polisirreducible():
counts[2] += 1
[zk, d] = nf.nfbasis_d()
if d <= B:
if verbose:
print "has discriminant", d,
# Find a minimal lattice element
counts[3] += 1
ng = pari([nf, zk]).polredabs()
# Check if K is contained in the list.
found = False
ind = bisect.bisect_left(S, [d, ng])
while ind < len(S) and S[ind][0] == d:
if S[ind][1] == ng:
if verbose:
print "but is not new"
found = True
break
ind += 1
if not found:
if verbose:
print "and is new!"
S.insert(ind, [d, ng])
Srel.insert(ind, Fx(f_out))
else:
if verbose:
print "has discriminant", abs(d), "> B"
else:
if verbose:
print "is not absolutely irreducible"
else:
if verbose:
print "has discriminant", abs(
d), "with no large enough square divisor"
else:
if verbose:
if d == 0:
print "is not squarefree"
else:
print "is not totally real"
if verbose == 2:
T.incr(f_out, verbose=verbose)
else:
T.incr(f_out)
# In the application of Smyth's theorem above, we exclude finitely
# many possibilities which we must now throw back in.
if m == 2:
if Fx([-1, 1, 1]).is_irreducible():
K = F.extension(Fx([-1, 1, 1]), 'tK')
Kabs = K.absolute_field('tKabs')
Kabs_pari = pari(Kabs.defining_polynomial())
d = K.absolute_discriminant()
if abs(d) <= B:
ng = Kabs_pari.polredabs()
ind = bisect.bisect_left(S, [d, ng])
S.insert(ind, [d, ng])
Srel.insert(ind, Fx([-1, 1, 1]))
elif F.degree() == 2:
for ff in [[1, -7, 13, -7, 1], [1, -8, 14, -7, 1]]:
f = Fx(ff).factor()[0][0]
K = F.extension(f, 'tK')
Kabs = K.absolute_field('tKabs')
Kabs_pari = pari(Kabs.defining_polynomial())
d = K.absolute_discriminant()
if abs(d) <= B:
ng = Kabs_pari.polredabs()
ind = bisect.bisect_left(S, [d, ng])
S.insert(ind, [d, ng])
Srel.insert(ind, f)
elif m == 3:
if Fx([-1, 6, -5, 1]).is_irreducible():
K = F.extension(Fx([-1, 6, -5, 1]), 'tK')
Kabs = K.absolute_field('tKabs')
Kabs_pari = pari(Kabs.defining_polynomial())
d = K.absolute_discriminant()
if abs(d) <= B:
ng = Kabs_pari.polredabs()
ind = bisect.bisect_left(S, [d, ng])
S.insert(ind, [d, ng])
Srel.insert(ind, Fx([-1, 6, -5, 1]))
# Now check for isomorphic fields
S = [[S[i][0], S[i][1], Srel[i]] for i in range(len(S))]
weed_fields(S)
# Output.
if verbose:
print "=" * 80
print "Polynomials tested:", counts[0]
print "Irreducible polynomials:", counts[1]
print "Polynomials with nfdisc <= B:", counts[2]
for i in range(len(S)):
print S[i]
if type(verbose) == str:
fsock.close()
sys.stdout = saveout
if return_seqs:
return [
counts, [[s[0], s[1].reverse().Vec(), s[2].coeffs()] for s in S]
]
else:
return S
def __init__(self, F, m, B, a=None):
r"""
Initialization routine (constructor).
INPUT:
- ``F`` -- number field, the base field
- ``m`` -- integer, the relative degree
- ``B`` -- integer, the discriminant bound
- ``a`` -- list (default: []), the coefficient list to begin with,
corresponding to ``a[len(a)]*x^n + ... + a[0]x^(n-len(a))``.
OUTPUT:
the data initialized to begin enumeration of totally real fields
with base field F, degree n, discriminant bounded by B, and starting
with coefficients a.
EXAMPLES::
sage: F.<t> = NumberField(x^2-2)
sage: T = sage.rings.number_field.totallyreal_rel.tr_data_rel(F, 2, 2000)
"""
if a is None: # don't make the stupid noob mistake of putting a=[]
a = [] # in the function signature above.
# Initialize constants.
self.m = m
d = F.degree()
self.d = d
self.n = m * d
self.B = B
self.gamma = hermite_constant(self.n - self.d)
self.F = F
self.Z_F = F.maximal_order()
self.Foo = F.real_embeddings()
self.dF = abs(F.disc())
self.Fx = PolynomialRing(F, 'xF')
self.beta = [[]] * m
self.gnk = [[]] * m
self.trace_elts = []
Z_Fbasis = self.Z_F.basis()
# Initialize variables.
if a == []:
# No starting input, all polynomials will be found; initialize to zero.
self.a = [0] * m + [1]
self.amaxvals = [[]] * m
anm1s = [[i] for i in range(0, m // 2 + 1)]
for i in range(1, self.d):
for j in range(len(anm1s)):
anm1s[j] = [anm1s[j] + [i] for i in range(m)]
anm1s = sum(anm1s, [])
anm1s = [
sum([Z_Fbasis[i] * a[i] for i in range(self.d)]) for a in anm1s
]
# Minimize trace in class.
import numpy
for i in range(len(anm1s)):
Q = [[v(m * x) for v in self.Foo] + [0]
for x in Z_Fbasis] + [[v(anm1s[i])
for v in self.Foo] + [10**6]]
pari_string = '[' + ';'.join(
[','.join(["%s" % ii for ii in row])
for row in zip(*Q)]) + ']'
adj = pari(pari_string).qflll()[self.d]
anm1s[i] += sum([
m * Z_Fbasis[ii] * int(adj[ii]) // int(adj[self.d])
for ii in range(self.d)
])
self.amaxvals[m - 1] = anm1s
self.a[m - 1] = self.amaxvals[m - 1].pop()
self.k = m - 2
bl = math.ceil(1.7719 * self.n)
br = max([1./m*(am1**2).trace() + \
self.gamma*(1./(m**d)*self.B/self.dF)**(1./(self.n-d)) for am1 in anm1s])
br = math.floor(br)
T2s = self.F._positive_integral_elements_with_trace([bl, br])
self.trace_elts.append([bl, br, T2s])
elif len(a) <= m + 1:
# First few coefficients have been specified.
# The value of k is the largest index of the coefficients of a which is
# currently unknown; e.g., if k == -1, then we can iterate
# over polynomials, and if k == n-1, then we have finished iterating.
if a[len(a) - 1] <> 1:
raise ValueError, "a[len(a)-1](=%s) must be 1 so polynomial is monic" % a[
len(a) - 1]
raise NotImplementedError, "These have not been checked."
k = m - len(a)
self.k = k
a = [0] * (k + 1) + a
self.amaxvals = [[]] * m
for i in range(0, n + 1):
self.a[i] = a[i]
# Bounds come from an application of Lagrange multipliers in degrees 2,3.
self.b_lower = [
-1. / m * (v(self.a[m - 1]) + (m - 1.) * math.sqrt(
v(self.a[m - 1])**2 - 2. *
(1 + 1. / (m - 1)) * v(self.a[m - 2]))) for v in self.Foo
]
self.b_upper = [
-1. / m * (v(self.a[m - 1]) - (m - 1.) * math.sqrt(
v(self.a[m - 1])**2 - 2. *
(1 + 1. / (m - 1)) * v(self.a[m - 2]))) for v in self.Foo
]
if k < m - 2:
bminmax = [
lagrange_degree_3(n, v(self.a[m - 1]), v(self.a[m - 2]),
v(self.a[m - 3])) for v in self.Foo
]
self.b_lower = bminmax[0]
self.b_upper = bminmax[1]
# Annoying, but must reverse coefficients for numpy.
gnk = [binomial(j, k + 2) * a[j] for j in range(k + 2, n + 1)]
self.beta[k + 1] = [[self.b_lower] + numpy.roots(
[v(gnk[i])
for i in range(len(gnk))].reverse()).tolist().sort() +
[self.b_upper] for v in self.Foo]
# Now to really initialize gnk.
self.gnk[k + 1] = [
[0] +
[binomial(j, k + 1) * v(a[j]) for j in range(k + 2, m + 1)]
for v in self.Foo
]
else:
# Bad input!
raise ValueError, "a has length %s > m+1" % len(a)
def pari(x):
"""
Return the pari object constructed from a Sage object.
The work is done by the __call__ method of the class PariInstance,
which in turn passes the work to any class which has its own
method _pari_().
EXAMPLES::
sage: pari([2,3,5])
[2, 3, 5]
sage: pari(Matrix(2,2,range(4)))
[0, 1; 2, 3]
sage: pari(x^2-3)
x^2 - 3
::
sage: a = pari(1); a, a.type()
(1, 't_INT')
sage: a = pari(1/2); a, a.type()
(1/2, 't_FRAC')
sage: a = pari(1/2); a, a.type()
(1/2, 't_FRAC')
Conversion from reals uses the real's own precision, here 53 bits (the default)::
sage: a = pari(1.2); a, a.type(), a.precision()
(1.20000000000000, 't_REAL', 4) # 32-bit
(1.20000000000000, 't_REAL', 3) # 64-bit
Conversion from strings uses the current pari real precision. By
default this is 4 words, 38 digits, 128 bits on 64-bit machines
and 5 words, 19 digits, 64 bits on 32-bit machines. ::
sage: a = pari('1.2'); a, a.type(), a.precision()
(1.20000000000000, 't_REAL', 5) # 32-bit
(1.20000000000000, 't_REAL', 4) # 64-bit
Conversion from matrices is supported, but not from vectors; use
lists instead::
sage: a = pari(matrix(2,3,[1,2,3,4,5,6])); a, a.type()
([1, 2, 3; 4, 5, 6], 't_MAT')
sage: v = vector([1.2,3.4,5.6])
sage: v.pari()
Traceback (most recent call last):
...
AttributeError: 'sage.modules.free_module_element.FreeModuleElement_generic_dense' object has no attribute 'pari'
sage: b = pari(list(v)); b,b.type()
([1.20000000000000, 3.40000000000000, 5.60000000000000], 't_VEC')
Some more exotic examples::
sage: K.<a> = NumberField(x^3 - 2)
sage: pari(K)
[y^3 - 2, [1, 1], -108, 1, [[1, 1.25992104989487, 1.58740105196820; 1, -0.629960524947437 - 1.09112363597172*I, -0.793700525984100 + 1.37472963699860*I], [1, 1.25992104989487, 1.58740105196820; 1, -1.72108416091916, 0.581029111014503; 1, 0.461163111024285, -2.16843016298270], [1, 1, 2; 1, -2, 1; 1, 0, -2], [3, 0, 0; 0, 0, 6; 0, 6, 0], [6, 0, 0; 0, 6, 0; 0, 0, 3], [2, 0, 0; 0, 0, 1; 0, 1, 0], [2, [0, 0, 2; 1, 0, 0; 0, 1, 0]]], [1.25992104989487, -0.629960524947437 - 1.09112363597172*I], [1, y, y^2], [1, 0, 0; 0, 1, 0; 0, 0, 1], [1, 0, 0, 0, 0, 2, 0, 2, 0; 0, 1, 0, 1, 0, 0, 0, 0, 2; 0, 0, 1, 0, 1, 0, 1, 0, 0]]
sage: E = EllipticCurve('37a1')
sage: pari(E)
[0, 0, 1, -1, 0, 0, -2, 1, -1, 48, -216, 37, 110592/37, [0.837565435283323, 0.269594436405445, -1.10715987168877]~, 2.99345864623196, -2.45138938198679*I, 0.942638555913623, 1.32703057887968*I, 7.33813274078958]
Conversion from basic Python types::
sage: pari(int(-5))
-5
sage: pari(long(2**150))
1427247692705959881058285969449495136382746624
sage: pari(float(pi))
3.14159265358979
sage: pari(complex(exp(pi*I/4)))
0.707106781186548 + 0.707106781186547*I
sage: pari(False)
0
sage: pari(True)
1
Some commands are just executed without returning a value::
sage: pari("dummy = 0; kill(dummy)")
sage: type(pari("dummy = 0; kill(dummy)"))
<type 'NoneType'>
"""
return gen.pari(x)
def enumerate_totallyreal_fields_all(n, B, verbose=0, return_seqs=False):
r"""
Enumerates *all* totally real fields of degree `n` with discriminant `\le B`,
primitive or otherwise.
EXAMPLES::
sage: enumerate_totallyreal_fields_all(4, 2000)
[[725, x^4 - x^3 - 3*x^2 + x + 1],
[1125, x^4 - x^3 - 4*x^2 + 4*x + 1],
[1600, x^4 - 6*x^2 + 4],
[1957, x^4 - 4*x^2 - x + 1],
[2000, x^4 - 5*x^2 + 5]]
In practice most of these will be found by :func:`~sage.rings.number_field.totallyreal.enumerate_totallyreal_fields_prim`, which is guaranteed to return all primitive fields but often returns many non-primitive ones as well. For instance, only one of the five fields in the example above is primitive, but :func:`~sage.rings.number_field.totallyreal.enumerate_totallyreal_fields_prim` finds four out of the five (the exception being `x^4 - 6x^2 + 4`).
"""
S = []
counts = [0, 0, 0]
if len(divisors(n)) > 4:
raise ValueError, "Only implemented for n = p*q with p,q prime"
for d in divisors(n):
if d > 1 and d < n:
Sds = enumerate_totallyreal_fields_prim(d, int(math.floor((1.0 * B) ** (1.0 * d / n))), verbose=verbose)
for i in range(len(Sds)):
if verbose:
print "=" * 80
print "Taking F =", Sds[i][1]
F = NumberField(ZZx(Sds[i][1]), "t")
T = enumerate_totallyreal_fields_rel(F, n / d, B, verbose=verbose, return_seqs=return_seqs)
if return_seqs:
for i in range(3):
counts[i] += T[0][i]
S += [[t[0], pari(t[1]).Polrev()] for t in T[1]]
else:
S += [[t[0], t[1]] for t in T]
j = i + 1
for E in enumerate_totallyreal_fields_prim(
n / d, int(math.floor((1.0 * B) ** (1.0 / d) / (1.0 * Sds[i][0]) ** (n * 1.0 / d ** 2)))
):
for EF in F.composite_fields(NumberField(ZZx(E[1]), "u")):
if EF.degree() == n and EF.disc() <= B:
S.append([EF.disc(), pari(EF.absolute_polynomial())])
S += enumerate_totallyreal_fields_prim(n, B, verbose=verbose)
S.sort()
weed_fields(S)
# Output.
if verbose:
saveout = sys.stdout
if type(verbose) == str:
fsock = open(verbose, "w")
sys.stdout = fsock
# Else, print to screen
print "=" * 80
print "Polynomials tested:", counts[0]
print "Irreducible polynomials:", counts[1]
print "Polynomials with nfdisc <= B:", counts[2]
for i in range(len(S)):
print S[i]
if type(verbose) == str:
fsock.close()
sys.stdout = saveout
if return_seqs:
return [counts, [[s[0], s[1].reverse().Vec()] for s in S]]
else:
return S
def enumerate_totallyreal_fields_rel(F, m, B, a=[], verbose=0, return_seqs=False):
r"""
This function enumerates (primitive) totally real field extensions of
degree `m>1` of the totally real field F with discriminant `d \leq B`;
optionally one can specify the first few coefficients, where the sequence ``a``
corresponds to a polynomial by
::
a[d]*x^n + ... + a[0]*x^(n-d)
if ``length(a) = d+1``, so in particular always ``a[d] = 1``.
If verbose == 1 (or 2), then print to the screen (really) verbosely; if
verbose is a string, then print verbosely to the file specified by verbose.
If return_seqs, then return the polynomials as sequences (for easier
exporting to a file).
NOTE:
This is guaranteed to give all primitive such fields, and
seems in practice to give many imprimitive ones.
INPUT:
- ``F`` -- number field, the base field
- ``m`` -- integer, the degree
- ``B`` -- integer, the discriminant bound
- ``a`` -- list (default: []), the coefficient list to begin with
- ``verbose`` -- boolean or string (default: 0)
- ``return_seqs`` -- boolean (default: False)
OUTPUT:
the list of fields with entries [d,fabs,f], where
d is the discriminant, fabs is an absolute defining polynomial,
and f is a defining polynomial relative to F,
sorted by discriminant.
EXAMPLES::
sage: ZZx = ZZ['x']
sage: F.<t> = NumberField(x^2-2)
sage: enumerate_totallyreal_fields_rel(F, 2, 2000)
[[1600, x^4 - 6*x^2 + 4, xF^2 + xF - 1]]
AUTHORS:
- John Voight (2007-11-01)
"""
if not isinstance(m, Integer):
try:
m = Integer(m)
except TypeError:
raise TypeError, "cannot coerce m (= %s) to an integer" % m
if m < 1:
raise ValueError, "m must be at least 1."
n = F.degree() * m
# Initialize
T = tr_data_rel(F, m, B, a)
S = []
Srel = []
dB_odlyzko = odlyzko_bound_totallyreal(n)
dB = math.ceil(40000 * dB_odlyzko ** n)
counts = [0, 0, 0, 0]
# Trivial case
if m == 1:
g = pari(F.defining_polynomial()).reverse().Vec()
if return_seqs:
return [[0, 0, 0, 0], [1, g, [-1, 1]]]
else:
return [[1, pari("x-1"), g]]
if verbose:
saveout = sys.stdout
if type(verbose) == str:
fsock = open(verbose, "w")
sys.stdout = fsock
# Else, print to screen
f_out = [0] * m + [1]
if verbose == 2:
T.incr(f_out, verbose)
else:
T.incr(f_out)
Fx = PolynomialRing(F, "xF")
nfF = pari(str(F.defining_polynomial()).replace("x", str(F.primitive_element())))
parit = pari(str(F.primitive_element()))
while f_out[m] <> 0:
counts[0] += 1
if verbose:
print "==>", f_out,
f_str = ""
for i in range(len(f_out)):
f_str += "(" + str(f_out[i]) + ")*x^" + str(i)
if i < len(f_out) - 1:
f_str += "+"
nf = pari(f_str)
if nf.poldegree("t") == 0:
nf = nf.subst("x", "x-t")
nf = nf.polresultant(nfF, parit)
d = nf.poldisc()
counts[0] += 1
if d > 0 and nf.polsturm_full() == n:
da = int_has_small_square_divisor(Integer(d))
if d > dB or d <= B * da:
counts[1] += 1
if nf.polisirreducible():
counts[2] += 1
[zk, d] = nf.nfbasis_d()
if d <= B:
if verbose:
print "has discriminant", d,
# Find a minimal lattice element
counts[3] += 1
ng = pari([nf, zk]).polredabs()
# Check if K is contained in the list.
found = False
ind = bisect.bisect_left(S, [d, ng])
while ind < len(S) and S[ind][0] == d:
if S[ind][1] == ng:
if verbose:
print "but is not new"
found = True
break
ind += 1
if not found:
if verbose:
print "and is new!"
S.insert(ind, [d, ng])
Srel.insert(ind, Fx(f_out))
else:
if verbose:
print "has discriminant", abs(d), "> B"
else:
if verbose:
print "is not absolutely irreducible"
else:
if verbose:
print "has discriminant", abs(d), "with no large enough square divisor"
else:
if verbose:
if d == 0:
print "is not squarefree"
else:
print "is not totally real"
if verbose == 2:
T.incr(f_out, verbose=verbose)
else:
T.incr(f_out)
# In the application of Smyth's theorem above, we exclude finitely
# many possibilities which we must now throw back in.
if m == 2:
if Fx([-1, 1, 1]).is_irreducible():
K = F.extension(Fx([-1, 1, 1]), "tK")
Kabs = K.absolute_field("tKabs")
Kabs_pari = pari(Kabs.defining_polynomial())
d = K.absolute_discriminant()
if abs(d) <= B:
ng = Kabs_pari.polredabs()
ind = bisect.bisect_left(S, [d, ng])
S.insert(ind, [d, ng])
Srel.insert(ind, Fx([-1, 1, 1]))
elif F.degree() == 2:
for ff in [[1, -7, 13, -7, 1], [1, -8, 14, -7, 1]]:
f = Fx(ff).factor()[0][0]
K = F.extension(f, "tK")
Kabs = K.absolute_field("tKabs")
Kabs_pari = pari(Kabs.defining_polynomial())
d = K.absolute_discriminant()
if abs(d) <= B:
ng = Kabs_pari.polredabs()
ind = bisect.bisect_left(S, [d, ng])
S.insert(ind, [d, ng])
Srel.insert(ind, f)
elif m == 3:
if Fx([-1, 6, -5, 1]).is_irreducible():
K = F.extension(Fx([-1, 6, -5, 1]), "tK")
Kabs = K.absolute_field("tKabs")
Kabs_pari = pari(Kabs.defining_polynomial())
d = K.absolute_discriminant()
if abs(d) <= B:
ng = Kabs_pari.polredabs()
ind = bisect.bisect_left(S, [d, ng])
S.insert(ind, [d, ng])
Srel.insert(ind, Fx([-1, 6, -5, 1]))
# Now check for isomorphic fields
S = [[S[i][0], S[i][1], Srel[i]] for i in range(len(S))]
weed_fields(S)
# Output.
if verbose:
print "=" * 80
print "Polynomials tested:", counts[0]
print "Irreducible polynomials:", counts[1]
print "Polynomials with nfdisc <= B:", counts[2]
for i in range(len(S)):
print S[i]
if type(verbose) == str:
fsock.close()
sys.stdout = saveout
if return_seqs:
return [counts, [[s[0], s[1].reverse().Vec(), s[2].coeffs()] for s in S]]
else:
return S
def __init__(self, F, m, B, a=None):
r"""
Initialization routine (constructor).
INPUT:
- ``F`` -- number field, the base field
- ``m`` -- integer, the relative degree
- ``B`` -- integer, the discriminant bound
- ``a`` -- list (default: []), the coefficient list to begin with,
corresponding to ``a[len(a)]*x^n + ... + a[0]x^(n-len(a))``.
OUTPUT:
the data initialized to begin enumeration of totally real fields
with base field F, degree n, discriminant bounded by B, and starting
with coefficients a.
EXAMPLES::
sage: F.<t> = NumberField(x^2-2)
sage: T = sage.rings.number_field.totallyreal_rel.tr_data_rel(F, 2, 2000)
"""
if a is None: # don't make the stupid noob mistake of putting a=[]
a = [] # in the function signature above.
# Initialize constants.
self.m = m
d = F.degree()
self.d = d
self.n = m * d
self.B = B
self.gamma = hermite_constant(self.n - self.d)
self.F = F
self.Z_F = F.maximal_order()
self.Foo = F.real_embeddings()
self.dF = abs(F.disc())
self.Fx = PolynomialRing(F, "xF")
self.beta = [[]] * m
self.gnk = [[]] * m
self.trace_elts = []
Z_Fbasis = self.Z_F.basis()
# Initialize variables.
if a == []:
# No starting input, all polynomials will be found; initialize to zero.
self.a = [0] * m + [1]
self.amaxvals = [[]] * m
anm1s = [[i] for i in range(0, m // 2 + 1)]
for i in range(1, self.d):
for j in range(len(anm1s)):
anm1s[j] = [anm1s[j] + [i] for i in range(m)]
anm1s = sum(anm1s, [])
anm1s = [sum([Z_Fbasis[i] * a[i] for i in range(self.d)]) for a in anm1s]
# Minimize trace in class.
import numpy
for i in range(len(anm1s)):
Q = [[v(m * x) for v in self.Foo] + [0] for x in Z_Fbasis] + [
[v(anm1s[i]) for v in self.Foo] + [10 ** 6]
]
pari_string = "[" + ";".join([",".join(["%s" % ii for ii in row]) for row in zip(*Q)]) + "]"
adj = pari(pari_string).qflll()[self.d]
anm1s[i] += sum([m * Z_Fbasis[ii] * int(adj[ii]) // int(adj[self.d]) for ii in range(self.d)])
self.amaxvals[m - 1] = anm1s
self.a[m - 1] = self.amaxvals[m - 1].pop()
self.k = m - 2
bl = math.ceil(1.7719 * self.n)
br = max(
[
1.0 / m * (am1 ** 2).trace()
+ self.gamma * (1.0 / (m ** d) * self.B / self.dF) ** (1.0 / (self.n - d))
for am1 in anm1s
]
)
br = math.floor(br)
T2s = self.F._positive_integral_elements_with_trace([bl, br])
self.trace_elts.append([bl, br, T2s])
elif len(a) <= m + 1:
# First few coefficients have been specified.
# The value of k is the largest index of the coefficients of a which is
# currently unknown; e.g., if k == -1, then we can iterate
# over polynomials, and if k == n-1, then we have finished iterating.
if a[len(a) - 1] <> 1:
raise ValueError, "a[len(a)-1](=%s) must be 1 so polynomial is monic" % a[len(a) - 1]
raise NotImplementedError, "These have not been checked."
k = m - len(a)
self.k = k
a = [0] * (k + 1) + a
self.amaxvals = [[]] * m
for i in range(0, n + 1):
self.a[i] = a[i]
# Bounds come from an application of Lagrange multipliers in degrees 2,3.
self.b_lower = [
-1.0
/ m
* (
v(self.a[m - 1])
+ (m - 1.0) * math.sqrt(v(self.a[m - 1]) ** 2 - 2.0 * (1 + 1.0 / (m - 1)) * v(self.a[m - 2]))
)
for v in self.Foo
]
self.b_upper = [
-1.0
/ m
* (
v(self.a[m - 1])
- (m - 1.0) * math.sqrt(v(self.a[m - 1]) ** 2 - 2.0 * (1 + 1.0 / (m - 1)) * v(self.a[m - 2]))
)
for v in self.Foo
]
if k < m - 2:
bminmax = [lagrange_degree_3(n, v(self.a[m - 1]), v(self.a[m - 2]), v(self.a[m - 3])) for v in self.Foo]
self.b_lower = bminmax[0]
self.b_upper = bminmax[1]
# Annoying, but must reverse coefficients for numpy.
gnk = [binomial(j, k + 2) * a[j] for j in range(k + 2, n + 1)]
self.beta[k + 1] = [
[self.b_lower]
+ numpy.roots([v(gnk[i]) for i in range(len(gnk))].reverse()).tolist().sort()
+ [self.b_upper]
for v in self.Foo
]
# Now to really initialize gnk.
self.gnk[k + 1] = [[0] + [binomial(j, k + 1) * v(a[j]) for j in range(k + 2, m + 1)] for v in self.Foo]
else:
# Bad input!
raise ValueError, "a has length %s > m+1" % len(a)
def pari(x):
"""
Return the pari object constructed from a Sage object.
The work is done by the __call__ method of the class PariInstance,
which in turn passes the work to any class which has its own
method _pari_().
EXAMPLES::
sage: pari([2,3,5])
[2, 3, 5]
sage: pari(Matrix(2,2,range(4)))
[0, 1; 2, 3]
sage: pari(x^2-3)
x^2 - 3
::
sage: a = pari(1); a, a.type()
(1, 't_INT')
sage: a = pari(1/2); a, a.type()
(1/2, 't_FRAC')
sage: a = pari(1/2); a, a.type()
(1/2, 't_FRAC')
Conversion from reals uses the real's own precision, here 53 bits (the default)::
sage: a = pari(1.2); a, a.type(), a.precision()
(1.20000000000000, 't_REAL', 4) # 32-bit
(1.20000000000000, 't_REAL', 3) # 64-bit
Conversion from strings uses the current pari real precision. By
default this is 4 words, 38 digits, 128 bits on 64-bit machines
and 5 words, 19 digits, 64 bits on 32-bit machines. ::
sage: a = pari('1.2'); a, a.type(), a.precision()
(1.20000000000000, 't_REAL', 5) # 32-bit
(1.20000000000000, 't_REAL', 4) # 64-bit
Conversion from matrices is supported, but not from vectors; use
lists instead::
sage: a = pari(matrix(2,3,[1,2,3,4,5,6])); a, a.type()
([1, 2, 3; 4, 5, 6], 't_MAT')
sage: v = vector([1.2,3.4,5.6])
sage: v.pari()
Traceback (most recent call last):
...
AttributeError: 'sage.modules.free_module_element.FreeModuleElement_generic_dense' object has no attribute 'pari'
sage: b = pari(list(v)); b,b.type()
([1.20000000000000, 3.40000000000000, 5.60000000000000], 't_VEC')
Some more exotic examples::
sage: K.<a> = NumberField(x^3 - 2)
sage: pari(K)
[y^3 - 2, [1, 1], -108, 1, [[1, 1.25992104989487, 1.58740105196820; 1, -0.629960524947437 - 1.09112363597172*I, -0.793700525984100 + 1.37472963699860*I], [1, 1.25992104989487, 1.58740105196820; 1, -1.72108416091916, 0.581029111014503; 1, 0.461163111024285, -2.16843016298270], [1, 1, 2; 1, -2, 1; 1, 0, -2], [3, 0, 0; 0, 0, 6; 0, 6, 0], [6, 0, 0; 0, 6, 0; 0, 0, 3], [2, 0, 0; 0, 0, 1; 0, 1, 0], [2, [0, 0, 2; 1, 0, 0; 0, 1, 0]]], [1.25992104989487, -0.629960524947437 - 1.09112363597172*I], [1, y, y^2], [1, 0, 0; 0, 1, 0; 0, 0, 1], [1, 0, 0, 0, 0, 2, 0, 2, 0; 0, 1, 0, 1, 0, 0, 0, 0, 2; 0, 0, 1, 0, 1, 0, 1, 0, 0]]
sage: E = EllipticCurve('37a1')
sage: pari(E)
[0, 0, 1, -1, 0, 0, -2, 1, -1, 48, -216, 37, 110592/37, [0.837565435283323, 0.269594436405445, -1.10715987168877]~, 2.99345864623196, -2.45138938198679*I, 0.942638555913623, 1.32703057887968*I, 7.33813274078958]
Conversion from basic Python types::
sage: pari(int(-5))
-5
sage: pari(long(2**150))
1427247692705959881058285969449495136382746624
sage: pari(float(pi))
3.14159265358979
sage: pari(complex(exp(pi*I/4)))
0.707106781186548 + 0.707106781186547*I
sage: pari(False)
0
sage: pari(True)
1
Some commands are just executed without returning a value::
sage: pari("dummy = 0; kill(dummy)")
sage: type(pari("dummy = 0; kill(dummy)"))
<type 'NoneType'>
"""
return gen.pari(x)
def classno(d):
"""Return the class number of the order of discriminant d."""
# There is currently no qfbclassno method in gen.pyx, hence the string.
return Integer(pari('qfbclassno(%s,%s)' % (d, 1 if proof else 0)))
def classno(d):
"""Return the class number of the order of discriminant d."""
# There is currently no qfbclassno method in gen.pyx, hence the string.
return Integer(pari('qfbclassno(%s,%s)'%(d, 1 if proof else 0)))