def decomposition(self, *args, **kwds):
"""
Return decomposition of this endomorphism, i.e., sequence of
subspaces obtained by finding invariant subspaces of self.
See the documentation for self.matrix().decomposition for more
details. All inputs to this function are passed onto the
matrix one.
EXAMPLES::
sage: V = ZZ^2; phi = V.hom([V.0+V.1, 2*V.1])
sage: phi.decomposition()
[
Free module of degree 2 and rank 1 over Integer Ring
Echelon basis matrix:
[0 1],
Free module of degree 2 and rank 1 over Integer Ring
Echelon basis matrix:
[ 1 -1]
]
"""
if not self.is_endomorphism():
raise ArithmeticError("Matrix morphism must be an endomorphism.")
D = self.domain()
E = self.matrix().decomposition(*args,**kwds)
if D.is_ambient():
return Sequence([D.submodule(V, check=False) for V, _ in E],
cr=True, check=False)
else:
B = D.basis_matrix()
R = D.base_ring()
return Sequence([D.submodule((V.basis_matrix() * B).row_module(R),
check=False) for V, _ in E],
cr=True, check=False)
def _load_from_csv(self, R):
r"""
EXAMPLES:
This indirectly tests ``_load_from_csv()``::
sage: filename = tmp_filename(ext='.csv')
sage: with open(filename,'w') as fobj:
....: _ = fobj.write("Date,Open,High,Low,Close,Volume\n1212405780,187.80,187.80,187.80,187.80,100\n1212407640,187.75,188.00,187.75,188.00,2000\n1212407700,188.00,188.00,188.00,188.00,1000\n1212408000,188.00,188.11,188.00,188.00,2877\n1212408060,188.00,188.00,188.00,188.00,687")
sage: finance.Stock('aapl').load_from_file(filename)
[
1212408060 188.00 188.00 188.00 188.00 687,
1212408000 188.00 188.11 188.00 188.00 2877,
1212407700 188.00 188.00 188.00 188.00 1000,
1212407640 187.75 188.00 187.75 188.00 2000,
1212405780 187.80 187.80 187.80 187.80 100
]
"""
R = R.splitlines()
hist_data = []
for x in reversed(R[1:]):
try:
timestamp, opn, high, low, close, volume = x.split(',')
ohlc = OHLC(timestamp, opn, high, low, close, volume)
hist_data.append(ohlc)
except ValueError:
pass
hist_data = Sequence(hist_data,
cr=True,
universe=lambda x: x,
immutable=True)
return hist_data
def gritsenko_lift_subspace(N, weight, precision):
"""
Return a list of data, which can be used by the Fourier expansion
generator, for the space of Gritsenko lifts.
INPUT:
- `N` -- the level of the paramodular group
- ``weight`` -- the weight of the space, that we consider
- ``precision`` -- A precision class
NOTE:
The lifts returned have to have integral Fourier coefficients.
"""
jf = JacobiFormsD1NN(QQ, JacobiFormD1NN_Gamma(
N,
weight), (4 * N**2 + precision._enveloping_discriminant_bound() - 1) //
(4 * N) + 1)
return Sequence([
gritsenko_lift_fourier_expansion(
g if i != 0 else (bernoulli(weight) /
(2 * weight)).denominator() * g, precision, True)
for (i, g) in enumerate(jf.gens())
],
universe=ParamodularFormD2FourierExpansionRing(ZZ, N),
immutable=True,
check=False)
def new_decomposition(self, verbose=False):
"""
Return complete irreducible Hecke decomposition of new subspace of self.
"""
V = self.degeneracy_matrix().kernel()
p = next_prime_of_characteristic_coprime_to(F.ideal(1), self.level())
T = self.hecke_matrix(p)
D = T.decomposition_of_subspace(V)
while len([X for X in D if not X[1]]) > 0:
p = next_prime_of_characteristic_coprime_to(p, self.level())
if verbose: print(p.norm())
T = self.hecke_matrix(p)
D2 = []
for X in D:
if X[1]:
D2.append(X)
else:
if verbose: print(T.restrict(X[0]).fcp())
for Z in T.decomposition_of_subspace(X[0]):
D2.append(Z)
D = D2
D = [self.subspace(X[0]) for X in D]
D.sort()
S = Sequence(D, immutable=True, cr=True, universe=int, check=False)
return S
def decomposition(self, B, verbose=False):
"""
Return Hecke decomposition of self using Hecke operators T_p
coprime to the level with norm(p) <= B.
"""
# TODO: rewrite to use primes_of_bounded_norm so that we
# iterate through primes ordered by *norm*, which is
# potentially vastly faster. Delete these functions
# involving characteristic!
p = next_prime_of_characteristic_coprime_to(F.ideal(1), self.level())
T = self.hecke_matrix(p)
D = T.decomposition()
while len([X for X in D if not X[1]]) > 0:
p = next_prime_of_characteristic_coprime_to(p, self.level())
if p.norm() > B:
break
if verbose: print(p.norm())
T = self.hecke_matrix(p)
D2 = []
for X in D:
if X[1]:
D2.append(X)
else:
if verbose: print(T.restrict(X[0]).fcp())
for Z in T.decomposition_of_subspace(X[0]):
D2.append(Z)
D = D2
D = [self.subspace(X[0]) for X in D]
D.sort()
S = Sequence(D, immutable=True, cr=True, universe=int, check=False)
return S
def lfsr_sequence(key, fill, n):
r"""
Create an LFSR sequence.
INPUT:
- ``key`` -- a list of finite field elements, `[c_0, c_1,\dots, c_k]`
- ``fill`` -- the list of the initial terms of the LFSR sequence, `[x_0,x_1,\dots,x_k]`
- ``n`` -- number of terms of the sequence that the function returns
OUTPUT:
The LFSR sequence defined by `x_{n+1} = c_kx_n+...+c_0x_{n-k}` for `n \geq k`.
EXAMPLES::
sage: F = GF(2); l = F(1); o = F(0)
sage: F = GF(2); S = LaurentSeriesRing(F,'x'); x = S.gen()
sage: fill = [l,l,o,l]; key = [1,o,o,l]; n = 20
sage: L = lfsr_sequence(key,fill,20); L
[1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0]
sage: from sage.matrix.berlekamp_massey import berlekamp_massey
sage: g = berlekamp_massey(L); g
x^4 + x^3 + 1
sage: (1)/(g.reverse()+O(x^20))
1 + x + x^2 + x^3 + x^5 + x^7 + x^8 + x^11 + x^15 + x^16 + x^17 + x^18 + O(x^20)
sage: (1+x^2)/(g.reverse()+O(x^20))
1 + x + x^4 + x^8 + x^9 + x^10 + x^11 + x^13 + x^15 + x^16 + x^19 + O(x^20)
sage: (1+x^2+x^3)/(g.reverse()+O(x^20))
1 + x + x^3 + x^5 + x^6 + x^9 + x^13 + x^14 + x^15 + x^16 + x^18 + O(x^20)
sage: fill = [l,l,o,l]; key = [l,o,o,o]; n = 20
sage: L = lfsr_sequence(key,fill,20); L
[1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1]
sage: g = berlekamp_massey(L); g
x^4 + 1
sage: (1+x)/(g.reverse()+O(x^20))
1 + x + x^4 + x^5 + x^8 + x^9 + x^12 + x^13 + x^16 + x^17 + O(x^20)
sage: (1+x+x^3)/(g.reverse()+O(x^20))
1 + x + x^3 + x^4 + x^5 + x^7 + x^8 + x^9 + x^11 + x^12 + x^13 + x^15 + x^16 + x^17 + x^19 + O(x^20)
"""
if not isinstance(key, list):
raise TypeError("key must be a list")
key = Sequence(key)
F = key.universe()
if not is_FiniteField(F):
raise TypeError("universe of sequence must be a finite field")
s = fill
k = len(fill)
L = []
for i in range(n):
s0 = copy.copy(s)
L.append(s[0])
s = s[1:k]
s.append(sum([key[i] * s0[i] for i in range(k)]))
return L
def vertices(self):
"""
EXAMPLES::
sage: P = polymake.convex_hull([[1,0,0,0], [1,0,0,1], [1,0,1,0], [1,0,1,1], [1,1,0,0], [1,1,0,1], [1,1,1,0], [1,1,1,1]]) # optional - polymake
sage: P.vertices() # optional - polymake
[(1, 0, 0, 0), (1, 0, 0, 1), (1, 0, 1, 0), (1, 0, 1, 1), (1, 1, 0, 0), (1, 1, 0, 1), (1, 1, 1, 0), (1, 1, 1, 1)]
"""
try:
return self.__vertices
except AttributeError:
pass
s = self.cmd('VERTICES')
s = s.rstrip().split('\n')[1:]
if len(s) == 0:
ans = Sequence([], immutable=True)
else:
n = len(s[0].split())
V = VectorSpace(QQ, n)
ans = Sequence((V(x.split()) for x in s), immutable=True)
self.__vertices = ans
return ans
def facets(self):
"""
EXAMPLES::
sage: P = polymake.convex_hull([[1,0,0,0], [1,0,0,1], [1,0,1,0], [1,0,1,1], [1,1,0,0], [1,1,0,1], [1,1,1,0], [1,1,1,1]]) # optional: needs polymake
sage: P.facets() # optional
[(0, 0, 0, 1), (0, 1, 0, 0), (0, 0, 1, 0), (1, 0, 0, -1), (1, 0, -1, 0), (1, -1, 0, 0)]
"""
try:
return self.__facets
except AttributeError:
pass
s = self.cmd('FACETS')
s = s.rstrip().split('\n')[1:]
if len(s) == 0:
ans = Sequence([], immutable=True)
else:
n = len(s[0].split())
V = VectorSpace(QQ, n)
ans = Sequence((V(x.split()) for x in s), immutable=True)
self.__facets = ans
return ans
def elliptic_curve_factors(self):
D = [X for X in self.new_decomposition() if X.dimension() == 1]
# Have to get rid of the Eisenstein factor
p = next_prime_of_characteristic_coprime_to(F.ideal(1), self.level())
while True:
q = p.residue_field().cardinality() + 1
E = [A for A in D if A.hecke_matrix(p)[0,0] == q]
if len(E) == 0:
break
elif len(E) == 1:
D = [A for A in D if A != E[0]]
break
else:
p = next_prime_of_characteristic_coprime_to(p, self.level())
return Sequence([EllipticCurveFactor(X, number) for number, X in enumerate(D)],
immutable=True, cr=True, universe=int, check=False)
def generators(self, K, precision):
"""
A list of Fourier expansions of forms that generate the ring
or module of modular forms.
INPUT:
- `K` -- A ring or module; The ring of Fourier coefficients.
- ``precision`` -- A precision class; The precision of the Fourier
expansions.
OUTPUT:
A sequence of equivariant monoid power series.
TESTS::
sage: from hermitianmodularforms import *
sage: from hermitianmodularforms.hermitianmodularformd2_fourierexpansion import HermitianModularFormD2Filter_diagonal
sage: from hermitianmodularforms.hermitianmodularformd2_fourierexpansion import HermitianModularFormD2FourierExpansionRing
sage: HermitianModularFormD2_Gamma(-3).generators(QQ, HermitianModularFormD2Filter_diagonal(2, -3)).universe() == HermitianModularFormD2FourierExpansionRing(QQ, -3)
True
sage: HermitianModularFormD2_Gamma(-3).generators(GF(2), HermitianModularFormD2Filter_diagonal(2, -3))
Traceback (most recent call last):
...
NotImplementedError: Only Fourier coefficients in a number fields are implemented.
"""
if self.__D == -3:
if K is QQ or K in NumberFields():
return Sequence(
[
self._e4_D3(precision),
self._e6_D3(precision),
self._e10_D3(precision),
self._e12_D3(precision),
self._phi9_D3(precision)
],
universe=HermitianModularFormD2FourierExpansionRing(
QQ, -3))
raise NotImplementedError(
"Only Fourier coefficients in a number fields are implemented."
)
raise NotImplementedError("Discriminant %s is not implemented." %
(self.__D, ))
def generators(self, K, precision):
"""
The generators for modular forms with coefficients in `K`.
INPUT:
- `K` -- A ring.
- ``precision`` -- An instance of :class:~`fourier_expansion_framework.monoidpowerseries.monoidpowerseries_basicmonoids.NNFilter`.
OUTPUT:
A sequence of elements of :class:`~fourier_expansion_framework.monoidpowerseries.monoidpowerseries_ring.EquivariantMonoidPowerSeriesModule`.
TESTS::
sage: from psage.modform.fourier_expansion_framework.monoidpowerseries.monoidpowerseries_basicmonoids import *
sage: from psage.modform.fourier_expansion_framework.modularforms.modularform_testtype import *
sage: t = ModularFormTestType_vectorvalued()
sage: t.generators(QQ, NNFilter(4))
[Equivariant monoid power series in Module of equivariant monoid power series over NN, Equivariant monoid power series in Module of equivariant monoid power series over NN, Equivariant monoid power series in Module of equivariant monoid power series over NN]
"""
if K.has_coerce_map_from(ZZ):
return Sequence([self._g(i, precision) for i in range(self.rank)],
universe=TestExpansionAmbient_vv(K))
raise NotImplementedError
def Conic(base_field, F=None, names=None, unique=True):
r"""
Return the plane projective conic curve defined by ``F``
over ``base_field``.
The input form ``Conic(F, names=None)`` is also accepted,
in which case the fraction field of the base ring of ``F``
is used as base field.
INPUT:
- ``base_field`` -- The base field of the conic.
- ``names`` -- a list, tuple, or comma separated string
of three variable names specifying the names
of the coordinate functions of the ambient
space `\Bold{P}^3`. If not specified or read
off from ``F``, then this defaults to ``'x,y,z'``.
- ``F`` -- a polynomial, list, matrix, ternary quadratic form,
or list or tuple of 5 points in the plane.
If ``F`` is a polynomial or quadratic form,
then the output is the curve in the projective plane
defined by ``F = 0``.
If ``F`` is a polynomial, then it must be a polynomial
of degree at most 2 in 2 variables, or a homogeneous
polynomial in of degree 2 in 3 variables.
If ``F`` is a matrix, then the output is the zero locus
of `(x,y,z) F (x,y,z)^t`.
If ``F`` is a list of coefficients, then it has
length 3 or 6 and gives the coefficients of
the monomials `x^2, y^2, z^2` or all 6 monomials
`x^2, xy, xz, y^2, yz, z^2` in lexicographic order.
If ``F`` is a list of 5 points in the plane, then the output
is a conic through those points.
- ``unique`` -- Used only if ``F`` is a list of points in the plane.
If the conic through the points is not unique, then
raise ``ValueError`` if and only if ``unique`` is True
OUTPUT:
A plane projective conic curve defined by ``F`` over a field.
EXAMPLES:
Conic curves given by polynomials ::
sage: X,Y,Z = QQ['X,Y,Z'].gens()
sage: Conic(X^2 - X*Y + Y^2 - Z^2)
Projective Conic Curve over Rational Field defined by X^2 - X*Y + Y^2 - Z^2
sage: x,y = GF(7)['x,y'].gens()
sage: Conic(x^2 - x + 2*y^2 - 3, 'U,V,W')
Projective Conic Curve over Finite Field of size 7 defined by U^2 + 2*V^2 - U*W - 3*W^2
Conic curves given by matrices ::
sage: Conic(matrix(QQ, [[1, 2, 0], [4, 0, 0], [7, 0, 9]]), 'x,y,z')
Projective Conic Curve over Rational Field defined by x^2 + 6*x*y + 7*x*z + 9*z^2
sage: x,y,z = GF(11)['x,y,z'].gens()
sage: C = Conic(x^2+y^2-2*z^2); C
Projective Conic Curve over Finite Field of size 11 defined by x^2 + y^2 - 2*z^2
sage: Conic(C.symmetric_matrix(), 'x,y,z')
Projective Conic Curve over Finite Field of size 11 defined by x^2 + y^2 - 2*z^2
Conics given by coefficients ::
sage: Conic(QQ, [1,2,3])
Projective Conic Curve over Rational Field defined by x^2 + 2*y^2 + 3*z^2
sage: Conic(GF(7), [1,2,3,4,5,6], 'X')
Projective Conic Curve over Finite Field of size 7 defined by X0^2 + 2*X0*X1 - 3*X1^2 + 3*X0*X2 - 2*X1*X2 - X2^2
The conic through a set of points ::
sage: C = Conic(QQ, [[10,2],[3,4],[-7,6],[7,8],[9,10]]); C
Projective Conic Curve over Rational Field defined by x^2 + 13/4*x*y - 17/4*y^2 - 35/2*x*z + 91/4*y*z - 37/2*z^2
sage: C.rational_point()
(10 : 2 : 1)
sage: C.point([3,4])
(3 : 4 : 1)
sage: a=AffineSpace(GF(13),2)
sage: Conic([a([x,x^2]) for x in range(5)])
Projective Conic Curve over Finite Field of size 13 defined by x^2 - y*z
"""
if not (is_IntegralDomain(base_field) or base_field == None):
if names is None:
names = F
F = base_field
base_field = None
if isinstance(F, (list,tuple)):
if len(F) == 1:
return Conic(base_field, F[0], names)
if names == None:
names = 'x,y,z'
if len(F) == 5:
L=[]
for f in F:
if isinstance(f, SchemeMorphism_point_affine):
C = Sequence(f, universe = base_field)
if len(C) != 2:
raise TypeError, "points in F (=%s) must be planar"%F
C.append(1)
elif isinstance(f, SchemeMorphism_point_projective_field):
C = Sequence(f, universe = base_field)
elif isinstance(f, (list, tuple)):
C = Sequence(f, universe = base_field)
if len(C) == 2:
C.append(1)
else:
raise TypeError, "F (=%s) must be a sequence of planar " \
"points" % F
if len(C) != 3:
raise TypeError, "points in F (=%s) must be planar" % F
P = C.universe()
if not is_IntegralDomain(P):
raise TypeError, "coordinates of points in F (=%s) must " \
"be in an integral domain" % F
L.append(Sequence([C[0]**2, C[0]*C[1], C[0]*C[2], C[1]**2,
C[1]*C[2], C[2]**2], P.fraction_field()))
M=Matrix(L)
if unique and M.rank() != 5:
raise ValueError, "points in F (=%s) do not define a unique " \
"conic" % F
con = Conic(base_field, Sequence(M.right_kernel().gen()), names)
con.point(F[0])
return con
F = Sequence(F, universe = base_field)
base_field = F.universe().fraction_field()
temp_ring = PolynomialRing(base_field, 3, names)
(x,y,z) = temp_ring.gens()
if len(F) == 3:
return Conic(F[0]*x**2 + F[1]*y**2 + F[2]*z**2)
if len(F) == 6:
return Conic(F[0]*x**2 + F[1]*x*y + F[2]*x*z + F[3]*y**2 + \
F[4]*y*z + F[5]*z**2)
raise TypeError, "F (=%s) must be a sequence of 3 or 6" \
"coefficients" % F
if is_QuadraticForm(F):
F = F.matrix()
if is_Matrix(F) and F.is_square() and F.ncols() == 3:
if names == None:
names = 'x,y,z'
temp_ring = PolynomialRing(F.base_ring(), 3, names)
F = vector(temp_ring.gens()) * F * vector(temp_ring.gens())
if not is_MPolynomial(F):
raise TypeError, "F (=%s) must be a three-variable polynomial or " \
"a sequence of points or coefficients" % F
if F.total_degree() != 2:
raise TypeError, "F (=%s) must have degree 2" % F
if base_field == None:
base_field = F.base_ring()
if not is_IntegralDomain(base_field):
raise ValueError, "Base field (=%s) must be a field" % base_field
base_field = base_field.fraction_field()
if names == None:
names = F.parent().variable_names()
pol_ring = PolynomialRing(base_field, 3, names)
if F.parent().ngens() == 2:
(x,y,z) = pol_ring.gens()
F = pol_ring(F(x/z,y/z)*z**2)
if F == 0:
raise ValueError, "F must be nonzero over base field %s" % base_field
if F.total_degree() != 2:
raise TypeError, "F (=%s) must have degree 2 over base field %s" % \
(F, base_field)
if F.parent().ngens() == 3:
P2 = ProjectiveSpace(2, base_field, names)
if is_PrimeFiniteField(base_field):
return ProjectiveConic_prime_finite_field(P2, F)
if is_FiniteField(base_field):
return ProjectiveConic_finite_field(P2, F)
if is_RationalField(base_field):
return ProjectiveConic_rational_field(P2, F)
if is_NumberField(base_field):
return ProjectiveConic_number_field(P2, F)
return ProjectiveConic_field(P2, F)
raise TypeError, "Number of variables of F (=%s) must be 2 or 3" % F
def victor_miller_basis(k, prec=10, cusp_only=False, var='q'):
r"""
Compute and return the Victor Miller basis for modular forms of
weight `k` and level 1 to precision `O(q^{prec})`. If
``cusp_only`` is True, return only a basis for the cuspidal
subspace.
INPUT:
- ``k`` -- an integer
- ``prec`` -- (default: 10) a positive integer
- ``cusp_only`` -- bool (default: False)
- ``var`` -- string (default: 'q')
OUTPUT:
A sequence whose entries are power series in ``ZZ[[var]]``.
EXAMPLES::
sage: victor_miller_basis(1, 6)
[]
sage: victor_miller_basis(0, 6)
[
1 + O(q^6)
]
sage: victor_miller_basis(2, 6)
[]
sage: victor_miller_basis(4, 6)
[
1 + 240*q + 2160*q^2 + 6720*q^3 + 17520*q^4 + 30240*q^5 + O(q^6)
]
sage: victor_miller_basis(6, 6, var='w')
[
1 - 504*w - 16632*w^2 - 122976*w^3 - 532728*w^4 - 1575504*w^5 + O(w^6)
]
sage: victor_miller_basis(6, 6)
[
1 - 504*q - 16632*q^2 - 122976*q^3 - 532728*q^4 - 1575504*q^5 + O(q^6)
]
sage: victor_miller_basis(12, 6)
[
1 + 196560*q^2 + 16773120*q^3 + 398034000*q^4 + 4629381120*q^5 + O(q^6),
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 + O(q^6)
]
sage: victor_miller_basis(12, 6, cusp_only=True)
[
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 + O(q^6)
]
sage: victor_miller_basis(24, 6, cusp_only=True)
[
q + 195660*q^3 + 12080128*q^4 + 44656110*q^5 + O(q^6),
q^2 - 48*q^3 + 1080*q^4 - 15040*q^5 + O(q^6)
]
sage: victor_miller_basis(24, 6)
[
1 + 52416000*q^3 + 39007332000*q^4 + 6609020221440*q^5 + O(q^6),
q + 195660*q^3 + 12080128*q^4 + 44656110*q^5 + O(q^6),
q^2 - 48*q^3 + 1080*q^4 - 15040*q^5 + O(q^6)
]
sage: victor_miller_basis(32, 6)
[
1 + 2611200*q^3 + 19524758400*q^4 + 19715347537920*q^5 + O(q^6),
q + 50220*q^3 + 87866368*q^4 + 18647219790*q^5 + O(q^6),
q^2 + 432*q^3 + 39960*q^4 - 1418560*q^5 + O(q^6)
]
sage: victor_miller_basis(40,200)[1:] == victor_miller_basis(40,200,cusp_only=True)
True
sage: victor_miller_basis(200,40)[1:] == victor_miller_basis(200,40,cusp_only=True)
True
AUTHORS:
- William Stein, Craig Citro: original code
- Martin Raum (2009-08-02): use FLINT for polynomial arithmetic (instead of NTL)
"""
k = Integer(k)
if k % 2 == 1 or k == 2:
return Sequence([])
elif k < 0:
raise ValueError("k must be non-negative")
elif k == 0:
return Sequence([PowerSeriesRing(ZZ, var)(1).add_bigoh(prec)], cr=True)
e = k.mod(12)
if e == 2: e += 12
n = (k - e) // 12
if n == 0 and cusp_only:
return Sequence([])
# If prec is less than or equal to the dimension of the space of
# cusp forms, which is just n, then we know the answer, and we
# simply return it.
if prec <= n:
q = PowerSeriesRing(ZZ, var).gen(0)
err = bigO(q**prec)
ls = [0] * (n + 1)
if not cusp_only:
ls[0] = 1 + err
for i in range(1, prec):
ls[i] = q**i + err
for i in range(prec, n + 1):
ls[i] = err
return Sequence(ls, cr=True)
F6 = eisenstein_series_poly(6, prec)
if e == 0:
A = Fmpz_poly(1)
elif e == 4:
A = eisenstein_series_poly(4, prec)
elif e == 6:
A = F6
elif e == 8:
A = eisenstein_series_poly(8, prec)
elif e == 10:
A = eisenstein_series_poly(10, prec)
else: # e == 14
A = eisenstein_series_poly(14, prec)
if A[0] == -1:
A = -A
if n == 0:
return Sequence([PowerSeriesRing(ZZ, var)(A.list()).add_bigoh(prec)],
cr=True)
F6_squared = F6**2
F6_squared._unsafe_mutate_truncate(prec)
D = _delta_poly(prec)
Fprod = F6_squared
Dprod = D
if cusp_only:
ls = [Fmpz_poly(0)] + [A] * n
else:
ls = [A] * (n + 1)
for i in xrange(1, n + 1):
ls[n - i] *= Fprod
ls[i] *= Dprod
ls[n - i]._unsafe_mutate_truncate(prec)
ls[i]._unsafe_mutate_truncate(prec)
Fprod *= F6_squared
Dprod *= D
Fprod._unsafe_mutate_truncate(prec)
Dprod._unsafe_mutate_truncate(prec)
P = PowerSeriesRing(ZZ, var)
if cusp_only:
for i in xrange(1, n + 1):
for j in xrange(1, i):
ls[j] = ls[j] - ls[j][i] * ls[i]
return Sequence([P(l.list()).add_bigoh(prec) for l in ls[1:]], cr=True)
else:
for i in xrange(1, n + 1):
for j in xrange(i):
ls[j] = ls[j] - ls[j][i] * ls[i]
return Sequence([P(l.list()).add_bigoh(prec) for l in ls], cr=True)
def Curve(F, A=None):
"""
Return the plane or space curve defined by ``F``, where ``F`` can be either
a multivariate polynomial, a list or tuple of polynomials, or an algebraic
scheme.
If no ambient space is passed in for ``A``, and if ``F`` is not an
algebraic scheme, a new ambient space is constructed.
Also not specifying an ambient space will cause the curve to be defined in
either affine or projective space based on properties of ``F``. In
particular, if ``F`` contains a nonhomogenous polynomial, the curve is
affine, and if ``F`` consists of homogenous polynomials, then the curve is
projective.
INPUT:
- ``F`` -- a multivariate polynomial, or a list or tuple of polynomials, or an algebraic scheme.
- ``A`` -- (default: None) an ambient space in which to create the curve.
EXAMPLES: A projective plane curve. ::
sage: x,y,z = QQ['x,y,z'].gens()
sage: C = Curve(x^3 + y^3 + z^3); C
Projective Plane Curve over Rational Field defined by x^3 + y^3 + z^3
sage: C.genus()
1
Affine plane curves. ::
sage: x,y = GF(7)['x,y'].gens()
sage: C = Curve(y^2 + x^3 + x^10); C
Affine Plane Curve over Finite Field of size 7 defined by x^10 + x^3 + y^2
sage: C.genus()
0
sage: x, y = QQ['x,y'].gens()
sage: Curve(x^3 + y^3 + 1)
Affine Plane Curve over Rational Field defined by x^3 + y^3 + 1
A projective space curve. ::
sage: x,y,z,w = QQ['x,y,z,w'].gens()
sage: C = Curve([x^3 + y^3 - z^3 - w^3, x^5 - y*z^4]); C
Projective Curve over Rational Field defined by x^3 + y^3 - z^3 - w^3, x^5 - y*z^4
sage: C.genus()
13
An affine space curve. ::
sage: x,y,z = QQ['x,y,z'].gens()
sage: C = Curve([y^2 + x^3 + x^10 + z^7, x^2 + y^2]); C
Affine Curve over Rational Field defined by x^10 + z^7 + x^3 + y^2, x^2 + y^2
sage: C.genus()
47
We can also make non-reduced non-irreducible curves. ::
sage: x,y,z = QQ['x,y,z'].gens()
sage: Curve((x-y)*(x+y))
Projective Conic Curve over Rational Field defined by x^2 - y^2
sage: Curve((x-y)^2*(x+y)^2)
Projective Plane Curve over Rational Field defined by x^4 - 2*x^2*y^2 + y^4
A union of curves is a curve. ::
sage: x,y,z = QQ['x,y,z'].gens()
sage: C = Curve(x^3 + y^3 + z^3)
sage: D = Curve(x^4 + y^4 + z^4)
sage: C.union(D)
Projective Plane Curve over Rational Field defined by
x^7 + x^4*y^3 + x^3*y^4 + y^7 + x^4*z^3 + y^4*z^3 + x^3*z^4 + y^3*z^4 + z^7
The intersection is not a curve, though it is a scheme. ::
sage: X = C.intersection(D); X
Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
x^3 + y^3 + z^3,
x^4 + y^4 + z^4
Note that the intersection has dimension 0. ::
sage: X.dimension()
0
sage: I = X.defining_ideal(); I
Ideal (x^3 + y^3 + z^3, x^4 + y^4 + z^4) of Multivariate Polynomial Ring in x, y, z over Rational Field
If only a polynomial in three variables is given, then it must be
homogeneous such that a projective curve is constructed. ::
sage: x,y,z = QQ['x,y,z'].gens()
sage: Curve(x^2+y^2)
Projective Conic Curve over Rational Field defined by x^2 + y^2
sage: Curve(x^2+y^2+z)
Traceback (most recent call last):
...
TypeError: x^2 + y^2 + z is not a homogeneous polynomial
An ambient space can be specified to construct a space curve in an affine
or a projective space. ::
sage: A.<x,y,z> = AffineSpace(QQ, 3)
sage: C = Curve([y - x^2, z - x^3], A)
sage: C
Affine Curve over Rational Field defined by -x^2 + y, -x^3 + z
sage: A == C.ambient_space()
True
The defining polynomial must be nonzero unless the ambient space itself is
of dimension 1. ::
sage: P1.<x,y> = ProjectiveSpace(1,GF(5))
sage: S = P1.coordinate_ring()
sage: Curve(S(0), P1)
Projective Line over Finite Field of size 5
sage: Curve(P1)
Projective Line over Finite Field of size 5
::
sage: A1.<x> = AffineSpace(1, QQ)
sage: R = A1.coordinate_ring()
sage: Curve(R(0), A1)
Affine Line over Rational Field
sage: Curve(A1)
Affine Line over Rational Field
"""
if A is None:
if is_AmbientSpace(F) and F.dimension() == 1:
return Curve(F.coordinate_ring().zero(), F)
if is_AlgebraicScheme(F):
return Curve(F.defining_polynomials(), F.ambient_space())
if isinstance(F, (list, tuple)):
P = Sequence(F).universe()
if not is_MPolynomialRing(P):
raise TypeError("universe of F must be a multivariate polynomial ring")
for f in F:
if not f.is_homogeneous():
A = AffineSpace(P.ngens(), P.base_ring(), names=P.variable_names())
A._coordinate_ring = P
break
else:
A = ProjectiveSpace(P.ngens()-1, P.base_ring(), names=P.variable_names())
A._coordinate_ring = P
elif is_MPolynomial(F): # define a plane curve
P = F.parent()
k = F.base_ring()
if not k.is_field():
if k.is_integral_domain(): # upgrade to a field
P = P.change_ring(k.fraction_field())
F = P(F)
k = F.base_ring()
else:
raise TypeError("not a multivariate polynomial over a field or an integral domain")
if F.parent().ngens() == 2:
if F == 0:
raise ValueError("defining polynomial of curve must be nonzero")
A = AffineSpace(2, P.base_ring(), names=P.variable_names())
A._coordinate_ring = P
elif F.parent().ngens() == 3:
if F == 0:
raise ValueError("defining polynomial of curve must be nonzero")
# special case: construct a conic curve
if F.total_degree() == 2 and k.is_field():
return Conic(k, F)
A = ProjectiveSpace(2, P.base_ring(), names=P.variable_names())
A._coordinate_ring = P
elif F.parent().ngens() == 1:
if not F.is_zero():
raise ValueError("defining polynomial of curve must be zero "
"if the ambient space is of dimension 1")
A = AffineSpace(1, P.base_ring(), names=P.variable_names())
A._coordinate_ring = P
else:
raise TypeError("number of variables of F (={}) must be 2 or 3".format(F))
F = [F]
else:
raise TypeError("F (={}) must be a multivariate polynomial".format(F))
else:
if not is_AmbientSpace(A):
raise TypeError("ambient space must be either an affine or projective space")
if not isinstance(F, (list, tuple)):
F = [F]
if not all(f.parent() == A.coordinate_ring() for f in F):
raise TypeError("need a list of polynomials of the coordinate ring of {}".format(A))
n = A.dimension_relative()
if n < 1:
raise TypeError("ambient space should be an affine or projective space of positive dimension")
k = A.base_ring()
if is_AffineSpace(A):
if n != 2:
if is_FiniteField(k):
if A.coordinate_ring().ideal(F).is_prime():
return IntegralAffineCurve_finite_field(A, F)
if k in Fields():
if k == QQ and A.coordinate_ring().ideal(F).is_prime():
return IntegralAffineCurve(A, F)
return AffineCurve_field(A, F)
return AffineCurve(A, F)
if not (len(F) == 1 and F[0] != 0 and F[0].degree() > 0):
raise TypeError("need a single nonconstant polynomial to define a plane curve")
F = F[0]
if is_FiniteField(k):
if _is_irreducible_and_reduced(F):
return IntegralAffinePlaneCurve_finite_field(A, F)
return AffinePlaneCurve_finite_field(A, F)
if k in Fields():
if k == QQ and _is_irreducible_and_reduced(F):
return IntegralAffinePlaneCurve(A, F)
return AffinePlaneCurve_field(A, F)
return AffinePlaneCurve(A, F)
elif is_ProjectiveSpace(A):
if n != 2:
if not all(f.is_homogeneous() for f in F):
raise TypeError("polynomials defining a curve in a projective space must be homogeneous")
if is_FiniteField(k):
if A.coordinate_ring().ideal(F).is_prime():
return IntegralProjectiveCurve_finite_field(A, F)
if k in Fields():
if k == QQ and A.coordinate_ring().ideal(F).is_prime():
return IntegralProjectiveCurve(A, F)
return ProjectiveCurve_field(A, F)
return ProjectiveCurve(A, F)
# There is no dimension check when initializing a plane curve, so check
# here that F consists of a single nonconstant polynomial.
if not (len(F) == 1 and F[0] != 0 and F[0].degree() > 0):
raise TypeError("need a single nonconstant polynomial to define a plane curve")
F = F[0]
if not F.is_homogeneous():
raise TypeError("{} is not a homogeneous polynomial".format(F))
if is_FiniteField(k):
if _is_irreducible_and_reduced(F):
return IntegralProjectivePlaneCurve_finite_field(A, F)
return ProjectivePlaneCurve_finite_field(A, F)
if k in Fields():
if k == QQ and _is_irreducible_and_reduced(F):
return IntegralProjectivePlaneCurve(A, F)
return ProjectivePlaneCurve_field(A, F)
return ProjectivePlaneCurve(A, F)
else:
raise TypeError('ambient space neither affine nor projective')
def Curve(F, A=None):
"""
Return the plane or space curve defined by ``F``, where
``F`` can be either a multivariate polynomial, a list or
tuple of polynomials, or an algebraic scheme.
If no ambient space is passed in for ``A``, and if ``F`` is not
an algebraic scheme, a new ambient space is constructed.
Also not specifying an ambient space will cause the curve to be defined
in either affine or projective space based on properties of ``F``. In
particular, if ``F`` contains a nonhomogenous polynomial, the curve is
affine, and if ``F`` consists of homogenous polynomials, then the curve
is projective.
INPUT:
- ``F`` -- a multivariate polynomial, or a list or tuple of polynomials, or an algebraic scheme.
- ``A`` -- (default: None) an ambient space in which to create the curve.
EXAMPLE: A projective plane curve
::
sage: x,y,z = QQ['x,y,z'].gens()
sage: C = Curve(x^3 + y^3 + z^3); C
Projective Plane Curve over Rational Field defined by x^3 + y^3 + z^3
sage: C.genus()
1
EXAMPLE: Affine plane curves
::
sage: x,y = GF(7)['x,y'].gens()
sage: C = Curve(y^2 + x^3 + x^10); C
Affine Plane Curve over Finite Field of size 7 defined by x^10 + x^3 + y^2
sage: C.genus()
0
sage: x, y = QQ['x,y'].gens()
sage: Curve(x^3 + y^3 + 1)
Affine Plane Curve over Rational Field defined by x^3 + y^3 + 1
EXAMPLE: A projective space curve
::
sage: x,y,z,w = QQ['x,y,z,w'].gens()
sage: C = Curve([x^3 + y^3 - z^3 - w^3, x^5 - y*z^4]); C
Projective Curve over Rational Field defined by x^3 + y^3 - z^3 - w^3, x^5 - y*z^4
sage: C.genus()
13
EXAMPLE: An affine space curve
::
sage: x,y,z = QQ['x,y,z'].gens()
sage: C = Curve([y^2 + x^3 + x^10 + z^7, x^2 + y^2]); C
Affine Curve over Rational Field defined by x^10 + z^7 + x^3 + y^2, x^2 + y^2
sage: C.genus()
47
EXAMPLE: We can also make non-reduced non-irreducible curves.
::
sage: x,y,z = QQ['x,y,z'].gens()
sage: Curve((x-y)*(x+y))
Projective Conic Curve over Rational Field defined by x^2 - y^2
sage: Curve((x-y)^2*(x+y)^2)
Projective Plane Curve over Rational Field defined by x^4 - 2*x^2*y^2 + y^4
EXAMPLE: A union of curves is a curve.
::
sage: x,y,z = QQ['x,y,z'].gens()
sage: C = Curve(x^3 + y^3 + z^3)
sage: D = Curve(x^4 + y^4 + z^4)
sage: C.union(D)
Projective Plane Curve over Rational Field defined by
x^7 + x^4*y^3 + x^3*y^4 + y^7 + x^4*z^3 + y^4*z^3 + x^3*z^4 + y^3*z^4 + z^7
The intersection is not a curve, though it is a scheme.
::
sage: X = C.intersection(D); X
Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
x^3 + y^3 + z^3,
x^4 + y^4 + z^4
Note that the intersection has dimension `0`.
::
sage: X.dimension()
0
sage: I = X.defining_ideal(); I
Ideal (x^3 + y^3 + z^3, x^4 + y^4 + z^4) of Multivariate Polynomial Ring in x, y, z over Rational Field
EXAMPLE: In three variables, the defining equation must be
homogeneous.
If the parent polynomial ring is in three variables, then the
defining ideal must be homogeneous.
::
sage: x,y,z = QQ['x,y,z'].gens()
sage: Curve(x^2+y^2)
Projective Conic Curve over Rational Field defined by x^2 + y^2
sage: Curve(x^2+y^2+z)
Traceback (most recent call last):
...
TypeError: x^2 + y^2 + z is not a homogeneous polynomial
The defining polynomial must always be nonzero::
sage: P1.<x,y> = ProjectiveSpace(1,GF(5))
sage: Curve(0*x)
Traceback (most recent call last):
...
ValueError: defining polynomial of curve must be nonzero
::
sage: A.<x,y,z> = AffineSpace(QQ, 3)
sage: C = Curve([y - x^2, z - x^3], A)
sage: A == C.ambient_space()
True
"""
if not A is None:
if not isinstance(F, (list, tuple)):
return Curve([F], A)
if not is_AmbientSpace(A):
raise TypeError(
"A (=%s) must be either an affine or projective space" % A)
if not all([f.parent() == A.coordinate_ring() for f in F]):
raise TypeError("F (=%s) must be a list or tuple of polynomials of the coordinate ring of " \
"A (=%s)"%(F, A))
n = A.dimension_relative()
if n < 2:
raise TypeError(
"A (=%s) must be either an affine or projective space of dimension > 1"
% A)
# there is no dimension check when initializing a plane curve, so check here that F consists
# of a single nonconstant polynomial
if n == 2:
if len(F) != 1 or F[0] == 0 or not is_MPolynomial(F[0]):
raise TypeError(
"F (=%s) must consist of a single nonconstant polynomial to define a plane curve"
% (F, ))
if is_AffineSpace(A):
if n > 2:
return AffineCurve(A, F)
k = A.base_ring()
if is_FiniteField(k):
if k.is_prime_field():
return AffinePlaneCurve_prime_finite_field(A, F[0])
return AffinePlaneCurve_finite_field(A, F[0])
return AffinePlaneCurve(A, F[0])
elif is_ProjectiveSpace(A):
if not all([f.is_homogeneous() for f in F]):
raise TypeError(
"polynomials defining a curve in a projective space must be homogeneous"
)
if n > 2:
return ProjectiveCurve(A, F)
k = A.base_ring()
if is_FiniteField(k):
if k.is_prime_field():
return ProjectivePlaneCurve_prime_finite_field(A, F[0])
return ProjectivePlaneCurve_finite_field(A, F[0])
return ProjectivePlaneCurve(A, F[0])
if is_AlgebraicScheme(F):
return Curve(F.defining_polynomials(), F.ambient_space())
if isinstance(F, (list, tuple)):
if len(F) == 1:
return Curve(F[0])
F = Sequence(F)
P = F.universe()
if not is_MPolynomialRing(P):
raise TypeError(
"universe of F must be a multivariate polynomial ring")
for f in F:
if not f.is_homogeneous():
A = AffineSpace(P.ngens(), P.base_ring())
A._coordinate_ring = P
return AffineCurve(A, F)
A = ProjectiveSpace(P.ngens() - 1, P.base_ring())
A._coordinate_ring = P
return ProjectiveCurve(A, F)
if not is_MPolynomial(F):
raise TypeError("F (=%s) must be a multivariate polynomial" % F)
P = F.parent()
k = F.base_ring()
if F.parent().ngens() == 2:
if F == 0:
raise ValueError("defining polynomial of curve must be nonzero")
A2 = AffineSpace(2, P.base_ring())
A2._coordinate_ring = P
if is_FiniteField(k):
if k.is_prime_field():
return AffinePlaneCurve_prime_finite_field(A2, F)
else:
return AffinePlaneCurve_finite_field(A2, F)
else:
return AffinePlaneCurve(A2, F)
elif F.parent().ngens() == 3:
if F == 0:
raise ValueError("defining polynomial of curve must be nonzero")
P2 = ProjectiveSpace(2, P.base_ring())
P2._coordinate_ring = P
if F.total_degree() == 2 and k.is_field():
return Conic(F)
if is_FiniteField(k):
if k.is_prime_field():
return ProjectivePlaneCurve_prime_finite_field(P2, F)
else:
return ProjectivePlaneCurve_finite_field(P2, F)
else:
return ProjectivePlaneCurve(P2, F)
else:
raise TypeError("Number of variables of F (=%s) must be 2 or 3" % F)
def Curve(F):
"""
Return the plane or space curve defined by `F`, where
`F` can be either a multivariate polynomial, a list or
tuple of polynomials, or an algebraic scheme.
If `F` is in two variables the curve is affine, and if it
is homogenous in `3` variables, then the curve is
projective.
EXAMPLE: A projective plane curve
::
sage: x,y,z = QQ['x,y,z'].gens()
sage: C = Curve(x^3 + y^3 + z^3); C
Projective Curve over Rational Field defined by x^3 + y^3 + z^3
sage: C.genus()
1
EXAMPLE: Affine plane curves
::
sage: x,y = GF(7)['x,y'].gens()
sage: C = Curve(y^2 + x^3 + x^10); C
Affine Curve over Finite Field of size 7 defined by x^10 + x^3 + y^2
sage: C.genus()
0
sage: x, y = QQ['x,y'].gens()
sage: Curve(x^3 + y^3 + 1)
Affine Curve over Rational Field defined by x^3 + y^3 + 1
EXAMPLE: A projective space curve
::
sage: x,y,z,w = QQ['x,y,z,w'].gens()
sage: C = Curve([x^3 + y^3 - z^3 - w^3, x^5 - y*z^4]); C
Projective Space Curve over Rational Field defined by x^3 + y^3 - z^3 - w^3, x^5 - y*z^4
sage: C.genus()
13
EXAMPLE: An affine space curve
::
sage: x,y,z = QQ['x,y,z'].gens()
sage: C = Curve([y^2 + x^3 + x^10 + z^7, x^2 + y^2]); C
Affine Space Curve over Rational Field defined by x^10 + z^7 + x^3 + y^2, x^2 + y^2
sage: C.genus()
47
EXAMPLE: We can also make non-reduced non-irreducible curves.
::
sage: x,y,z = QQ['x,y,z'].gens()
sage: Curve((x-y)*(x+y))
Projective Conic Curve over Rational Field defined by x^2 - y^2
sage: Curve((x-y)^2*(x+y)^2)
Projective Curve over Rational Field defined by x^4 - 2*x^2*y^2 + y^4
EXAMPLE: A union of curves is a curve.
::
sage: x,y,z = QQ['x,y,z'].gens()
sage: C = Curve(x^3 + y^3 + z^3)
sage: D = Curve(x^4 + y^4 + z^4)
sage: C.union(D)
Projective Curve over Rational Field defined by
x^7 + x^4*y^3 + x^3*y^4 + y^7 + x^4*z^3 + y^4*z^3 + x^3*z^4 + y^3*z^4 + z^7
The intersection is not a curve, though it is a scheme.
::
sage: X = C.intersection(D); X
Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
x^3 + y^3 + z^3,
x^4 + y^4 + z^4
Note that the intersection has dimension `0`.
::
sage: X.dimension()
0
sage: I = X.defining_ideal(); I
Ideal (x^3 + y^3 + z^3, x^4 + y^4 + z^4) of Multivariate Polynomial Ring in x, y, z over Rational Field
EXAMPLE: In three variables, the defining equation must be
homogeneous.
If the parent polynomial ring is in three variables, then the
defining ideal must be homogeneous.
::
sage: x,y,z = QQ['x,y,z'].gens()
sage: Curve(x^2+y^2)
Projective Conic Curve over Rational Field defined by x^2 + y^2
sage: Curve(x^2+y^2+z)
Traceback (most recent call last):
...
TypeError: x^2 + y^2 + z is not a homogeneous polynomial!
The defining polynomial must always be nonzero::
sage: P1.<x,y> = ProjectiveSpace(1,GF(5))
sage: Curve(0*x)
Traceback (most recent call last):
...
ValueError: defining polynomial of curve must be nonzero
"""
if is_AlgebraicScheme(F):
return Curve(F.defining_polynomials())
if isinstance(F, (list, tuple)):
if len(F) == 1:
return Curve(F[0])
F = Sequence(F)
P = F.universe()
if not is_MPolynomialRing(P):
raise TypeError, "universe of F must be a multivariate polynomial ring"
for f in F:
if not f.is_homogeneous():
A = AffineSpace(P.ngens(), P.base_ring())
A._coordinate_ring = P
return AffineSpaceCurve_generic(A, F)
A = ProjectiveSpace(P.ngens() - 1, P.base_ring())
A._coordinate_ring = P
return ProjectiveSpaceCurve_generic(A, F)
if not is_MPolynomial(F):
raise TypeError, "F (=%s) must be a multivariate polynomial" % F
P = F.parent()
k = F.base_ring()
if F.parent().ngens() == 2:
if F == 0:
raise ValueError, "defining polynomial of curve must be nonzero"
A2 = AffineSpace(2, P.base_ring())
A2._coordinate_ring = P
if is_FiniteField(k):
if k.is_prime_field():
return AffineCurve_prime_finite_field(A2, F)
else:
return AffineCurve_finite_field(A2, F)
else:
return AffineCurve_generic(A2, F)
elif F.parent().ngens() == 3:
if F == 0:
raise ValueError, "defining polynomial of curve must be nonzero"
P2 = ProjectiveSpace(2, P.base_ring())
P2._coordinate_ring = P
if F.total_degree() == 2 and k.is_field():
return Conic(F)
if is_FiniteField(k):
if k.is_prime_field():
return ProjectiveCurve_prime_finite_field(P2, F)
else:
return ProjectiveCurve_finite_field(P2, F)
else:
return ProjectiveCurve_generic(P2, F)
else:
raise TypeError, "Number of variables of F (=%s) must be 2 or 3" % F
def eisenstein_series(self):
"""
Return the Eisenstein series that span this space (over the
algebraic closure).
EXAMPLES::
sage: EisensteinForms(11,2).eisenstein_series()
[
5/12 + q + 3*q^2 + 4*q^3 + 7*q^4 + 6*q^5 + O(q^6)
]
sage: EisensteinForms(1,4).eisenstein_series()
[
1/240 + q + 9*q^2 + 28*q^3 + 73*q^4 + 126*q^5 + O(q^6)
]
sage: EisensteinForms(1,24).eisenstein_series()
[
236364091/131040 + q + 8388609*q^2 + 94143178828*q^3 + 70368752566273*q^4 + 11920928955078126*q^5 + O(q^6)
]
sage: EisensteinForms(5,4).eisenstein_series()
[
1/240 + q + 9*q^2 + 28*q^3 + 73*q^4 + 126*q^5 + O(q^6),
1/240 + q^5 + O(q^6)
]
sage: EisensteinForms(13,2).eisenstein_series()
[
1/2 + q + 3*q^2 + 4*q^3 + 7*q^4 + 6*q^5 + O(q^6)
]
sage: E = EisensteinForms(Gamma1(7),2)
sage: E.set_precision(4)
sage: E.eisenstein_series()
[
1/4 + q + 3*q^2 + 4*q^3 + O(q^4),
1/7*zeta6 - 3/7 + q + (-2*zeta6 + 1)*q^2 + (3*zeta6 - 2)*q^3 + O(q^4),
q + (-zeta6 + 2)*q^2 + (zeta6 + 2)*q^3 + O(q^4),
-1/7*zeta6 - 2/7 + q + (2*zeta6 - 1)*q^2 + (-3*zeta6 + 1)*q^3 + O(q^4),
q + (zeta6 + 1)*q^2 + (-zeta6 + 3)*q^3 + O(q^4)
]
sage: eps = DirichletGroup(13).0^2
sage: ModularForms(eps,2).eisenstein_series()
[
-7/13*zeta6 - 11/13 + q + (2*zeta6 + 1)*q^2 + (-3*zeta6 + 1)*q^3 + (6*zeta6 - 3)*q^4 - 4*q^5 + O(q^6),
q + (zeta6 + 2)*q^2 + (-zeta6 + 3)*q^3 + (3*zeta6 + 3)*q^4 + 4*q^5 + O(q^6)
]
sage: M = ModularForms(19,3).eisenstein_subspace()
sage: M.eisenstein_series()
[
]
sage: M = ModularForms(DirichletGroup(13).0, 1)
sage: M.eisenstein_series()
[
-1/13*zeta12^3 + 6/13*zeta12^2 + 4/13*zeta12 + 2/13 + q + (zeta12 + 1)*q^2 + zeta12^2*q^3 + (zeta12^2 + zeta12 + 1)*q^4 + (-zeta12^3 + 1)*q^5 + O(q^6)
]
sage: M = ModularForms(GammaH(15, [4]), 4)
sage: M.eisenstein_series()
[
1/240 + q + 9*q^2 + 28*q^3 + 73*q^4 + 126*q^5 + O(q^6),
1/240 + q^3 + O(q^6),
1/240 + q^5 + O(q^6),
1/240 + O(q^6),
1 + q - 7*q^2 - 26*q^3 + 57*q^4 + q^5 + O(q^6),
1 + q^3 + O(q^6),
q + 7*q^2 + 26*q^3 + 57*q^4 + 125*q^5 + O(q^6),
q^3 + O(q^6)
]
"""
P = self.parameters()
E = Sequence([
element.EisensteinSeries(self.change_ring(chi.base_ring()), None,
t, chi, psi) for chi, psi, t in P
],
immutable=True,
cr=True,
universe=Objects())
assert len(
E) == self.dimension(), "bug in enumeration of Eisenstein series."
return E
def lfsr_sequence(key, fill, n):
r"""
This function creates an lfsr sequence.
INPUT:
- ``key`` - a list of finite field elements,
[c_0,c_1,...,c_k].
- ``fill`` - the list of the initial terms of the lfsr
sequence, [x_0,x_1,...,x_k].
- ``n`` - number of terms of the sequence that the
function returns.
OUTPUT: The lfsr sequence defined by
`x_{n+1} = c_kx_n+...+c_0x_{n-k}`, for
`n \leq k`.
EXAMPLES::
sage: F = GF(2); l = F(1); o = F(0)
sage: F = GF(2); S = LaurentSeriesRing(F,'x'); x = S.gen()
sage: fill = [l,l,o,l]; key = [1,o,o,l]; n = 20
sage: L = lfsr_sequence(key,fill,20); L
[1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0]
sage: g = berlekamp_massey(L); g
x^4 + x^3 + 1
sage: (1)/(g.reverse()+O(x^20))
1 + x + x^2 + x^3 + x^5 + x^7 + x^8 + x^11 + x^15 + x^16 + x^17 + x^18 + O(x^20)
sage: (1+x^2)/(g.reverse()+O(x^20))
1 + x + x^4 + x^8 + x^9 + x^10 + x^11 + x^13 + x^15 + x^16 + x^19 + O(x^20)
sage: (1+x^2+x^3)/(g.reverse()+O(x^20))
1 + x + x^3 + x^5 + x^6 + x^9 + x^13 + x^14 + x^15 + x^16 + x^18 + O(x^20)
sage: fill = [l,l,o,l]; key = [l,o,o,o]; n = 20
sage: L = lfsr_sequence(key,fill,20); L
[1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1]
sage: g = berlekamp_massey(L); g
x^4 + 1
sage: (1+x)/(g.reverse()+O(x^20))
1 + x + x^4 + x^5 + x^8 + x^9 + x^12 + x^13 + x^16 + x^17 + O(x^20)
sage: (1+x+x^3)/(g.reverse()+O(x^20))
1 + x + x^3 + x^4 + x^5 + x^7 + x^8 + x^9 + x^11 + x^12 + x^13 + x^15 + x^16 + x^17 + x^19 + O(x^20)
AUTHORS:
- Timothy Brock (2005-11): with code modified from Python
Cookbook, http://aspn.activestate.com/ASPN/Python/Cookbook/
"""
if not isinstance(key, list):
raise TypeError("key must be a list")
key = Sequence(key)
F = key.universe()
if not is_FiniteField(F):
raise TypeError("universe of sequence must be a finite field")
s = fill
k = len(fill)
L = []
for i in range(n):
s0 = copy.copy(s)
L.append(s[0])
s = s[1:k]
s.append(sum([key[i] * s0[i] for i in range(k)]))
return L
