def getLfunctionPlot(request, arg1, arg2, arg3, arg4, arg5, arg6, arg7, arg8, arg9):
pythonL = generateLfunctionFromUrl(
arg1, arg2, arg3, arg4, arg5, arg6, arg7, arg8, arg9, to_dict(request.args))
if not pythonL:
return ""
plotrange = 30
if hasattr(pythonL, 'plotpoints'):
F = p2sage(pythonL.plotpoints)
plotrange = min(plotrange, F[-1][0]) # F[-1][0] is the highest t-coordinated that we have a value for L
else:
# obsolete, because lfunc_data comes from DB?
L = pythonL.sageLfunction
if not hasattr(L, "hardy_z_function"):
return None
plotStep = .1
if pythonL._Ltype not in ["riemann", "maass", "ellipticmodularform", "ellipticcurve"]:
plotrange = 12
F = [(i, L.hardy_z_function(i).real()) for i in srange(-1*plotrange, plotrange, plotStep)]
interpolation = spline(F)
F_interp = [(i, interpolation(i)) for i in srange(-1*plotrange, plotrange, 0.05)]
p = line(F_interp)
# p = line(F) # temporary hack while the correct interpolation is being implemented
styleLfunctionPlot(p, 10)
fn = tempfile.mktemp(suffix=".png")
p.save(filename=fn)
data = file(fn).read()
os.remove(fn)
return data
def fourier_series_partial_sum(cls, self, parameters, variable, N, L):
r"""
Returns the partial sum
.. math::
f(x) \sim \frac{a_0}{2} + \sum_{n=1}^N [a_n\cos(\frac{n\pi x}{L}) + b_n\sin(\frac{n\pi x}{L})],
as a string.
EXAMPLE::
sage: f(x) = x^2
sage: f = piecewise([[(-1,1),f]])
sage: f.fourier_series_partial_sum(3,1)
cos(2*pi*x)/pi^2 - 4*cos(pi*x)/pi^2 + 1/3
sage: f1(x) = -1
sage: f2(x) = 2
sage: f = piecewise([[(-pi,pi/2),f1],[(pi/2,pi),f2]])
sage: f.fourier_series_partial_sum(3,pi)
-3*cos(x)/pi - 3*sin(2*x)/pi + 3*sin(x)/pi - 1/4
"""
from sage.all import pi, sin, cos, srange
x = self.default_variable()
a0 = self.fourier_series_cosine_coefficient(0, L)
result = a0 / 2 + sum(
[(self.fourier_series_cosine_coefficient(n, L) *
cos(n * pi * x / L) + self.fourier_series_sine_coefficient(
n, L) * sin(n * pi * x / L)) for n in srange(1, N)])
return SR(result).expand()
def fourier_series_partial_sum(cls, self, parameters, variable, N, L):
r"""
Returns the partial sum
.. math::
f(x) \sim \frac{a_0}{2} + \sum_{n=1}^N [a_n\cos(\frac{n\pi x}{L}) + b_n\sin(\frac{n\pi x}{L})],
as a string.
EXAMPLE::
sage: f(x) = x^2
sage: f = piecewise([[(-1,1),f]])
sage: f.fourier_series_partial_sum(3,1)
cos(2*pi*x)/pi^2 - 4*cos(pi*x)/pi^2 + 1/3
sage: f1(x) = -1
sage: f2(x) = 2
sage: f = piecewise([[(-pi,pi/2),f1],[(pi/2,pi),f2]])
sage: f.fourier_series_partial_sum(3,pi)
-3*cos(x)/pi - 3*sin(2*x)/pi + 3*sin(x)/pi - 1/4
"""
from sage.all import pi, sin, cos, srange
x = self.default_variable()
a0 = self.fourier_series_cosine_coefficient(0,L)
result = a0/2 + sum([(self.fourier_series_cosine_coefficient(n,L)*cos(n*pi*x/L) +
self.fourier_series_sine_coefficient(n,L)*sin(n*pi*x/L))
for n in srange(1,N)])
return SR(result).expand()
def make_hash(p, N1, N2, np=20):
plist = [next_prime(400000)]
while len(plist) < np:
plist.append(next_prime(plist[-1]))
hashtab = {}
nc = 0
for E in cremona_optimal_curves(srange(N1, N2 + 1)):
nc += 1
h = hash1(p, E, plist)
lab = E.label()
if nc % 1000 == 0:
print(lab)
#print("{} has hash = {}".format(lab,h))
if h in hashtab:
hashtab[h].append(lab)
#print("new set {}".format(hashtab[h]))
else:
hashtab[h] = [lab]
ns = 0
for s in hashtab.values():
if len(s) > 1:
ns += 1
#print s
print("{} sets of conjugate curves".format(ns))
return hashtab
def make_hash_many(plist, N1, N2, nq=20):
qlist = [next_prime(400000)]
while len(qlist) < nq:
qlist.append(next_prime(qlist[-1]))
hashtabs = dict([(p, dict()) for p in plist])
nc = 0
for E in cremona_optimal_curves(srange(N1, N2 + 1)):
nc += 1
lab = E.label()
if nc % 1000 == 0:
print(lab)
h = hash1many(plist, E, qlist)
for p in plist:
hp = h[p]
if hp in hashtabs[p]:
hashtabs[p][hp].append(lab)
print("p={}, new set {}".format(p, hashtabs[p][hp]))
else:
hashtabs[p][hp] = [lab]
ns = dict([(p, len([v for v in hashtabs[p].values() if len(v) > 1]))
for p in plist])
for p in plist:
print("p={}: {} nontrivial sets of congruent curves".format(p, ns[p]))
return hashtabs
def find_cong_db(Nmin, Nmax, p, ofile=None, verbose=False):
if ofile:
ofile = open(ofile, mode='a')
nc = 0
for N in srange(Nmin, Nmax + 1):
if N % 100 == 0:
print(N)
curves_N = get_curves(N)
if curves_N:
if verbose:
print("N1 = {} ({} classes)".format(N, len(curves_N)))
for M in srange(11, N + 1):
#for M in srange(11,42000):
curves_M = get_curves(M)
if curves_M and verbose:
pass
print("...N2 = {} ({} classes)".format(M, len(curves_M)))
for c1 in curves_N:
ap1 = c1['aplist']
iso1 = c1['iso_nlabel']
lab1 = str(c1['label'])
curves2 = [
c2 for c2 in curves_M
if M < N or c2['iso_nlabel'] < iso1
]
curves2 = [
c2 for c2 in curves2
if test2(N, ap1, M, c2['aplist'], p)
]
for c2 in curves2:
lab2 = str(c2['label'])
print("Candidate for congruence mod {}: {} {}".format(
p, lab1, lab2))
E1 = EllipticCurve(lab1)
E2 = EllipticCurve(lab2)
res, reason = test_cong(p, E1, E2)
report(res, reason, p, lab1, lab2)
if res:
nc += 1
if ofile:
ofile.write("{} {} {}\n".format(p, lab1, lab2))
if ofile:
ofile.close()
print("{} pairs of congruent curves found".format(nc))
def min_formula(N,t):
"""
Function MinFormula in Mark his code
"""
N = ZZ(N); t = QQ(t)
if N < 2:
raise ValueError
if N == 2:
return 4 * t -1
if N == 3:
return 9 * min(t, ZZ(1)/3) - 8*t
return sum(N * Phi(gcd(i,N)) * (min(t, i/N) - 4*(i/N)*(1-i/N)*t) for i in srange(1,(N-1)//2+1))
def prod_plot_values(factor_plot_deltas, factor_values):
assert len(factor_plot_deltas) == len(factor_values)
halfdegree = len(factor_values)
if halfdegree == 1:
return factor_plot_deltas[0], factor_values[0]
factor_plot_values = [ [ ( j * factor_plot_deltas[k], z) for j, z in enumerate(values) ] for k, values in enumerate(factor_values)]
interpolations = [spline(elt) for elt in factor_plot_values]
max_delta = max(factor_plot_deltas)
new_delta = max_delta/halfdegree
plot_range = min( [elt[-1][0] for elt in factor_plot_values] )
values = [prod([elt(i) for elt in interpolations]) for i in srange(0, plot_range, new_delta)]
return new_delta, values
def genTemplate(name,d):
if name == 'coef':
assert(len(d.values())==1)
coefVal = d.values()[0]
template = None if coefVal == 0 else '(%s)'%str(coefVal)
else:
idxVals= ['[%s]'%d[name+str(idx)] for idx in srange(len(d)-1)]
if d['coef'] == -1:
coefStr = '-'
elif d['coef'] == 1:
coefStr = ''
else:
coefStr = "({}) *".format(d['coef'])
template = '(%s%s%s)'%(coefStr,name,list_str(idxVals,''))
return template
def download_Rub_data():
import gridfs
label = (request.args.get('label'))
limit = (request.args.get('limit'))
C = base.getDBConnection()
fs = gridfs.GridFS(C.quadratic_twists, 'isogeny')
isogeny = C.quadratic_twists.isogeny.files
filename = isogeny.find_one({'label': label})['filename']
d = fs.get_last_version(filename)
if limit is None:
response = flask.Response(d.__iter__())
response.headers['Content-disposition'] = 'attachment; filename=%s' % label
response.content_length = d.length
else:
limit = int(limit)
response = make_response(''.join(str(d.readline()) for i in srange(limit)))
response.headers['Content-type'] = 'text/plain'
return response
def download_Rub_data():
import gridfs
label = (request.args.get('label'))
limit = (request.args.get('limit'))
C = base.getDBConnection()
fs = gridfs.GridFS(C.quadratic_twists, 'isogeny')
isogeny = C.quadratic_twists.isogeny.files
filename = isogeny.find_one({'label': label})['filename']
d = fs.get_last_version(filename)
if limit is None:
response = flask.Response(d.__iter__())
response.headers['Content-disposition'] = 'attachment; filename=%s' % label
response.content_length = d.length
else:
limit = int(limit)
response = make_response(''.join(str(d.readline()) for i in srange(limit)))
response.headers['Content-type'] = 'text/plain'
return response
def compute_traces(aname, acontents, ainfo, tsinfo):
vi = Miscs.travel(acontents)
vals = Miscs.getVals(vi)
idxs = Miscs.getIdxs(vi)
aname = str(aname)
newvars = [var(aname + '_' + list_str(idx, '_')) for idx in idxs]
if aname not in tsinfo:
tsinfo[aname] = newvars
else:
assert tsinfo[aname] == newvars
dVals = dict(zip(newvars,vals)) #{A_0_0_1:'w'}
for nv,idx in zip(newvars,idxs):
if nv not in ainfo:
idx_ = zip([var('{}{}'.format(aname,li))
for li in srange(len(idx))],idx)
ainfo[nv]={'name':aname, 'idx_':idx_}
return dVals
def rqf_iterator(d1, d2):
from lmfdb.WebNumberField import is_fundamental_discriminant
for d in srange(d1, d2 + 1):
if is_fundamental_discriminant(d):
yield d, '2.2.%s.1' % d
def fourier_series_partial_sum(self, parameters, variable, N,
L=None):
r"""
Returns the partial sum up to a given order of the Fourier series
of the periodic function `f` extending the piecewise-defined
function ``self``.
The Fourier partial sum of order `N` is defined as
.. MATH::
S_{N}(x) = \frac{a_0}{2} + \sum_{n=1}^{N} \left[
a_n\cos\left(\frac{n\pi x}{L}\right)
+ b_n\sin\left(\frac{n\pi x}{L}\right)\right],
where `L` is the half-period of `f` and the `a_n`'s and `b_n`'s
are respectively the cosine coefficients and sine coefficients
of the Fourier series of `f` (cf.
:meth:`fourier_series_cosine_coefficient` and
:meth:`fourier_series_sine_coefficient`).
INPUT:
- ``N`` -- a positive integer; the order of the partial sum
- ``L`` -- (default: ``None``) the half-period of `f`; if none
is provided, `L` is assumed to be the half-width of the domain
of ``self``
OUTPUT:
- the partial sum `S_{N}(x)`, as a symbolic expression
EXAMPLES:
A square wave function of period 2::
sage: f = piecewise([((-1,0), -1), ((0,1), 1)])
sage: f.fourier_series_partial_sum(5)
4/5*sin(5*pi*x)/pi + 4/3*sin(3*pi*x)/pi + 4*sin(pi*x)/pi
If the domain of the piecewise-defined function encompasses
more than one period, the half-period must be passed as the
second argument; for instance::
sage: f2 = piecewise([((-1,0), -1), ((0,1), 1),
....: ((1,2), -1), ((2,3), 1)])
sage: bool(f2.restriction((-1,1)) == f) # f2 extends f on (-1,3)
True
sage: f2.fourier_series_partial_sum(5, 1) # half-period = 1
4/5*sin(5*pi*x)/pi + 4/3*sin(3*pi*x)/pi + 4*sin(pi*x)/pi
sage: bool(f2.fourier_series_partial_sum(5, 1) ==
....: f.fourier_series_partial_sum(5))
True
The default half-period is 2, so that skipping the second
argument yields a different result::
sage: f2.fourier_series_partial_sum(5) # half-period = 2
4*sin(pi*x)/pi
An example of partial sum involving both cosine and sine terms::
sage: f = piecewise([((-1,0), 0), ((0,1/2), 2*x),
....: ((1/2,1), 2*(1-x))])
sage: f.fourier_series_partial_sum(5)
-2*cos(2*pi*x)/pi^2 + 4/25*sin(5*pi*x)/pi^2
- 4/9*sin(3*pi*x)/pi^2 + 4*sin(pi*x)/pi^2 + 1/4
"""
from sage.all import pi, sin, cos, srange
if not L:
L = (self.domain().sup() - self.domain().inf()) / 2
x = self.default_variable()
a0 = self.fourier_series_cosine_coefficient(0, L)
result = a0/2 + sum([(self.fourier_series_cosine_coefficient(n, L)*cos(n*pi*x/L) +
self.fourier_series_sine_coefficient(n, L)*sin(n*pi*x/L))
for n in srange(1, N+1)])
return SR(result).expand()
def divisor_F_bc(N,k):
"""
Function Divisor_F_bc in Mark his code, return the divisor of F_k as function on X_1(N)
"""
N = ZZ(N); k = ZZ(k)
return vector([min_formula(k, i/N) * inverse_gcd(i,N) for i in srange(0,N//2+1)])
def fourier_series_partial_sum(self, parameters, variable, N, L=None):
r"""
Returns the partial sum up to a given order of the Fourier series
of the periodic function `f` extending the piecewise-defined
function ``self``.
The Fourier partial sum of order `N` is defined as
.. MATH::
S_{N}(x) = \frac{a_0}{2} + \sum_{n=1}^{N} \left[
a_n\cos\left(\frac{n\pi x}{L}\right)
+ b_n\sin\left(\frac{n\pi x}{L}\right)\right],
where `L` is the half-period of `f` and the `a_n`'s and `b_n`'s
are respectively the cosine coefficients and sine coefficients
of the Fourier series of `f` (cf.
:meth:`fourier_series_cosine_coefficient` and
:meth:`fourier_series_sine_coefficient`).
INPUT:
- ``N`` -- a positive integer; the order of the partial sum
- ``L`` -- (default: ``None``) the half-period of `f`; if none
is provided, `L` is assumed to be the half-width of the domain
of ``self``
OUTPUT:
- the partial sum `S_{N}(x)`, as a symbolic expression
EXAMPLES:
A square wave function of period 2::
sage: f = piecewise([((-1,0), -1), ((0,1), 1)])
sage: f.fourier_series_partial_sum(5)
4/5*sin(5*pi*x)/pi + 4/3*sin(3*pi*x)/pi + 4*sin(pi*x)/pi
If the domain of the piecewise-defined function encompasses
more than one period, the half-period must be passed as the
second argument; for instance::
sage: f2 = piecewise([((-1,0), -1), ((0,1), 1),
....: ((1,2), -1), ((2,3), 1)])
sage: bool(f2.restriction((-1,1)) == f) # f2 extends f on (-1,3)
True
sage: f2.fourier_series_partial_sum(5, 1) # half-period = 1
4/5*sin(5*pi*x)/pi + 4/3*sin(3*pi*x)/pi + 4*sin(pi*x)/pi
sage: bool(f2.fourier_series_partial_sum(5, 1) ==
....: f.fourier_series_partial_sum(5))
True
The default half-period is 2, so that skipping the second
argument yields a different result::
sage: f2.fourier_series_partial_sum(5) # half-period = 2
4*sin(pi*x)/pi
An example of partial sum involving both cosine and sine terms::
sage: f = piecewise([((-1,0), 0), ((0,1/2), 2*x),
....: ((1/2,1), 2*(1-x))])
sage: f.fourier_series_partial_sum(5)
-2*cos(2*pi*x)/pi^2 + 4/25*sin(5*pi*x)/pi^2
- 4/9*sin(3*pi*x)/pi^2 + 4*sin(pi*x)/pi^2 + 1/4
"""
from sage.all import pi, sin, cos, srange
if not L:
L = (self.domain().sup() - self.domain().inf()) / 2
x = self.default_variable()
a0 = self.fourier_series_cosine_coefficient(0, L)
result = a0 / 2 + sum(
[(self.fourier_series_cosine_coefficient(n, L) *
cos(n * pi * x / L) + self.fourier_series_sine_coefficient(
n, L) * sin(n * pi * x / L)) for n in srange(1, N + 1)])
return SR(result).expand()
def rqf_iterator(d1, d2):
from lmfdb.WebNumberField import is_fundamental_discriminant
for d in srange(d1, d2 + 1):
if is_fundamental_discriminant(d):
yield d, '2.2.%s.1' % d
def ideals_iterator(K, minnorm=1, maxnorm=Infinity):
r""" Return an iterator over all ideals of norm n up to maxnorm (sorted).
"""
for n in srange(minnorm, maxnorm + 1):
for I in ideals_of_norm(K, n):
yield I
def ideals_iterator(K,minnorm=1,maxnorm=Infinity):
r""" Return an iterator over all ideals of norm n up to maxnorm (sorted).
"""
for n in srange(minnorm,maxnorm+1):
for I in ideals_of_norm(K,n):
yield I
def LB_c(N):
N = ZZ(N);
return [divisor_F_bc(N,k) for k in srange(2,N//2+2)]
