def lfuncEPhtml(L,fmt):
""" Euler product as a formula and a table of local factors.
"""
texform_gen = "\[L(s) = " # "\[L(A,s) = "
texform_gen += "\prod_{p \\text{ prime}} F_p(p^{-s})^{-1} \]\n"
pfactors = prime_divisors(L.level)
if len(pfactors) == 1: #i.e., the conductor is prime
pgoodset = "$p \\neq " + str(pfactors[0]) + "$"
pbadset = "$p = " + str(pfactors[0]) + "$"
else:
badset = "\\{" + str(pfactors[0])
for j in range(1,len(pfactors)):
badset += ",\\;"
badset += str(pfactors[j])
badset += "\\}"
pgoodset = "$p \\notin " + badset + "$"
pbadset = "$p \\in " + badset + "$"
ans = ""
ans += texform_gen + "where, for " + pgoodset + ",\n"
if L.degree == 4 and L.motivic_weight == 1:
ans += "\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]"
ans += "with $b_p = a_p^2 - a_{p^2}$. "
elif L.degree == 2 and L.motivic_weight == 1:
ans += "\[F_p(T) = 1 - a_p T + p T^2 .\]"
else:
ans += "\(F_p\) is a polynomial of degree " + str(L.degree) + ". "
ans += "If " + pbadset + ", then $F_p$ is a polynomial of degree at most "
ans += str(L.degree - 1) + ". "
bad_primes = []
for lf in L.bad_lfactors:
bad_primes.append(lf[0])
eulerlim = 25
good_primes = []
for j in range(0, eulerlim):
this_prime = Primes().unrank(j)
if this_prime not in bad_primes:
good_primes.append(this_prime)
eptable = "<table id='eptable' class='ntdata euler'>\n"
eptable += "<thead>"
eptable += "<tr class='space'><th class='weight'></th><th class='weight'>$p$</th><th class='weight'>$F_p$</th>"
if L.degree > 2:
eptable += "<th class='weight galois'>$\Gal(F_p)$</th>"
eptable += "</tr>\n"
eptable += "</thead>"
goodorbad = "bad"
for lf in L.bad_lfactors:
try:
thispolygal = list_to_factored_poly_otherorder(lf[1], galois=True)
eptable += ("<tr><td>" + goodorbad + "</td><td>" + str(lf[0]) + "</td><td>" +
"$" + thispolygal[0] + "$" +
"</td>")
if L.degree > 2:
eptable += "<td class='galois'>"
this_gal_group = thispolygal[1]
if this_gal_group[0]==[0,0]:
pass # do nothing, because the local faco is 1
elif this_gal_group[0]==[1,1]:
eptable += group_display_knowl(this_gal_group[0][0],this_gal_group[0][1],'$C_1$')
else:
eptable += group_display_knowl(this_gal_group[0][0],this_gal_group[0][1])
for j in range(1,len(thispolygal[1])):
eptable += "$\\times$"
eptable += group_display_knowl(this_gal_group[j][0],this_gal_group[j][1])
eptable += "</td>"
eptable += "</tr>\n"
except IndexError:
eptable += "<tr><td></td><td>" + str(j) + "</td><td>" + "not available" + "</td></tr>\n"
goodorbad = ""
goodorbad = "good"
firsttime = " class='first'"
good_primes1 = good_primes[:9]
good_primes2 = good_primes[9:]
for j in good_primes1:
this_prime_index = prime_pi(j) - 1
thispolygal = list_to_factored_poly_otherorder(L.localfactors[this_prime_index],galois=True)
eptable += ("<tr" + firsttime + "><td>" + goodorbad + "</td><td>" + str(j) + "</td><td>" +
"$" + thispolygal[0] + "$" +
"</td>")
if L.degree > 2:
eptable += "<td class='galois'>"
this_gal_group = thispolygal[1]
eptable += group_display_knowl(this_gal_group[0][0],this_gal_group[0][1])
for j in range(1,len(thispolygal[1])):
eptable += "$\\times$"
eptable += group_display_knowl(this_gal_group[j][0],this_gal_group[j][1])
eptable += "</td>"
eptable += "</tr>\n"
# eptable += "<td>" + group_display_knowl(4,1) + "</td>"
# eptable += "</tr>\n"
goodorbad = ""
firsttime = ""
firsttime = " id='moreep'"
for j in good_primes2:
this_prime_index = prime_pi(j) - 1
thispolygal = list_to_factored_poly_otherorder(L.localfactors[this_prime_index],galois=True)
eptable += ("<tr" + firsttime + " class='more nodisplay'" + "><td>" + goodorbad + "</td><td>" + str(j) + "</td><td>" +
"$" + list_to_factored_poly_otherorder(L.localfactors[this_prime_index], galois=True)[0] + "$" +
"</td>")
if L.degree > 2:
this_gal_group = thispolygal[1]
eptable += "<td class='galois'>"
eptable += group_display_knowl(this_gal_group[0][0],this_gal_group[0][1])
for j in range(1,len(thispolygal[1])):
eptable += "$\\times$"
eptable += group_display_knowl(this_gal_group[j][0],this_gal_group[j][1])
eptable += "</td>"
eptable += "</tr>\n"
firsttime = ""
eptable += "<tr class='less toggle'><td></td><td></td><td> <a onclick='"
eptable += 'show_moreless("more"); return true' + "'"
eptable += ' href="#moreep" '
eptable += ">show more</a></td></tr>\n"
eptable += "<tr class='more toggle nodisplay'><td></td><td></td><td> <a onclick='"
eptable += 'show_moreless("less"); return true' + "'"
eptable += ' href="#eptable" '
eptable += ">show less</a></td></tr>\n"
eptable += "</table>\n"
ans += "\n" + eptable
return(ans)
def lfuncEPhtml(L, fmt):
""" Euler product as a formula and a table of local factors.
"""
texform_gen = "\[L(s) = " # "\[L(A,s) = "
texform_gen += "\prod_{p \\text{ prime}} F_p(p^{-s})^{-1} \]\n"
pfactors = prime_divisors(L.level)
if len(pfactors) == 1: #i.e., the conductor is prime
pgoodset = "$p \\neq " + str(pfactors[0]) + "$"
pbadset = "$p = " + str(pfactors[0]) + "$"
else:
badset = "\\{" + str(pfactors[0])
for j in range(1, len(pfactors)):
badset += ",\\;"
badset += str(pfactors[j])
badset += "\\}"
pgoodset = "$p \\notin " + badset + "$"
pbadset = "$p \\in " + badset + "$"
ans = ""
ans += texform_gen + "where, for " + pgoodset + ",\n"
if L.degree == 4 and L.motivic_weight == 1:
ans += "\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]"
ans += "with $b_p = a_p^2 - a_{p^2}$. "
elif L.degree == 2 and L.motivic_weight == 1:
ans += "\[F_p(T) = 1 - a_p T + p T^2 .\]"
else:
ans += "\(F_p\) is a polynomial of degree " + str(L.degree) + ". "
ans += "If " + pbadset + ", then $F_p$ is a polynomial of degree at most "
ans += str(L.degree - 1) + ". "
bad_primes = []
for lf in L.bad_lfactors:
bad_primes.append(lf[0])
eulerlim = 25
good_primes = []
for j in range(0, eulerlim):
this_prime = Primes().unrank(j)
if this_prime not in bad_primes:
good_primes.append(this_prime)
eptable = "<table id='eptable' class='ntdata euler'>\n"
eptable += "<thead>"
eptable += "<tr class='space'><th class='weight'></th><th class='weight'>$p$</th><th class='weight'>$F_p$</th>"
if L.degree > 2:
eptable += "<th class='weight galois'>$\Gal(F_p)$</th>"
eptable += "</tr>\n"
eptable += "</thead>"
goodorbad = "bad"
for lf in L.bad_lfactors:
try:
thispolygal = list_to_factored_poly_otherorder(lf[1], galois=True)
eptable += ("<tr><td>" + goodorbad + "</td><td>" + str(lf[0]) +
"</td><td>" + "$" + thispolygal[0] + "$" + "</td>")
if L.degree > 2:
eptable += "<td class='galois'>"
this_gal_group = thispolygal[1]
if this_gal_group[0] == [0, 0]:
pass # do nothing, because the local faco is 1
elif this_gal_group[0] == [1, 1]:
eptable += group_display_knowl(this_gal_group[0][0],
this_gal_group[0][1],
'$C_1$')
else:
eptable += group_display_knowl(this_gal_group[0][0],
this_gal_group[0][1])
for j in range(1, len(thispolygal[1])):
eptable += "$\\times$"
eptable += group_display_knowl(this_gal_group[j][0],
this_gal_group[j][1])
eptable += "</td>"
eptable += "</tr>\n"
except IndexError:
eptable += "<tr><td></td><td>" + str(
j) + "</td><td>" + "not available" + "</td></tr>\n"
goodorbad = ""
goodorbad = "good"
firsttime = " class='first'"
good_primes1 = good_primes[:9]
good_primes2 = good_primes[9:]
for j in good_primes1:
this_prime_index = prime_pi(j) - 1
thispolygal = list_to_factored_poly_otherorder(
L.localfactors[this_prime_index], galois=True)
eptable += ("<tr" + firsttime + "><td>" + goodorbad + "</td><td>" +
str(j) + "</td><td>" + "$" + thispolygal[0] + "$" +
"</td>")
if L.degree > 2:
eptable += "<td class='galois'>"
this_gal_group = thispolygal[1]
eptable += group_display_knowl(this_gal_group[0][0],
this_gal_group[0][1])
for j in range(1, len(thispolygal[1])):
eptable += "$\\times$"
eptable += group_display_knowl(this_gal_group[j][0],
this_gal_group[j][1])
eptable += "</td>"
eptable += "</tr>\n"
# eptable += "<td>" + group_display_knowl(4,1) + "</td>"
# eptable += "</tr>\n"
goodorbad = ""
firsttime = ""
firsttime = " id='moreep'"
for j in good_primes2:
this_prime_index = prime_pi(j) - 1
thispolygal = list_to_factored_poly_otherorder(
L.localfactors[this_prime_index], galois=True)
eptable += ("<tr" + firsttime + " class='more nodisplay'" + "><td>" +
goodorbad + "</td><td>" + str(j) + "</td><td>" + "$" +
list_to_factored_poly_otherorder(
L.localfactors[this_prime_index], galois=True)[0] +
"$" + "</td>")
if L.degree > 2:
this_gal_group = thispolygal[1]
eptable += "<td class='galois'>"
eptable += group_display_knowl(this_gal_group[0][0],
this_gal_group[0][1])
for j in range(1, len(thispolygal[1])):
eptable += "$\\times$"
eptable += group_display_knowl(this_gal_group[j][0],
this_gal_group[j][1])
eptable += "</td>"
eptable += "</tr>\n"
firsttime = ""
eptable += "<tr class='less toggle'><td></td><td></td><td> <a onclick='"
eptable += 'show_moreless("more"); return true' + "'"
eptable += ' href="#moreep" '
eptable += ">show more</a></td></tr>\n"
eptable += "<tr class='more toggle nodisplay'><td></td><td></td><td> <a onclick='"
eptable += 'show_moreless("less"); return true' + "'"
eptable += ' href="#eptable" '
eptable += ">show less</a></td></tr>\n"
eptable += "</table>\n"
ans += "\n" + eptable
return (ans)
def render_hgm_webpage(args):
data = None
info = {}
if 'label' in args:
label = clean_input(args['label'])
data = motivedb().find_one({'label': label})
if data is None:
bread = get_bread([("Search error", url_for('.search'))])
info['err'] = "Motive " + label + " was not found in the database."
info['label'] = label
return search_input_error(info, bread)
title = 'Hypergeometric Motive:' + label
A = data['A']
B = data['B']
myfam = familydb().find_one({'A': A, 'B': B})
if myfam is None:
det = 'data not computed'
else:
det = myfam['det']
det = [det[0],str(det[1])]
d1 = det[1]
d1 = re.sub(r'\s','', d1)
d1 = re.sub(r'(.)\(', r'\1*(', d1)
R = PolynomialRing(ZZ, 't')
if det[1]=='':
d2 = R(1)
else:
d2 = R(d1)
det = d2(QQ(data['t']))*det[0]
t = latex(QQ(data['t']))
typee = 'Orthogonal'
if (data['weight'] % 2) == 1 and (data['degree'] % 2) == 0:
typee = 'Symplectic'
primes = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71]
locinfo = data['locinfo']
for j in range(len(locinfo)):
locinfo[j] = [primes[j]] + locinfo[j]
#locinfo[j][2] = poly_with_factored_coeffs(locinfo[j][2], primes[j])
locinfo[j][2] = list_to_factored_poly_otherorder(locinfo[j][2], vari='x')
hodge = string2list(data['hodge'])
famhodge = string2list(data['famhodge'])
prop2 = [
('Degree', '\(%s\)' % data['degree']),
('Weight', '\(%s\)' % data['weight']),
('Hodge vector', '\(%s\)' % hodge),
('Conductor', '\(%s\)' % data['cond']),
]
# Now add factorization of conductor
Cond = ZZ(data['cond'])
if not (Cond.abs().is_prime() or Cond == 1):
data['cond'] = "%s=%s" % (str(Cond), factorint(data['cond']))
info.update({
'A': A,
'B': B,
't': t,
'degree': data['degree'],
'weight': data['weight'],
'sign': data['sign'],
'sig': data['sig'],
'hodge': hodge,
'famhodge': famhodge,
'cond': data['cond'],
'req': data['req'],
'lcms': data['lcms'],
'type': typee,
'det': det,
'locinfo': locinfo
})
AB_data, t_data = data["label"].split("_t")
friends = [("Motive family "+AB_data.replace("_"," "), url_for(".by_family_label", label = AB_data))]
friends.append(('L-function', url_for("l_functions.l_function_hgm_page", label=AB_data, t='t'+t_data)))
# if rffriend != '':
# friends.append(('Discriminant root field', rffriend))
bread = get_bread([(label, ' ')])
return render_template("hgm-show-motive.html", credit=HGM_credit, title=title, bread=bread, info=info, properties2=prop2, friends=friends, learnmore=learnmore_list())
def render_hgm_webpage(label):
data = None
info = {}
data = db.hgm_motives.lookup(label)
if data is None:
abort(404, "Hypergeometric motive " + label + " was not found in the database.")
title = 'Hypergeometric Motive:' + label
A = data['A']
B = data['B']
det = db.hgm_families.lucky({'A': A, 'B': B}, 'det')
if det is None:
det = 'data not computed'
else:
det = [det[0],str(det[1])]
d1 = det[1]
d1 = re.sub(r'\s','', d1)
d1 = re.sub(r'(.)\(', r'\1*(', d1)
R = PolynomialRing(ZZ, 't')
if det[1]=='':
d2 = R(1)
else:
d2 = R(d1)
det = d2(QQ(data['t']))*det[0]
t = latex(QQ(data['t']))
typee = 'Orthogonal'
if (data['weight'] % 2) == 1 and (data['degree'] % 2) == 0:
typee = 'Symplectic'
primes = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71]
locinfo = data['locinfo']
for j in range(len(locinfo)):
locinfo[j] = [primes[j]] + locinfo[j]
#locinfo[j][2] = poly_with_factored_coeffs(locinfo[j][2], primes[j])
locinfo[j][2] = list_to_factored_poly_otherorder(locinfo[j][2], vari='x')
hodge = data['hodge']
famhodge = data['famhodge']
prop2 = [
('Degree', '\(%s\)' % data['degree']),
('Weight', '\(%s\)' % data['weight']),
('Hodge vector', '\(%s\)' % hodge),
('Conductor', '\(%s\)' % data['cond']),
]
# Now add factorization of conductor
Cond = ZZ(data['cond'])
if not (Cond.abs().is_prime() or Cond == 1):
data['cond'] = "%s=%s" % (str(Cond), factorint(data['cond']))
info.update({
'A': A,
'B': B,
't': t,
'degree': data['degree'],
'weight': data['weight'],
'sign': data['sign'],
'sig': data['sig'],
'hodge': hodge,
'famhodge': famhodge,
'cond': data['cond'],
'req': data['req'],
'lcms': data['lcms'],
'type': typee,
'det': det,
'locinfo': locinfo
})
AB_data, t_data = data["label"].split("_t")
friends = [("Motive family "+AB_data.replace("_"," "), url_for(".by_family_label", label = AB_data))]
friends.append(('L-function', url_for("l_functions.l_function_hgm_page", label=AB_data, t='t'+t_data)))
# if rffriend != '':
# friends.append(('Discriminant root field', rffriend))
bread = get_bread([(label, ' ')])
return render_template("hgm-show-motive.html", credit=HGM_credit, title=title, bread=bread, info=info, properties2=prop2, friends=friends, learnmore=learnmore_list())
def make_class(self):
self.decompositioninfo = self.decomposition_display()
self.basechangeinfo = self.basechange_display()
self.formatted_polynomial = list_to_factored_poly_otherorder(
self.polynomial, galois=False, vari='x')
def make_class(self):
self.decompositioninfo = self.decomposition_display()
self.basechangeinfo = self.basechange_display()
self.formatted_polynomial = list_to_factored_poly_otherorder(self.polynomial,galois=False,vari = 'x')
