This code is still pretty slow, but it seems to work! Test against Cremona's tables to get a sense if it's working.
|
|
s = server(directory='/tmp/a')
db.nosqlite.Client('/
c = client(s.port)
database = c.db
collection = database.foo; collection
Traceback (click to the left of this block for traceback) ... SyntaxError: EOL while scanning string literal Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "_sage_input_3.py", line 10, in <module>
exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("cyA9IHNlcnZlcihkaXJlY3Rvcnk9Jy90bXAvYScpCmRiLm5vc3FsaXRlLkNsaWVudCgnLwpjID0gY2xpZW50KHMucG9ydCkKZGF0YWJhc2UgPSBjLmRiCmNvbGxlY3Rpb24gPSBkYXRhYmFzZS5mb287IGNvbGxlY3Rpb24="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
File "", line 1, in <module>
File "/tmp/tmpCiYad7/___code___.py", line 3
db.nosqlite.Client('/
^
SyntaxError: EOL while scanning string literal
|
collection.insert({'a':123, 'b':'hello', 'Matrix':Matrix(3)})
list(collection.find('a<3'))
Traceback (click to the left of this block for traceback) ... NameError: name 'collection' is not defined Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "_sage_input_4.py", line 10, in <module>
exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("Y29sbGVjdGlvbi5pbnNlcnQoeydhJzoxMjMsICdiJzonaGVsbG8nLCAnTWF0cml4JzpNYXRyaXgoMyl9KQpsaXN0KGNvbGxlY3Rpb24uZmluZCgnYTwzJykp"),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
File "", line 1, in <module>
File "/tmp/tmpETBmF_/___code___.py", line 3, in <module>
collection.insert({'a':_sage_const_123 , 'b':'hello', 'Matrix':Matrix(_sage_const_3 )})
NameError: name 'collection' is not defined
|
dictiona = {};dictiona['1']=3;dictiona
{'1': 3}
{'1': 3}
|
|
|
x = var('x')
K.<a> = NumberField(x^2-x-1)
import psage.modform.hilbert.sqrt5.tables as sqrt5
|
|
K(0).denominator_ideal()
Traceback (click to the left of this block for traceback) ... ValueError: denominator ideal of 0 is not defined. Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "_sage_input_8.py", line 10, in <module>
exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("SygwKS5kZW5vbWluYXRvcl9pZGVhbCgp"),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
File "", line 1, in <module>
File "/tmp/tmpq6CnTm/___code___.py", line 3, in <module>
exec compile(u'K(_sage_const_0 ).denominator_ideal()
File "", line 1, in <module>
File "number_field_element.pyx", line 3005, in sage.rings.number_field.number_field_element.NumberFieldElement.denominator_ideal (sage/rings/number_field/number_field_element.cpp:19281)
ValueError: denominator ideal of 0 is not defined.
|
@cached_function
def ap(E,p):
return E.change_ring(p.residue_field()).trace_of_frobenius()
R.<ch> = GF(2)[]
def frob(E,p):
t = ap(E,p)
return ch^2 - ap(E, p)*ch + int(p.norm())
def disc(E, p):
t = ap(E, p)
return t^2 - 4*p.norm()
def isogeny_primes(E, norm_bound, isog_degree_bound): #Returns prime for which E has an isogeny
P = [p for p in sqrt5.ideals_of_bounded_norm(norm_bound) if p.is_prime() and E.has_good_reduction(p)]
w = set(primes(isog_degree_bound+1))
i = 0
w.remove(2)
while len(w) > 0 and i < len(P):
d = disc(E, P[i])
w = [ell for ell in w if not (legendre_symbol(d,ell) == -1)]
i = i +1
i = 0
while i < len(P):
if frob(E,P[i]).is_irreducible():
break
i = i+1
if i == len(P):
w.insert(0,2)
return w
|
|
#def isogeny_class_computation(E,p): #Returns the equations for curves in p-iso class of E
# if p != 2:
# E = E.short_weierstrass_model()
# F = E.division_polynomial(p).change_ring(K)
# f = F.factor()
# i=0
# S.<x> = K[]
# t=f[0][0].degree()
# w=[S(1)]
# while t < (p+1)/2:
# for j in w:
# if (j*f[i][0]).degree() < (p + 1)/2 and (j*f[i][0]).divides(F):
# w.append(j*f[i][0])
# t = f[i+1][0].degree()
# i+=1
# v=[]
# o = 1
# while o < len(w):
# try:
# v.append(E.isogeny(w[o]).codomain())
# except ValueError:
# pass
# o+=1
# v = [F.change_ring(K).global_minimal_model() for F in v]
# return v
# else:
# w = [Q for Q in E.torsion_subgroup() if order(Q)==2]
# v = [E.isogeny(E(Q)).codomain() for Q in w]
# return v
|
|
def isogeny_class_computation(E,p):
if p != 2:
E = E.short_weierstrass_model()
F = E.division_polynomial(p).change_ring(K)
v = []
for f in [f for f in divisors(F) if f.degree() == (p-1)/2]:
try:
v.append(E.change_ring(K).isogeny(f).codomain())
except ValueError:
pass
v = [F.change_ring(K).global_minimal_model() for F in v]
return v
else:
w = [Q for Q in E.torsion_subgroup() if order(Q)==2]
v = [E.isogeny(E(Q)).codomain() for Q in w]
return v
|
|
#def isogeny_class_computation2(E,p): #William's
# if p != 2:
# E = E.short_weierstrass_model()
# F = E.division_polynomial(p)
# print F.parent(), type(F), F
# F = F.change_ring(K)
# try:
# w = (f for f in F.divisors() if f.degree() == (p-1)/2)
# except AttributeError:
# print F.parent(), type(F)
# raise AttributeError
# for f in w:
# try:
# v.append(E.change_ring(K).isogeny(f).codomain())
# except ValueError:
# pass
# v = [F.change_ring(K).global_minimal_model() for F in v]
# return v
# else:
# w = [Q for Q in E.torsion_subgroup() if order(Q)==2]
# v = [E.isogeny(E(Q)).codomain() for Q in w]
# return v
|
|
#def curve_isogeny_classes(E):
# isolist = isogeny_primes(E,1000,1000)
# for i in isolist:
# print i,'- isogeny class: '
# for j in isogeny_class_computation(E,i):
# print ' ',j.a_invariants()
|
|
def curve_isogeny_vector(E): #Returns isogeny class and adjacency matrix
curve_list = [E]
i = 0
Adj = matrix(50)
ins = 1
while i < len(curve_list):
isolist = isogeny_primes(curve_list[i],500,500)
for p in isolist:
for F in isogeny_class_computation(curve_list[i],p):
bool = True
for G in curve_list:
if F.is_isomorphic(G):
bool = False
Adj[i,curve_list.index(G)]=p #if a curve in the isogeny class computation is isom
Adj[curve_list.index(G),i]=p #to a curve already in the list, we want a line
if bool:
curve_list.append(F)
Adj[i,ins]=p
Adj[ins,i]=p
ins += 1
i+=1
Adj = Adj.submatrix(nrows=len(curve_list),ncols=len(curve_list))
return curve_list, Adj
|
|
walltime(
Traceback (click to the left of this block for traceback) ... SyntaxError: invalid syntax Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "_sage_input_10.py", line 10, in <module>
exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("d2FsbHRpbWUo"),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
File "", line 1, in <module>
File "/tmp/tmpjsw9ix/___code___.py", line 3
^
SyntaxError: invalid syntax
|
##############
|
|
|
|
|
|
def isogeny_printer(E,fsize=(5,4)):
X,Y=curve_isogeny_vector(E)
counter = 0
for i in X:
print 'curve',counter,': ',i.a_invariants(), i.conductor().norm()
counter +=1
show(Y)
show(Graph(Y),figsize=fsize)
|
|
E=EllipticCurve(K,[1,a+1,a,a,0])
isogeny_printer(E)
curve 0 : (1, a + 1, a, a, 0) 31 curve 1 : (1, a + 1, a, a - 5, 3*a - 5) 31 curve 2 : (1, a + 1, a, 41*a - 70, 170*a - 276) 31 curve 3 : (1, a + 1, a, -39*a - 20, 108*a + 46) 31 curve 4 : (1, a + 1, a, 31*a - 75, 141*a - 303) 31 curve 5 : (1, a + 1, a, 691*a - 1105, 10487*a - 16933) 31 curve 0 : (1, a + 1, a, a, 0) 31 curve 1 : (1, a + 1, a, a - 5, 3*a - 5) 31 curve 2 : (1, a + 1, a, 41*a - 70, 170*a - 276) 31 curve 3 : (1, a + 1, a, -39*a - 20, 108*a + 46) 31 curve 4 : (1, a + 1, a, 31*a - 75, 141*a - 303) 31 curve 5 : (1, a + 1, a, 691*a - 1105, 10487*a - 16933) 31 |
EL = EllipticCurve(K,[0,-1,1,0,0])
isogeny_printer(EL)
curve 0 : (0, -1, 1, 0, 0) 121 curve 1 : (0, -1, 1, -10, -20) 121 curve 2 : (0, -1, 1, -7820, -263580) 121 curve 3 : (0, a + 1, 1, -543074*a - 362966, -261178146*a - 163231804) 652098911521882337515422475 curve 4 : (0, a + 1, 1, -39824*a - 27466, -3448146*a - 2150554) 83026402989233316959225 curve 5 : (0, a + 1, 1, -562324*a - 373966, -260524746*a - 162832504) 652098911521882337515422475 curve 6 : (0, a + 1, 1, -59074*a - 38466, -2794746*a - 1751254) 83026402989233316959225 curve 7 : (0, -1, 1, -110*a + 320, 47030*a - 76094) 174643179603485750245 curve 8 : (0, -1, 1, 390*a + 300, 4322*a + 908) 107026473484575847961 curve 9 : (0, -1, 1, -390*a + 690, -4322*a + 5230) 107026473484575847961 curve 10 : (0, -1, 1, 110*a + 210, -47030*a - 29064) 174643179603485750245 curve 0 : (0, -1, 1, 0, 0) 121 curve 1 : (0, -1, 1, -10, -20) 121 curve 2 : (0, -1, 1, -7820, -263580) 121 curve 3 : (0, a + 1, 1, -543074*a - 362966, -261178146*a - 163231804) 652098911521882337515422475 curve 4 : (0, a + 1, 1, -39824*a - 27466, -3448146*a - 2150554) 83026402989233316959225 curve 5 : (0, a + 1, 1, -562324*a - 373966, -260524746*a - 162832504) 652098911521882337515422475 curve 6 : (0, a + 1, 1, -59074*a - 38466, -2794746*a - 1751254) 83026402989233316959225 curve 7 : (0, -1, 1, -110*a + 320, 47030*a - 76094) 174643179603485750245 curve 8 : (0, -1, 1, 390*a + 300, 4322*a + 908) 107026473484575847961 curve 9 : (0, -1, 1, -390*a + 690, -4322*a + 5230) 107026473484575847961 curve 10 : (0, -1, 1, 110*a + 210, -47030*a - 29064) 174643179603485750245 |
##############
|
|
w=walltime()
CA31 = EllipticCurve(K,[1,-a-1,a,0,0])
time isogeny_printer(CA31)
walltime(w)
curve 0 : (1, -a - 1, a, 0, 0) 31 curve 1 : (1, -a - 1, a, -5, -3*a + 3) 31 curve 2 : (1, -a - 1, a, -40*a - 30, -130*a - 76) 31 curve 3 : (1, -a - 1, a, 40*a - 60, -148*a + 214) 31 curve 4 : (1, -a - 1, a, -690*a - 415, -9797*a - 6031) 31 curve 5 : (1, -a - 1, a, -30*a - 45, -111*a - 117) 31 curve 0 : (1, -a - 1, a, 0, 0) 31 curve 1 : (1, -a - 1, a, -5, -3*a + 3) 31 curve 2 : (1, -a - 1, a, -40*a - 30, -130*a - 76) 31 curve 3 : (1, -a - 1, a, 40*a - 60, -148*a + 214) 31 curve 4 : (1, -a - 1, a, -690*a - 415, -9797*a - 6031) 31 curve 5 : (1, -a - 1, a, -30*a - 45, -111*a - 117) 31 |
Time: CPU 41.73 s, Wall: 41.85 s
41.875041007995605CA = EllipticCurve(K,[1,a+1,a,a,0])
time isogeny_printer(CA)
curve 0 : (1, a + 1, a, a, 0) 31 curve 1 : (1, a + 1, a, a - 5, 3*a - 5) 31 curve 2 : (1, a + 1, a, 41*a - 70, 170*a - 276) 31 curve 3 : (1, a + 1, a, -39*a - 20, 108*a + 46) 31 curve 4 : (1, a + 1, a, 31*a - 75, 141*a - 303) 31 curve 5 : (1, a + 1, a, 691*a - 1105, 10487*a - 16933) 31 curve 0 : (1, a + 1, a, a, 0) 31 curve 1 : (1, a + 1, a, a - 5, 3*a - 5) 31 curve 2 : (1, a + 1, a, 41*a - 70, 170*a - 276) 31 curve 3 : (1, a + 1, a, -39*a - 20, 108*a + 46) 31 curve 4 : (1, a + 1, a, 31*a - 75, 141*a - 303) 31 curve 5 : (1, a + 1, a, 691*a - 1105, 10487*a - 16933) 31 |
Time: CPU 43.53 s, Wall: 43.76 sCE = EllipticCurve(K,[a,-a+1,1,-1,0])
time isogeny_printer(CE)
curve 0 : (a, -a + 1, 1, -1, 0) curve 1 : (a, -a + 1, 1, -5*a + 9, -8*a + 6) curve 2 : (a, -a + 1, 1, -45*a - 151, -264*a - 858) curve 3 : (a, -a + 1, 1, -3365*a - 12621, -229016*a - 564210) curve 0 : (a, -a + 1, 1, -1, 0) curve 1 : (a, -a + 1, 1, -5*a + 9, -8*a + 6) curve 2 : (a, -a + 1, 1, -45*a - 151, -264*a - 858) curve 3 : (a, -a + 1, 1, -3365*a - 12621, -229016*a - 564210) |
Time: CPU 22.44 s, Wall: 22.60 stime isogeny_printer(CE)
curve 0 : (a, -a + 1, 1, -1, 0) curve 1 : (a, -a + 1, 1, -5*a + 9, -8*a + 6) curve 2 : (a, -a + 1, 1, -45*a - 151, -264*a - 858) curve 3 : (a, -a + 1, 1, -3365*a - 12621, -229016*a - 564210) curve 0 : (a, -a + 1, 1, -1, 0) curve 1 : (a, -a + 1, 1, -5*a + 9, -8*a + 6) curve 2 : (a, -a + 1, 1, -45*a - 151, -264*a - 858) curve 3 : (a, -a + 1, 1, -3365*a - 12621, -229016*a - 564210) |
Time: CPU 7.63 s, Wall: 7.63 sCF = EllipticCurve(K,[0,a,1,1,0])
isogeny_printer(CF,fsize=(7,5))
curve 0 : (0, a, 1, 1, 0) curve 1 : (0, a, 1, 30*a - 59, 102*a - 186) curve 0 : (0, a, 1, 1, 0) curve 1 : (0, a, 1, 30*a - 59, 102*a - 186) |
print ap(CF,3)
|
|
CG = EllipticCurve(K,[1,-1,a,-2*a,a])
time isogeny_printer(CG)
|
|
CH = EllipticCurve(K,[1,-1,1,-5,-7]) #126 A in Cremona's tables
isogeny_printer(CH)
curve 0 : (1, -1, 1, -5, -7) curve 1 : (1, -1, 1, -95, -331) curve 2 : (1, -1, 1, 40, 155) curve 3 : (1, -1, 1, -320, 1883) curve 4 : (1, -1, 1, -1535, 23591) curve 5 : (1, -1, 1, -24575, 1488935) curve 0 : (1, -1, 1, -5, -7) curve 1 : (1, -1, 1, -95, -331) curve 2 : (1, -1, 1, 40, 155) curve 3 : (1, -1, 1, -320, 1883) curve 4 : (1, -1, 1, -1535, 23591) curve 5 : (1, -1, 1, -24575, 1488935) |
J = EllipticCurve(K,[a+1,-a-1,1,-2*a-2,-a-1])
time isogeny_printer(J,fsize=(20,20))
curve 0 : (a + 1, -a - 1, 1, -2*a - 2, -a - 1) curve 1 : (a + 1, -a - 1, 1, -41617*a - 41617, 5859391*a + 4394543) curve 0 : (a + 1, -a - 1, 1, -2*a - 2, -a - 1) curve 1 : (a + 1, -a - 1, 1, -41617*a - 41617, 5859391*a + 4394543) |
Time: CPU 189.24 s, Wall: 189.25 sCT = EllipticCurve(K,[0,0,0,6,7]) #144 B in Cremona's tables
time isogeny_printer(CT)
curve 0 : (0, 0, 0, 6, 7) curve 1 : (0, 0, 0, -39, 70) curve 2 : (0, 0, 0, -219, -1190) curve 3 : (0, 0, 0, -579, 5362) curve 4 : (0, 0, 0, -3459, -78302) curve 5 : (0, 0, 0, 141, -4718) curve 0 : (0, 0, 0, 6, 7) curve 1 : (0, 0, 0, -39, 70) curve 2 : (0, 0, 0, -219, -1190) curve 3 : (0, 0, 0, -579, 5362) curve 4 : (0, 0, 0, -3459, -78302) curve 5 : (0, 0, 0, 141, -4718) |
Time: CPU 23.56 s, Wall: 23.56 sCW = EllipticCurve(K,[1,1,1,37,281]) #150 C in Cremona's Tables
time isogeny_printer(CW)
curve 0 : (1, 1, 1, 37, 281) curve 1 : (1, 1, 1, -463, 3281) curve 2 : (1, 0, 1, -14, -64) curve 3 : (1, 1, 1, -1713, -24219) curve 4 : (1, 1, 1, -7213, 232781) curve 5 : (1, 0, 1, -334, -2368) curve 6 : (1, 0, 1, -5334, -150368) curve 7 : (1, 0, 1, -454, -544) curve 0 : (1, 1, 1, 37, 281) curve 1 : (1, 1, 1, -463, 3281) curve 2 : (1, 0, 1, -14, -64) curve 3 : (1, 1, 1, -1713, -24219) curve 4 : (1, 1, 1, -7213, 232781) curve 5 : (1, 0, 1, -334, -2368) curve 6 : (1, 0, 1, -5334, -150368) curve 7 : (1, 0, 1, -454, -544) |
Time: CPU 37.67 s, Wall: 37.67 s|
|
################################################################################
|
|
import nosqlite
db = nosqlite
|
|
'[a,2,1,43,3]' == '[a,2,1,43,3]'
True True |
f = open(
|
|
def grabber(s):
return '['+s.split('[')[1].split(']')[0]+']' #returns a-invariants
def dictionary_entry_maker(s):
|
|
def grabber(s):
return '['+s.split('[')[1].split(']')[0]+']' #returns a-invariants
data=[]
isowidth = 0
ainvwidth = 0
#primes_in_table = set([])
for line in f.readlines():
isooutput = isogeny_primes_readin(200,1000,line)
isostring = '['
astring = grabber(line)
for i in isooutput:
#primes_in_table.add(i)
isostring += str(i)
isostring += ','
if len(isooutput)>=1:
isostring = isostring[:-1]
isostring += ']'
if len(isostring)>isowidth:
isowidth = len(isostring)
if len(astring)>ainvwidth:
ainvwidth = len(astring)
data.append(line[0:20]+isostring+(12-len(isostring))*' '+astring)
lnwidth = 32 + ainvwidth + 3
#print primes_in_table
#awidth = lnwidth+ainvwidth+2
g = open('/Users/bleveque/Sage/IsogenyTable7.txt','w')
for current_data in data:
returnstr = current_data
returnstr += (lnwidth - len(current_data))*' '
returnstr += '\n'
g.write(returnstr)
|
|
co = 4
dictiona['%s'%co]=3
dictiona
{'1': 3, '4': 3}
{'1': 3, '4': 3}
|
f = 'adsfsda adfasdfaewf adfawefawef efwfaefef rfwearewfwef rfewfaergaer'
f.split()
['adsfsda', 'adfasdfaewf', 'adfawefawef', 'efwfaefef', 'rfwearewfwef', 'rfewfaergaer'] ['adsfsda', 'adfasdfaewf', 'adfawefawef', 'efwfaefef', 'rfwearewfwef', 'rfewfaergaer'] |
f.split()[0]
'adsfsda' 'adsfsda' |
K.ideal(5).denominator_ideal()
Traceback (click to the left of this block for traceback) ... AttributeError: 'NumberFieldFractionalIdeal' object has no attribute 'denominator_ideal' Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "_sage_input_5.py", line 10, in <module>
exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("Sy5pZGVhbCg1KS5kZW5vbWluYXRvcl9pZGVhbCgp"),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
File "", line 1, in <module>
File "/tmp/tmp79vjQP/___code___.py", line 3, in <module>
exec compile(u'K.ideal(_sage_const_5 ).denominator_ideal()
File "", line 1, in <module>
File "element.pyx", line 328, in sage.structure.element.Element.__getattr__ (sage/structure/element.c:2790)
File "parent.pyx", line 277, in sage.structure.parent.getattr_from_other_class (sage/structure/parent.c:2930)
File "parent.pyx", line 177, in sage.structure.parent.raise_attribute_error (sage/structure/parent.c:2726)
AttributeError: 'NumberFieldFractionalIdeal' object has no attribute 'denominator_ideal'
|
|
|
def isogeny_primes(E, norm_bound, isog_degree_bound): #Returns prime for which E has an isogeny
P = [p for p in sqrt5.ideals_of_bounded_norm(norm_bound) if p.is_prime() and E.has_good_reduction(p)]
w = set(primes(isog_degree_bound+1))
i = 0
w.remove(2)
while len(w) > 0 and i < len(P):
d = disc(E, P[i])
w = [ell for ell in w if not (legendre_symbol(d,ell) == -1)]
i = i +1
i = 0
while i < len(P):
if frob(E,P[i]).is_irreducible():
break
i = i+1
if i == len(P):
w.insert(0,2)
return w
|
|
E = EllipticCurve(K,[0,a+1,1,-562324*a-373966,-260524746*a-162832504])
isogeny_primes(E,1000,1000)
[] [] |
F = EllipticCurve(K,[0,-1,1,0,0])
isogeny_primes(F,1000,1000)
[5] [5] |
F.is_isogenous(E)
False False |
|
|