Coverage for local/lib/python2.7/site-packages/sage/databases/jones.py : 4%

r""" 

John Jones's tables of number fields 

In order to use the Jones database, the optional database package 

must be installed using the Sage command !sage -i 

database_jones_numfield 

This is a table of number fields with bounded ramification and 

degree `\leq 6`. You can query the database for all number 

fields in Jones's tables with bounded ramification and degree. 

EXAMPLES: First load the database:: 

sage: J = JonesDatabase() 

sage: J 

John Jones's table of number fields with bounded ramification and degree <= 6 

List the degree and discriminant of all fields in the database that 

have ramification at most at 2:: 

sage: [(k.degree(), k.disc()) for k in J.unramified_outside([2])] # optional - database_jones_numfield 

[(1, 1), (2, -4), (2, -8), (2, 8), (4, 256), (4, 512), (4, -1024), (4, -2048), (4, 2048), (4, 2048), (4, 2048)] 

List the discriminants of the fields of degree exactly 2 unramified 

outside 2:: 

sage: [k.disc() for k in J.unramified_outside([2],2)] # optional - database_jones_numfield 

[-4, -8, 8] 

List the discriminants of cubic field in the database ramified 

exactly at 3 and 5:: 

sage: [k.disc() for k in J.ramified_at([3,5],3)] # optional - database_jones_numfield 

[-135, -675, -6075, -6075] 

sage: factor(6075) 

3^5 * 5^2 

sage: factor(675) 

3^3 * 5^2 

sage: factor(135) 

3^3 * 5 

List all fields in the database ramified at 101:: 

sage: J.ramified_at(101) # optional - database_jones_numfield 

[Number Field in a with defining polynomial x^2 - 101, 

Number Field in a with defining polynomial x^4 - x^3 + 13*x^2 - 19*x + 361, 

Number Field in a with defining polynomial x^5 + x^4 - 6*x^3 - x^2 + 18*x + 4, 

Number Field in a with defining polynomial x^5 + 2*x^4 + 7*x^3 + 4*x^2 + 11*x - 6, 

Number Field in a with defining polynomial x^5 - x^4 - 40*x^3 - 93*x^2 - 21*x + 17] 

""" 

#***************************************************************************** 

# Sage: System for Algebra and Geometry Experimentation 

# 

# Copyright (C) 2005 William Stein <wstein@gmail.com> 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# 

# This code is distributed in the hope that it will be useful, 

# but WITHOUT ANY WARRANTY; without even the implied warranty of 

# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU 

# General Public License for more details. 

# 

# The full text of the GPL is available at: 

# 

# http://www.gnu.org/licenses/ 

#***************************************************************************** 

import os 

from sage.rings.all import NumberField, RationalField, PolynomialRing 

from sage.misc.misc import powerset 

from sage.env import SAGE_SHARE 

from sage.structure.sage_object import load, save 

from sage.misc.package import PackageNotFoundError 

JONESDATA = os.path.join(SAGE_SHARE, 'jones') 

def sortkey(K): 

""" 

A completely deterministic sorting key for number fields. 

EXAMPLES:: 

sage: from sage.databases.jones import sortkey 

sage: sortkey(QuadraticField(-3)) 

(2, 3, False, x^2 + 3) 

""" 

return K.degree(), abs(K.discriminant()), K.discriminant() > 0, K.polynomial() 

class JonesDatabase: 

def __init__(self): 

self.root = None 

def __repr__(self): 

return "John Jones's table of number fields with bounded ramification and degree <= 6" 

def _load(self, path, filename): 

print(filename) 

i = 0 

while filename[i].isalpha(): 

i += 1 

j = len(filename) - 1 

while filename[j].isalpha() or filename[j] in [".", "_"]: 

j -= 1 

S = sorted([eval(z) for z in filename[i:j + 1].split("-")]) 

data = open(path + "/" + filename).read() 

data = data.replace("^", "**") 

x = PolynomialRing(RationalField(), 'x').gen() 

v = eval(data) 

s = tuple(S) 

if s in self.root: 

self.root[s] += v 

self.root[s].sort() 

else: 

self.root[s] = v 

def _init(self, path): 

""" 

Create the database from scratch from the PARI files on John Jones's 

web page, downloaded (e.g., via wget) to a local directory, which 

is specified as path above. 

INPUT: 

- ``path`` - (default works on William Stein install.) 

path must be the path to Jones's Number_Fields directory 

http://hobbes.la.asu.edu/Number_Fields These files should have 

been downloaded using wget. 

EXAMPLES: This is how to create the database from scratch, assuming 

that the number fields are in the default directory above: From a 

cold start of Sage:: 

sage: J = JonesDatabase() 

sage: J._init() # not tested 

... 

This takes about 5 seconds. 

""" 

from sage.misc.misc import sage_makedirs 

n = 0 

x = PolynomialRing(RationalField(), 'x').gen() 

self.root = {} 

self.root[tuple([])] = [x - 1] 

if not os.path.exists(path): 

raise IOError("Path %s does not exist." % path) 

for X in os.listdir(path): 

if X[-4:] == "solo": 

Z = path + "/" + X 

print(X) 

for Y in os.listdir(Z): 

if Y[-3:] == ".gp": 

self._load(Z, Y) 

sage_makedirs(JONESDATA) 

save(self.root, JONESDATA + "/jones.sobj") 

def unramified_outside(self, S, d=None, var='a'): 

""" 

Return all fields in the database of degree d unramified 

outside S. If d is omitted, return fields of any degree up to 6. 

The fields are ordered by degree and discriminant. 

INPUT: 

- ``S`` - list or set of primes, or a single prime 

- ``d`` - None (default, in which case all fields of degree <= 6 are returned) 

or a positive integer giving the degree of the number fields returned. 

- ``var`` - the name used for the generator of the number fields (default 'a'). 

EXAMPLES:: 

sage: J = JonesDatabase() # optional - database_jones_numfield 

sage: J.unramified_outside([101,109]) # optional - database_jones_numfield 

[Number Field in a with defining polynomial x - 1, 

Number Field in a with defining polynomial x^2 - 101, 

Number Field in a with defining polynomial x^2 - 109, 

Number Field in a with defining polynomial x^3 - x^2 - 36*x + 4, 

Number Field in a with defining polynomial x^4 - x^3 + 13*x^2 - 19*x + 361, 

Number Field in a with defining polynomial x^4 - x^3 + 14*x^2 + 34*x + 393, 

Number Field in a with defining polynomial x^5 + x^4 - 6*x^3 - x^2 + 18*x + 4, 

Number Field in a with defining polynomial x^5 + 2*x^4 + 7*x^3 + 4*x^2 + 11*x - 6, 

Number Field in a with defining polynomial x^5 - x^4 - 40*x^3 - 93*x^2 - 21*x + 17] 

""" 

try: 

S = list(S) 

except TypeError: 

S = [S] 

Z = [] 

for X in powerset(S): 

Z += self.ramified_at(X, d=d, var=var) 

return sorted(Z, key=sortkey) 

def __getitem__(self, S): 

return self.get(S) 

def get(self, S, var='a'): 

""" 

Return all fields in the database ramified exactly at 

the primes in S. 

INPUT: 

- ``S`` - list or set of primes, or a single prime 

- ``var`` - the name used for the generator of the number fields (default 'a'). 

EXAMPLES:: 

sage: J = JonesDatabase() # optional - database_jones_numfield 

sage: J.get(163, var='z') # optional - database_jones_numfield 

[Number Field in z with defining polynomial x^2 + 163, 

Number Field in z with defining polynomial x^3 - x^2 - 54*x + 169, 

Number Field in z with defining polynomial x^4 - x^3 - 7*x^2 + 2*x + 9] 

sage: J.get([3, 4]) # optional - database_jones_numfield 

Traceback (most recent call last): 

... 

ValueError: S must be a list of primes 

""" 

if self.root is None: 

if os.path.exists(JONESDATA + "/jones.sobj"): 

self.root = load(JONESDATA + "/jones.sobj") 

else: 

raise PackageNotFoundError("database_jones_numfield") 

try: 

S = list(S) 

except TypeError: 

S = [S] 

if not all([p.is_prime() for p in S]): 

raise ValueError("S must be a list of primes") 

S.sort() 

s = tuple(S) 

if s not in self.root: 

return [] 

return [NumberField(f, var, check=False) for f in self.root[s]] 

def ramified_at(self, S, d=None, var='a'): 

""" 

Return all fields in the database of degree d ramified exactly at 

the primes in S. The fields are ordered by degree and discriminant. 

INPUT: 

- ``S`` - list or set of primes 

- ``d`` - None (default, in which case all fields of degree <= 6 are returned) 

or a positive integer giving the degree of the number fields returned. 

- ``var`` - the name used for the generator of the number fields (default 'a'). 

EXAMPLES:: 

sage: J = JonesDatabase() # optional - database_jones_numfield 

sage: J.ramified_at([101,109]) # optional - database_jones_numfield 

[] 

sage: J.ramified_at([109]) # optional - database_jones_numfield 

[Number Field in a with defining polynomial x^2 - 109, 

Number Field in a with defining polynomial x^3 - x^2 - 36*x + 4, 

Number Field in a with defining polynomial x^4 - x^3 + 14*x^2 + 34*x + 393] 

sage: J.ramified_at(101) # optional - database_jones_numfield 

[Number Field in a with defining polynomial x^2 - 101, 

Number Field in a with defining polynomial x^4 - x^3 + 13*x^2 - 19*x + 361, 

Number Field in a with defining polynomial x^5 + x^4 - 6*x^3 - x^2 + 18*x + 4, 

Number Field in a with defining polynomial x^5 + 2*x^4 + 7*x^3 + 4*x^2 + 11*x - 6, 

Number Field in a with defining polynomial x^5 - x^4 - 40*x^3 - 93*x^2 - 21*x + 17] 

sage: J.ramified_at((2, 5, 29), 3, 'c') # optional - database_jones_numfield 

[Number Field in c with defining polynomial x^3 - x^2 - 8*x - 28, 

Number Field in c with defining polynomial x^3 - x^2 + 10*x + 102, 

Number Field in c with defining polynomial x^3 - x^2 - 48*x - 188, 

Number Field in c with defining polynomial x^3 - x^2 + 97*x - 333] 

""" 

Z = self.get(S, var=var) 

if d is not None: 

Z = [k for k in Z if k.degree() == d] 

return sorted(Z, key=sortkey)