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r""" The Stein-Watkins table of elliptic curves
Sage gives access to the Stein-Watkins table of elliptic curves, via an optional package that you must install. This is a huge database of elliptic curves. You can install the database (a 2.6GB package) with the command
::
sage -i database_stein_watkins
You can also automatically download a small version, which takes much less time, using the command
::
sage -i database_stein_watkins_mini
This database covers a wide range of conductors, but unlike the :mod:`Cremona database <sage.databases.cremona>`, this database need not list all curves of a given conductor. It lists the curves whose coefficients are not "too large" (see [SW2002]_).
- The command ``SteinWatkinsAllData(n)`` returns an iterator over the curves in the `n`-th Stein-Watkins table, which contains elliptic curves of conductor between `n10^5` and `(n+1)10^5`. Here `n` can be between 0 and 999, inclusive.
- The command ``SteinWatkinsPrimeData(n)`` returns an iterator over the curves in the `n^{th}` Stein-Watkins prime table, which contains prime conductor elliptic curves of conductor between `n10^6` and `(n+1)10^6`. Here `n` varies between 0 and 99, inclusive.
EXAMPLES: We obtain the first table of elliptic curves.
::
sage: d = SteinWatkinsAllData(0) sage: d Stein-Watkins Database a.0 Iterator
We type ``next(d)`` to get each isogeny class of curves from ``d``::
sage: C = next(d) # optional - database_stein_watkins sage: C # optional - database_stein_watkins Stein-Watkins isogeny class of conductor 11 sage: next(d) # optional - database_stein_watkins Stein-Watkins isogeny class of conductor 14 sage: next(d) # optional - database_stein_watkins Stein-Watkins isogeny class of conductor 15
An isogeny class has a number of attributes that give data about the isogeny class, such as the rank, equations of curves, conductor, leading coefficient of `L`-function, etc.
::
sage: C.data # optional - database_stein_watkins ['11', '[11]', '0', '0.253842', '25', '+*1'] sage: C.curves # optional - database_stein_watkins [[[0, -1, 1, 0, 0], '(1)', '1', '5'], [[0, -1, 1, -10, -20], '(5)', '1', '5'], [[0, -1, 1, -7820, -263580], '(1)', '1', '1']] sage: C.conductor # optional - database_stein_watkins 11 sage: C.leading_coefficient # optional - database_stein_watkins '0.253842' sage: C.modular_degree # optional - database_stein_watkins '+*1' sage: C.rank # optional - database_stein_watkins 0 sage: C.isogeny_number # optional - database_stein_watkins '25'
If we were to continue typing ``next(d)`` we would iterate over all curves in the Stein-Watkins database up to conductor `10^5`. We could also type ``for C in d: ...``
To access the data file starting at `10^5` do the following::
sage: d = SteinWatkinsAllData(1) sage: C = next(d) # optional - database_stein_watkins sage: C # optional - database_stein_watkins Stein-Watkins isogeny class of conductor 100002 sage: C.curves # optional - database_stein_watkins [[[1, 1, 0, 112, 0], '(8,1,2,1)', 'X', '2'], [[1, 1, 0, -448, -560], '[4,2,1,2]', 'X', '2']]
Next we access the prime-conductor data::
sage: d = SteinWatkinsPrimeData(0) sage: C = next(d) # optional - database_stein_watkins sage: C # optional - database_stein_watkins Stein-Watkins isogeny class of conductor 11
Each call ``next(d)`` gives another elliptic curve of prime conductor::
sage: C = next(d) # optional - database_stein_watkins sage: C # optional - database_stein_watkins Stein-Watkins isogeny class of conductor 17 sage: C.curves # optional - database_stein_watkins [[[1, -1, 1, -1, 0], '[1]', '1', '4'], [[1, -1, 1, -6, -4], '[2]', '1', '2x'], [[1, -1, 1, -1, -14], '(4)', '1', '4'], [[1, -1, 1, -91, -310], '[1]', '1', '2']] sage: C = next(d) # optional - database_stein_watkins sage: C # optional - database_stein_watkins Stein-Watkins isogeny class of conductor 19
REFERENCE:
- [SW2002]_ """
#***************************************************************************** # # Sage: 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/ #***************************************************************************** from __future__ import print_function from six.moves import range
import bz2 import os
from sage.env import SAGE_SHARE
class SteinWatkinsIsogenyClass: def __init__(self, conductor): self.conductor = conductor
def __repr__(self): return "Stein-Watkins isogeny class of conductor %s"%self.conductor
def __len__(self): try: return len(self.curves) except AttributeError: return 0
def __iter__(self): try: for E in self.curves: yield E except AttributeError: return
def _lines(s): while True: i = s.find("\n") if i == -1: yield "" return line = s[:i] s = s[i+1:] yield line
class SteinWatkinsAllData: """ Class for iterating through one of the Stein-Watkins database files for all conductors. """ def __init__(self, num): raise RuntimeError("num (=%s) must be a nonnegative integer"%num)
def __repr__(self): """ EXAMPLES::
sage: d = SteinWatkinsAllData(1) sage: d Stein-Watkins Database a.1 Iterator """
def __iter__(self): """ EXAMPLES::
sage: d = SteinWatkinsAllData(0) sage: d = d[10:20] # optional - database_stein_watkins; long time sage: for C in d: # optional - database_stein_watkins; long time ....: print(C) Stein-Watkins isogeny class of conductor 11 Stein-Watkins isogeny class of conductor 14 Stein-Watkins isogeny class of conductor 15 Stein-Watkins isogeny class of conductor 17 Stein-Watkins isogeny class of conductor 19 Stein-Watkins isogeny class of conductor 20 """ try: file = bz2.BZ2File(self._file, 'r') except IOError: raise IOError("The Stein-Watkins data file %s must be installed."%self._file) C = None for L in file: if len(L) == 0: continue if L[0] != '[': # new curve if C is not None: yield C x = L.split() N = int(x[0]) C = SteinWatkinsIsogenyClass(N) C.rank = int(x[2]) C.leading_coefficient = x[3] C.isogeny_number = x[4] C.modular_degree = x[5] C.curves = [] C.data = x else: w = L.split() C.curves.append([eval(w[0]), w[1], w[2], w[3]]) yield C
def __next__(self): return next(self._iter)
next = __next__
def __getitem__(self, N): """ Return the curves of conductor N in this table. (Very slow!) Return all data about curves between the given levels in this database file.
EXAMPLES::
sage: d = SteinWatkinsAllData(0) sage: d[15:18] # optional - database_stein_watkins; long time [Stein-Watkins isogeny class of conductor 15, Stein-Watkins isogeny class of conductor 17] """ X = [] if isinstance(N, slice): min_level, max_level, step = N.indices(len(list(self))) for C in self: M = C.conductor if M >= min_level and M <= max_level: X.append(C) elif M > max_level: return X else: for C in self: M = C.conductor if M == N: X.append(C) elif M > N: return X return X
def iter_levels(self): """ Iterate through the curve classes, but grouped into lists by level.
EXAMPLES::
sage: d = SteinWatkinsAllData(1) sage: E = d.iter_levels() sage: next(E) # optional - database_stein_watkins [Stein-Watkins isogeny class of conductor 100002] sage: next(E) # optional - database_stein_watkins [Stein-Watkins isogeny class of conductor 100005, Stein-Watkins isogeny class of conductor 100005] sage: next(E) # optional - database_stein_watkins [Stein-Watkins isogeny class of conductor 100007] """ it = iter(self) C = [] N = 0 while True: try: E = next(it) except StopIteration: if C != []: yield C return if E.conductor != N: if C != []: yield C C = [E] N = E.conductor else: C.append(E) yield C
class SteinWatkinsPrimeData(SteinWatkinsAllData): def __init__(self, num): raise RuntimeError("num (=%s) must be a nonnegative integer"%num)
def __repr__(self): """ EXAMPLES::
sage: d = SteinWatkinsPrimeData(1) sage: d Stein-Watkins Prime Conductor Database p.1 Iterator """
def ecdb_num_curves(max_level=200000): r""" Return a list whose `N`-th entry, for ``0 <= N <= max_level``, is the number of elliptic curves of conductor `N` in the database.
EXAMPLES::
sage: sage.databases.stein_watkins.ecdb_num_curves(100) # optional - database_stein_watkins [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 6, 8, 0, 4, 0, 3, 4, 6, 0, 0, 6, 0, 5, 4, 0, 0, 8, 0, 4, 4, 4, 3, 4, 4, 5, 4, 4, 0, 6, 1, 2, 8, 2, 0, 6, 4, 8, 2, 2, 1, 6, 4, 6, 7, 3, 0, 0, 1, 4, 6, 4, 2, 12, 1, 0, 2, 4, 0, 6, 2, 0, 12, 1, 6, 4, 1, 8, 0, 2, 1, 6, 2, 0, 0, 1, 3, 16, 4, 3, 0, 2, 0, 8, 0, 6, 11, 4] """ i = 0 N = 1 d = SteinWatkinsAllData(i) v = [int(0) for _ in range(max_level + 1)] while True: try: C = next(d) except StopIteration: i += 1 d = SteinWatkinsAllData(i) continue N = C.conductor if N > max_level: break v[N] += len(C.curves) return v
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