rem stringlengths 1 322k | add stringlengths 0 2.05M | context stringlengths 4 228k | meta stringlengths 156 215 |
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descent. Default strongloy recommended. | descent. *Default strongly recommended.* | def two_descent(self, verbose = True, selmer_only = False, first_limit = 20, second_limit = 8, n_aux = -1, second_descent = True): """ Compute 2-descent data for this curve. | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
Nothing -- nothing is returned | Nothing -- nothing is returned. | def two_descent(self, verbose = True, selmer_only = False, first_limit = 20, second_limit = 8, n_aux = -1, second_descent = True): """ Compute 2-descent data for this curve. | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
second_descent = int(second_descent) | second_descent = int(second_descent) | def two_descent(self, verbose = True, selmer_only = False, first_limit = 20, second_limit = 8, n_aux = -1, second_descent = True): """ Compute 2-descent data for this curve. | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
Returns the rank of this curve, computed using 2-descent. | Returns the rank of this curve, computed using :meth:`two_descent()`. | def rank(self): """ Returns the rank of this curve, computed using 2-descent. | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
upper bound may be obtained using the function rank_bound(). | upper bound may be obtained using the function :meth:`rank_bound()`. | def rank(self): """ Returns the rank of this curve, computed using 2-descent. | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
the method :meth:`certain`. | the method :meth:`certain()`. | def rank(self): """ Returns the rank of this curve, computed using 2-descent. | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
using 2-descent. | using :meth:`two_descent()`. | def rank_bound(self): """ Returns an upper bound for the rank of this curve, computed using 2-descent. | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
of order 4, we only obtain an upper bound of 2: | of order 4, we only obtain an upper bound of 2:: | def rank_bound(self): """ Returns an upper bound for the rank of this curve, computed using 2-descent. | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
In this case the value returned by :meth:`rank` is only a lower bound in general (though in this is correct):: | In this case the value returned by :meth:`rank()` is only a lower bound in general (though this is correct):: | def rank_bound(self): """ Returns an upper bound for the rank of this curve, computed using 2-descent. | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
the computation again but turn off the second descent:: | the computation again but turn off ``second_descent``:: | def selmer_rank(self): r""" Returns the rank of the 2-Selmer group of the curve. EXAMPLES: | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
but with no 2-torsion, the selmer rank is strictly greater | but with no 2-torsion, the Selmer rank is strictly greater | def selmer_rank(self): r""" Returns the rank of the 2-Selmer group of the curve. EXAMPLES: | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
primes up to bound. | primes up to ``bound``. | def saturate(self, bound=-1): """ Compute the saturation of the Mordell-Weil group at all primes up to bound. | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
- `bound` (int, default -1) -- Use `-1` (the default) to | - ``bound`` (int, default -1) -- Use `-1` (the default) to | def saturate(self, bound=-1): """ Compute the saturation of the Mordell-Weil group at all primes up to bound. | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
True if the last :meth:`two_descent` call provably correctly computed the rank. If :meth:`two_descent` hasn't been called, then it is first called by :meth:`certain` | Returns ``True`` if the last :meth:`two_descent()` call provably correctly computed the rank. If :meth:`two_descent()` hasn't been called, then it is first called by :meth:`certain()` | def certain(self): r""" True if the last :meth:`two_descent` call provably correctly computed the rank. If :meth:`two_descent` hasn't been called, then it is first called by :meth:`certain` using the default parameters. | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
The result is true if and only if the results of the methods :meth:`rank` and :meth:`rank_bound` are equal. | The result is ``True`` if and only if the results of the methods :meth:`rank()` and :meth:`rank_bound()` are equal. | def certain(self): r""" True if the last :meth:`two_descent` call provably correctly computed the rank. If :meth:`two_descent` hasn't been called, then it is first called by :meth:`certain` using the default parameters. | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
for the curve `y^2 + y = x^3 - x^2 - 120x - 2183`.:: | for the curve `y^2 + y = x^3 - x^2 - 120x - 2183`:: | def certain(self): r""" True if the last :meth:`two_descent` call provably correctly computed the rank. If :meth:`two_descent` hasn't been called, then it is first called by :meth:`certain` using the default parameters. | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
computing the L-function), but Sha has order 4 and the $2$-torsion is trivial, so mwrank cannot conclusively | computing the `L`-function), but Sha has order 4 and the 2-torsion is trivial, so mwrank cannot conclusively | def certain(self): r""" True if the last :meth:`two_descent` call provably correctly computed the rank. If :meth:`two_descent` hasn't been called, then it is first called by :meth:`certain` using the default parameters. | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
floating point number $B$ such that if $P$ is a point on the curve, then the naive logarithmic height $h(P)$ is less than $B+\hat{h}(P)$, where $\hat{h}(P)$ is the canonical height of $P$. | floating point number `B` such that if `P` is a point on the curve, then the naive logarithmic height `h(P)` is less than `B+\hat{h}(P)`, where `\hat{h}(P)` is the canonical height of `P`. | def CPS_height_bound(self): r""" Return the Cremona-Prickett-Siksek height bound. This is a floating point number $B$ such that if $P$ is a point on the curve, then the naive logarithmic height $h(P)$ is less than $B+\hat{h}(P)$, where $\hat{h}(P)$ is the canonical height of $P$. | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
number $B$ such that if $P$ is a point on the curve, then the naive logarithmic height $h(P)$ is less than $B+\hat{h}(P)$, where $\hat{h}(P)$ is the canonical height of $P$. | number `B` such that if `P` is a point on the curve, then the naive logarithmic height `h(P)` is less than `B+\hat{h}(P)`, where `\hat{h}(P)` is the canonical height of `P`. | def silverman_bound(self): r""" Return the Silverman height bound. This is a floating point number $B$ such that if $P$ is a point on the curve, then the naive logarithmic height $h(P)$ is less than $B+\hat{h}(P)$, where $\hat{h}(P)$ is the canonical height of $P$. | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
INPUT: - `curve` (:class:`mwrank_EllipticCurve`) -- the underlying elliptic curve. - `verbose` (bool, default False) -- verbosity flag (controls amount of output produced in point searches) - `pp` (int, default 1) -- process points flag (if nonzero, the points found are processed, so that at all times only a `\ZZ`-b... | See the docstring of this class for full documentation. | def __init__(self, curve, verbose=True, pp=1, maxr=999): r""" Constructor for the :class:`mwrank_MordellWeil` class. | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
Subgroup of Mordell Weil group: [] sage: EQ.search(2) The previous command produces the following output:: P1 = [0:1:0] is torsion point, order 1 P1 = [1:-1:1] is torsion point, order 2 P1 = [2:2:1] is torsion point, order 3 P1 = [9:23:1] is torsion point, order 6 sage: E = mwrank_EllipticCurve([0,0,1,-7,6]) sag... | Subgroup of Mordell-Weil group: [] | def __init__(self, curve, verbose=True, pp=1, maxr=999): r""" Constructor for the :class:`mwrank_MordellWeil` class. | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
'Subgroup of Mordell Weil group: []' | 'Subgroup of Mordell-Weil group: []' | def __repr__(self): r""" String representation of this Mordell-Weil subgroup. | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
'Subgroup of Mordell Weil group: [[1:-1:1], [-2:3:1], [-14:25:8]]' """ return "Subgroup of Mordell Weil group: %s"%self.__mw | 'Subgroup of Mordell-Weil group: [[1:-1:1], [-2:3:1], [-14:25:8]]' """ return "Subgroup of Mordell-Weil group: %s"%self.__mw | def __repr__(self): r""" String representation of this Mordell-Weil subgroup. | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
This function allows one to add points to a mwrank_MordellWeil object. Process points in the list v, with saturation at primes up to sat. If sat = 0 (the default), do no saturation. | This function allows one to add points to a :class:`mwrank_MordellWeil` object. Process points in the list ``v``, with saturation at primes up to ``sat``. If ``sat`` is zero (the default), do no saturation. | def process(self, v, sat=0): """ This function allows one to add points to a mwrank_MordellWeil object. | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
- `v` (list of 3-tuples or lists of ints or Integers) -- a | - ``v`` (list of 3-tuples or lists of ints or Integers) -- a | def process(self, v, sat=0): """ This function allows one to add points to a mwrank_MordellWeil object. | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
- `sat` (int, default 0) --saturate at primes up to sat, or at all primes if sat=0. | - ``sat`` (int, default 0) -- saturate at primes up to ``sat``, or at *all* primes if ``sat`` is zero. | def process(self, v, sat=0): """ This function allows one to add points to a mwrank_MordellWeil object. | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
None. But note that if the verbose flag is set, then there | None. But note that if the ``verbose`` flag is set, then there | def process(self, v, sat=0): """ This function allows one to add points to a mwrank_MordellWeil object. | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
Example to illustrate the saturation parameter:: | Example to illustrate the saturation parameter ``sat``:: | def process(self, v, sat=0): """ This function allows one to add points to a mwrank_MordellWeil object. | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
raise TypeError, "v (=%s) must be a list of 3-tuples of ints"%v | raise TypeError, "v (=%s) must be a list of 3-tuples (or 3-element lists) of ints"%v | def process(self, v, sat=0): """ This function allows one to add points to a mwrank_MordellWeil object. | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
but the interface currently returns the output as a float. | but the interface currently returns the output as a ``float``. | def regulator(self): """ Return the regulator of the points in this subgroup of the Mordell-Weil group. | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
(float) the regulator of the points in this subgroup. | (float) The regulator of the points in this subgroup. | def regulator(self): """ Return the regulator of the points in this subgroup of the Mordell-Weil group. | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
OUTPUT: (int) The rank of this subgroup of the Mordell-Weil group. | def rank(self): """ Return the rank of this subgroup of the Mordell-Weil group. | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py | |
Subgroup of Mordell Weil group: [] | Subgroup of Mordell-Weil group: [] | def rank(self): """ Return the rank of this subgroup of the Mordell-Weil group. | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
Subgroup of Mordell Weil group: [[1:-1:1], [-2:3:1], [-14:25:8]] | Subgroup of Mordell-Weil group: [[1:-1:1], [-2:3:1], [-14:25:8]] | def rank(self): """ Return the rank of this subgroup of the Mordell-Weil group. | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
all primes up to `max_prime`. If `-1` (default) then an | all primes up to ``max_prime``. If `-1` (the default), an | def saturate(self, max_prime=-1, odd_primes_only=False): r""" Saturate this subgroup of the Mordell-Weil group. INPUT: - ``max_prime`` (int, default -1) -- saturation is performed for all primes up to `max_prime`. If `-1` (default) then an upper bound is computed for the primes at which the subgroup may not be satura... | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
computed bound is greater than a value set by the eclib | computed bound is greater than a value set by the ``eclib`` | def saturate(self, max_prime=-1, odd_primes_only=False): r""" Saturate this subgroup of the Mordell-Weil group. INPUT: - ``max_prime`` (int, default -1) -- saturation is performed for all primes up to `max_prime`. If `-1` (default) then an upper bound is computed for the primes at which the subgroup may not be satura... | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
- ``odd_primes_only`` (bool, default False) -- only do | - ``odd_primes_only`` (bool, default ``False``) -- only do | def saturate(self, max_prime=-1, odd_primes_only=False): r""" Saturate this subgroup of the Mordell-Weil group. INPUT: - ``max_prime`` (int, default -1) -- saturation is performed for all primes up to `max_prime`. If `-1` (default) then an upper bound is computed for the primes at which the subgroup may not be satura... | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
via 2-descent they should alreday be 2-saturated.) | via :meth:``two_descent()`` they should already be 2-saturated.) | def saturate(self, max_prime=-1, odd_primes_only=False): r""" Saturate this subgroup of the Mordell-Weil group. INPUT: - ``max_prime`` (int, default -1) -- saturation is performed for all primes up to `max_prime`. If `-1` (default) then an upper bound is computed for the primes at which the subgroup may not be satura... | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
- ``ok`` (bool) is True if and only if the saturation was | - ``ok`` (bool) -- ``True`` if and only if the saturation was | def saturate(self, max_prime=-1, odd_primes_only=False): r""" Saturate this subgroup of the Mordell-Weil group. INPUT: - ``max_prime`` (int, default -1) -- saturation is performed for all primes up to `max_prime`. If `-1` (default) then an upper bound is computed for the primes at which the subgroup may not be satura... | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
the computed saturation bound being too high, then True indicates that the subgroup is saturated at `\emph{all}` | the computed saturation bound being too high, then ``True`` indicates that the subgroup is saturated at *all* | def saturate(self, max_prime=-1, odd_primes_only=False): r""" Saturate this subgroup of the Mordell-Weil group. INPUT: - ``max_prime`` (int, default -1) -- saturation is performed for all primes up to `max_prime`. If `-1` (default) then an upper bound is computed for the primes at which the subgroup may not be satura... | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
- ``index`` (int) is the index of the group generated by the | - ``index`` (int) -- the index of the group generated by the | def saturate(self, max_prime=-1, odd_primes_only=False): r""" Saturate this subgroup of the Mordell-Weil group. INPUT: - ``max_prime`` (int, default -1) -- saturation is performed for all primes up to `max_prime`. If `-1` (default) then an upper bound is computed for the primes at which the subgroup may not be satura... | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
uses floating points methods based on elliptic logarithms to | uses floating point methods based on elliptic logarithms to | def saturate(self, max_prime=-1, odd_primes_only=False): r""" Saturate this subgroup of the Mordell-Weil group. INPUT: - ``max_prime`` (int, default -1) -- saturation is performed for all primes up to `max_prime`. If `-1` (default) then an upper bound is computed for the primes at which the subgroup may not be satura... | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
We emphasize that if this function returns True as the first return argument, and if the default was used for the parameter ``sat_bnd``, then the points in the basis after calling this function are saturated at `\emph{all}` primes, | We emphasize that if this function returns ``True`` as the first return argument (``ok``), and if the default was used for the parameter ``max_prime``, then the points in the basis after calling this function are saturated at *all* primes, | def saturate(self, max_prime=-1, odd_primes_only=False): r""" Saturate this subgroup of the Mordell-Weil group. INPUT: - ``max_prime`` (int, default -1) -- saturation is performed for all primes up to `max_prime`. If `-1` (default) then an upper bound is computed for the primes at which the subgroup may not be satura... | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
saturate up to, and that prime is `\leq` ``max_prime``. | saturate up to, and that prime might be smaller than ``max_prime``. | def saturate(self, max_prime=-1, odd_primes_only=False): r""" Saturate this subgroup of the Mordell-Weil group. INPUT: - ``max_prime`` (int, default -1) -- saturation is performed for all primes up to `max_prime`. If `-1` (default) then an upper bound is computed for the primes at which the subgroup may not be satura... | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
calling search. So calling search up to height 20 then calling saturate results in another search up to height 18. | calling :meth:`search()`. So calling :meth:`search()` up to height 20 then calling :meth:`saturate()` results in another search up to height 18. | def saturate(self, max_prime=-1, odd_primes_only=False): r""" Saturate this subgroup of the Mordell-Weil group. INPUT: - ``max_prime`` (int, default -1) -- saturation is performed for all primes up to `max_prime`. If `-1` (default) then an upper bound is computed for the primes at which the subgroup may not be satura... | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
Subgroup of Mordell Weil group: [[1547:-2967:343], [2707496766203306:864581029138191:2969715140223272], [-13422227300:-49322830557:12167000000]] | Subgroup of Mordell-Weil group: [[1547:-2967:343], [2707496766203306:864581029138191:2969715140223272], [-13422227300:-49322830557:12167000000]] | def saturate(self, max_prime=-1, odd_primes_only=False): r""" Saturate this subgroup of the Mordell-Weil group. INPUT: - ``max_prime`` (int, default -1) -- saturation is performed for all primes up to `max_prime`. If `-1` (default) then an upper bound is computed for the primes at which the subgroup may not be satura... | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
Subgroup of Mordell Weil group: [[-2:3:1], [2707496766203306:864581029138191:2969715140223272], [-13422227300:-49322830557:12167000000]] | Subgroup of Mordell-Weil group: [[-2:3:1], [2707496766203306:864581029138191:2969715140223272], [-13422227300:-49322830557:12167000000]] | def saturate(self, max_prime=-1, odd_primes_only=False): r""" Saturate this subgroup of the Mordell-Weil group. INPUT: - ``max_prime`` (int, default -1) -- saturation is performed for all primes up to `max_prime`. If `-1` (default) then an upper bound is computed for the primes at which the subgroup may not be satura... | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
Subgroup of Mordell Weil group: [[-2:3:1], [-14:25:8], [-13422227300:-49322830557:12167000000]] | Subgroup of Mordell-Weil group: [[-2:3:1], [-14:25:8], [-13422227300:-49322830557:12167000000]] | def saturate(self, max_prime=-1, odd_primes_only=False): r""" Saturate this subgroup of the Mordell-Weil group. INPUT: - ``max_prime`` (int, default -1) -- saturation is performed for all primes up to `max_prime`. If `-1` (default) then an upper bound is computed for the primes at which the subgroup may not be satura... | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
Subgroup of Mordell Weil group: [[-2:3:1], [-14:25:8], [1:-1:1]] | Subgroup of Mordell-Weil group: [[-2:3:1], [-14:25:8], [1:-1:1]] | def saturate(self, max_prime=-1, odd_primes_only=False): r""" Saturate this subgroup of the Mordell-Weil group. INPUT: - ``max_prime`` (int, default -1) -- saturation is performed for all primes up to `max_prime`. If `-1` (default) then an upper bound is computed for the primes at which the subgroup may not be satura... | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
Of course, the ``process`` function would have done all this | Of course, the :meth:`process()` function would have done all this | def saturate(self, max_prime=-1, odd_primes_only=False): r""" Saturate this subgroup of the Mordell-Weil group. INPUT: - ``max_prime`` (int, default -1) -- saturation is performed for all primes up to `max_prime`. If `-1` (default) then an upper bound is computed for the primes at which the subgroup may not be satura... | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
Subgroup of Mordell Weil group: [[-2:3:1], [-14:25:8], [1:-1:1]] | Subgroup of Mordell-Weil group: [[-2:3:1], [-14:25:8], [1:-1:1]] | def saturate(self, max_prime=-1, odd_primes_only=False): r""" Saturate this subgroup of the Mordell-Weil group. INPUT: - ``max_prime`` (int, default -1) -- saturation is performed for all primes up to `max_prime`. If `-1` (default) then an upper bound is computed for the primes at which the subgroup may not be satura... | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
But we would still need to use the ``saturate`` function to | But we would still need to use the :meth:`saturate()` function to | def saturate(self, max_prime=-1, odd_primes_only=False): r""" Saturate this subgroup of the Mordell-Weil group. INPUT: - ``max_prime`` (int, default -1) -- saturation is performed for all primes up to `max_prime`. If `-1` (default) then an upper bound is computed for the primes at which the subgroup may not be satura... | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
side-effect. It proves that the inde of the points in their | side-effect. It proves that the index of the points in their | def saturate(self, max_prime=-1, odd_primes_only=False): r""" Saturate this subgroup of the Mordell-Weil group. INPUT: - ``max_prime`` (int, default -1) -- saturation is performed for all primes up to `max_prime`. If `-1` (default) then an upper bound is computed for the primes at which the subgroup may not be satura... | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
by reducing the poits modulo all primes of good reduction up | by reducing the points modulo all primes of good reduction up | def saturate(self, max_prime=-1, odd_primes_only=False): r""" Saturate this subgroup of the Mordell-Weil group. INPUT: - ``max_prime`` (int, default -1) -- saturation is performed for all primes up to `max_prime`. If `-1` (default) then an upper bound is computed for the primes at which the subgroup may not be satura... | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
On 32-bit machines, this MUST be < 21.48 else $\exp(h_lim)>2^31$ and overflows. On 64-bit machines it must be at most 43.668. However, this bound is a logarithic | On 32-bit machines, this *must* be < 21.48 else `\exp(h_{\text{lim}}) > 2^{31}` and overflows. On 64-bit machines, it must be *at most* 43.668. However, this bound is a logarithmic | def search(self, height_limit=18, verbose=False): r""" Search for new points, and add them to this subgroup of the Mordell-Weil group. | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
Subgroup of Mordell Weil group: [[1:-1:1], [-2:3:1], [-14:25:8]] In the next example, a serach bound of 12 is needed to find a non-torsin point:: | Subgroup of Mordell-Weil group: [[1:-1:1], [-2:3:1], [-14:25:8]] In the next example, a search bound of 12 is needed to find a non-torsion point:: | def search(self, height_limit=18, verbose=False): r""" Search for new points, and add them to this subgroup of the Mordell-Weil group. | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
Subgroup of Mordell Weil group: [] | Subgroup of Mordell-Weil group: [] | def search(self, height_limit=18, verbose=False): r""" Search for new points, and add them to this subgroup of the Mordell-Weil group. | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
Subgroup of Mordell Weil group: [[4413270:10381877:27000]] | Subgroup of Mordell-Weil group: [[4413270:10381877:27000]] | def search(self, height_limit=18, verbose=False): r""" Search for new points, and add them to this subgroup of the Mordell-Weil group. | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
if height_limit >= 21.4: | if height_limit >= 21.4: | def search(self, height_limit=18, verbose=False): r""" Search for new points, and add them to this subgroup of the Mordell-Weil group. | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
verbose == bool(verbose) | verbose = bool(verbose) | def search(self, height_limit=18, verbose=False): r""" Search for new points, and add them to this subgroup of the Mordell-Weil group. | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
(list) a list of lists of length 3, each holding the | (list) A list of lists of length 3, each holding the | def points(self): """ Return a list of the generating points in this Mordell-Weil group. | 1cc4edc9aafb88d044d853b60331d1432546cf60 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/1cc4edc9aafb88d044d853b60331d1432546cf60/interface.py |
except: | except MIPSolverException: | def edge_coloring(g, value_only=False, vizing=False, hex_colors=False, log=0): r""" Properly colors the edges of a graph. See the URL http://en.wikipedia.org/wiki/Edge_coloring for further details on edge coloring. INPUT: - ``g`` -- a graph. - ``value_only`` -- (default: ``False``): - When set to ``True``, only the... | f8a4840b981a88293ff407d02e2b06df77e6383d /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/f8a4840b981a88293ff407d02e2b06df77e6383d/graph_coloring.py |
sage: c = c2 = 1 | sage: c == c2 calling __eq__ defined in Metaclass True | def metaclass(name, bases): """ Creates a new class in this metaclass INPUT:: - name: a string - bases: a tuple of classes EXAMPLES:: sage: from sage.misc.test_class_pickling import metaclass, bar sage: c = metaclass("foo2", (object, bar,)) constructing class sage: c <class 'sage.misc.test_class_pickling.foo2'> sage... | c91cf7cfa8df95b89a0ac34a7f63efd9ddc6cb69 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/c91cf7cfa8df95b89a0ac34a7f63efd9ddc6cb69/test_class_pickling.py |
Some cases that check on the negative twists:: | Some harder cases fail:: | def _find_scaling_L_ratio(self): r""" This function is use to set ``_scaling``, the factor used to adjust the scalar multiple of the modular symbol. If `[0]`, the modular symbol evaluated at 0, is non-zero, we can just scale it with respect to the approximation of the L-value. It is known that the quotient is a rationa... | ccd9827ad09ec152373415abbdaf9c16b21128bb /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/ccd9827ad09ec152373415abbdaf9c16b21128bb/ell_modular_symbols.py |
-2 | 1 | def _find_scaling_L_ratio(self): r""" This function is use to set ``_scaling``, the factor used to adjust the scalar multiple of the modular symbol. If `[0]`, the modular symbol evaluated at 0, is non-zero, we can just scale it with respect to the approximation of the L-value. It is known that the quotient is a rationa... | ccd9827ad09ec152373415abbdaf9c16b21128bb /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/ccd9827ad09ec152373415abbdaf9c16b21128bb/ell_modular_symbols.py |
Dlist = [5,8,12,13,17,21,24,28,29] | Dlist = [5,8,12,13,17,21,24,28,29, 33, 37, 40, 41, 44, 53, 56, 57, 60, 61, 65, 69, 73, 76, 77, 85, 88, 89, 92, 93, 97] | def _find_scaling_L_ratio(self): r""" This function is use to set ``_scaling``, the factor used to adjust the scalar multiple of the modular symbol. If `[0]`, the modular symbol evaluated at 0, is non-zero, we can just scale it with respect to the approximation of the L-value. It is known that the quotient is a rationa... | ccd9827ad09ec152373415abbdaf9c16b21128bb /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/ccd9827ad09ec152373415abbdaf9c16b21128bb/ell_modular_symbols.py |
while j < 9 and at0 == 0 : | while j < 30 and at0 == 0 : | def _find_scaling_L_ratio(self): r""" This function is use to set ``_scaling``, the factor used to adjust the scalar multiple of the modular symbol. If `[0]`, the modular symbol evaluated at 0, is non-zero, we can just scale it with respect to the approximation of the L-value. It is known that the quotient is a rationa... | ccd9827ad09ec152373415abbdaf9c16b21128bb /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/ccd9827ad09ec152373415abbdaf9c16b21128bb/ell_modular_symbols.py |
chtw = True for ell in prime_divisors(D): chtw = chtw and ( valuation(E.conductor(),ell)<= valuation(D,ell) ) if chtw : | if all( valuation(E.conductor(),ell)<= valuation(D,ell) for ell in prime_divisors(D) ) : | def _find_scaling_L_ratio(self): r""" This function is use to set ``_scaling``, the factor used to adjust the scalar multiple of the modular symbol. If `[0]`, the modular symbol evaluated at 0, is non-zero, we can just scale it with respect to the approximation of the L-value. It is known that the quotient is a rationa... | ccd9827ad09ec152373415abbdaf9c16b21128bb /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/ccd9827ad09ec152373415abbdaf9c16b21128bb/ell_modular_symbols.py |
if j == 9 and at0 == 0: sc = 1 if not self._use_eclib : msn = ModularSymbolSage(self._E,sign = -1,normalize = "L_ratio") sc = msn._scaling if sc == 0 or self._use_eclib : self.__scale_by_periods_only__() else : self._scaling = sc | if j == 30 and at0 == 0: self.__scale_by_periods_only__() | def _find_scaling_L_ratio(self): r""" This function is use to set ``_scaling``, the factor used to adjust the scalar multiple of the modular symbol. If `[0]`, the modular symbol evaluated at 0, is non-zero, we can just scale it with respect to the approximation of the L-value. It is known that the quotient is a rationa... | ccd9827ad09ec152373415abbdaf9c16b21128bb /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/ccd9827ad09ec152373415abbdaf9c16b21128bb/ell_modular_symbols.py |
verbose('scale modular symbols by %s'%(l1/at0)) | verbose('scale modular symbols by %s found at D=%s '%(l1/at0,D), level=2) | def _find_scaling_L_ratio(self): r""" This function is use to set ``_scaling``, the factor used to adjust the scalar multiple of the modular symbol. If `[0]`, the modular symbol evaluated at 0, is non-zero, we can just scale it with respect to the approximation of the L-value. It is known that the quotient is a rationa... | ccd9827ad09ec152373415abbdaf9c16b21128bb /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/ccd9827ad09ec152373415abbdaf9c16b21128bb/ell_modular_symbols.py |
Dlist = [-3,-4,-7,-8,-11,-15,-19,-20,-23,-24] | Dlist = [-3,-4,-7,-8,-11,-15,-19,-20,-23,-24, -31, -35, -39, -40, -43, -47, -51, -52, -55, -56, -59, -67, -68, -71, -79, -83, -84, -87, -88, -91] | def _find_scaling_L_ratio(self): r""" This function is use to set ``_scaling``, the factor used to adjust the scalar multiple of the modular symbol. If `[0]`, the modular symbol evaluated at 0, is non-zero, we can just scale it with respect to the approximation of the L-value. It is known that the quotient is a rationa... | ccd9827ad09ec152373415abbdaf9c16b21128bb /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/ccd9827ad09ec152373415abbdaf9c16b21128bb/ell_modular_symbols.py |
while j < 9 and at0 == 0 : | while j < 30 and at0 == 0 : | def _find_scaling_L_ratio(self): r""" This function is use to set ``_scaling``, the factor used to adjust the scalar multiple of the modular symbol. If `[0]`, the modular symbol evaluated at 0, is non-zero, we can just scale it with respect to the approximation of the L-value. It is known that the quotient is a rationa... | ccd9827ad09ec152373415abbdaf9c16b21128bb /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/ccd9827ad09ec152373415abbdaf9c16b21128bb/ell_modular_symbols.py |
chtw = True for ell in prime_divisors(D): chtw = chtw and ( valuation(E.conductor(),ell)<= valuation(D,ell) ) if chtw : | if all( valuation(E.conductor(),ell)<= valuation(D,ell) for ell in prime_divisors(D) ) : | def _find_scaling_L_ratio(self): r""" This function is use to set ``_scaling``, the factor used to adjust the scalar multiple of the modular symbol. If `[0]`, the modular symbol evaluated at 0, is non-zero, we can just scale it with respect to the approximation of the L-value. It is known that the quotient is a rationa... | ccd9827ad09ec152373415abbdaf9c16b21128bb /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/ccd9827ad09ec152373415abbdaf9c16b21128bb/ell_modular_symbols.py |
if j == 9 and at0 == 0: | if j == 30 and at0 == 0: | def _find_scaling_L_ratio(self): r""" This function is use to set ``_scaling``, the factor used to adjust the scalar multiple of the modular symbol. If `[0]`, the modular symbol evaluated at 0, is non-zero, we can just scale it with respect to the approximation of the L-value. It is known that the quotient is a rationa... | ccd9827ad09ec152373415abbdaf9c16b21128bb /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/ccd9827ad09ec152373415abbdaf9c16b21128bb/ell_modular_symbols.py |
def plot(self, *args, **kwds): """ Plot the real points of an affine patch of this projective plane curve. | 60fa2d7563b965c543d95334de8974fbe21499e1 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/60fa2d7563b965c543d95334de8974fbe21499e1/projective_curve.py | ||
def rational_points(self, algorithm="enum", sort=True): | def rational_points_iterator(self): | def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration. | 60fa2d7563b965c543d95334de8974fbe21499e1 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/60fa2d7563b965c543d95334de8974fbe21499e1/projective_curve.py |
Return the rational points on this curve computed via enumeration. INPUT: - ``algorithm`` (string, default: 'enum') -- the algorithm to use. Currently this is ignored. - ``sort`` (boolean, default ``True``) -- whether the output points should be sorted. If False, the order of the output is non-deterministic. | Return a generator object for the rational points on this curve. INPUT: - ``self`` -- a projective curve | def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration. | 60fa2d7563b965c543d95334de8974fbe21499e1 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/60fa2d7563b965c543d95334de8974fbe21499e1/projective_curve.py |
A list of all the rational points on the curve defined over its base field, possibly sorted. .. note:: This is a slow Python-level implementation. EXAMPLES:: sage: F = GF(5) | A generator of all the rational points on the curve defined over its base field. EXAMPLE:: sage: F = GF(37) sage: P2.<X,Y,Z> = ProjectiveSpace(F,2) sage: C = Curve(X^7+Y*X*Z^5*55+Y^7*12) sage: len(list(C.rational_points_iterator())) 37 :: sage: F = GF(2) | def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration. | 60fa2d7563b965c543d95334de8974fbe21499e1 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/60fa2d7563b965c543d95334de8974fbe21499e1/projective_curve.py |
sage: C = Curve(X^3+Y^3-Z^3) sage: C.rational_points() [(0 : 1 : 1), (1 : 0 : 1), (2 : 2 : 1), (3 : 4 : 1), (4 : 1 : 0), (4 : 3 : 1)] sage: C.rational_points(sort=False) [(4 : 1 : 0), (1 : 0 : 1), (0 : 1 : 1), (2 : 2 : 1), (4 : 3 : 1), (3 : 4 : 1)] :: sage: F = GF(1009) sage: P2.<X,Y,Z> = ProjectiveSpace(F,2) sage: C... | sage: C = Curve(X*Y*Z) sage: a = C.rational_points_iterator() sage: a.next() (1 : 0 : 0) sage: a.next() (0 : 1 : 0) sage: a.next() (1 : 1 : 0) sage: a.next() (0 : 0 : 1) sage: a.next() (1 : 0 : 1) sage: a.next() (0 : 1 : 1) sage: a.next() Traceback (most recent call last): ... StopIteration | def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration. | 60fa2d7563b965c543d95334de8974fbe21499e1 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/60fa2d7563b965c543d95334de8974fbe21499e1/projective_curve.py |
sage: F = GF(2^6,'a') sage: P2.<X,Y,Z> = ProjectiveSpace(F,2) sage: C = Curve(X^5+11*X*Y*Z^3 + X^2*Y^3 - 13*Y^2*Z^3) sage: len(C.rational_points()) 104 :: sage: R.<x,y,z> = GF(2)[] sage: f = x^3*y + y^3*z + x*z^3 sage: C = Curve(f); pts = C.rational_points() sage: pts [(0 : 0 : 1), (0 : 1 : 0), (1 : 0 : 0)] | sage: F = GF(3^2,'a') sage: P2.<X,Y,Z> = ProjectiveSpace(F,2) sage: C = Curve(X^3+5*Y^2*Z-33*X*Y*X) sage: b = C.rational_points_iterator() sage: b.next() (0 : 1 : 0) sage: b.next() (0 : 0 : 1) sage: b.next() (2*a + 2 : 2*a : 1) sage: b.next() (2 : a + 1 : 1) sage: b.next() (a + 1 : a + 2 : 1) sage: b.next() (1 : 2 : 1... | def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration. | 60fa2d7563b965c543d95334de8974fbe21499e1 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/60fa2d7563b965c543d95334de8974fbe21499e1/projective_curve.py |
points = [] | def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration. | 60fa2d7563b965c543d95334de8974fbe21499e1 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/60fa2d7563b965c543d95334de8974fbe21499e1/projective_curve.py | |
points.append(self.point([one,zero,zero])) except TypeError: | t = self.point([one,zero,zero]) yield(t) except TypeError: | def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration. | 60fa2d7563b965c543d95334de8974fbe21499e1 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/60fa2d7563b965c543d95334de8974fbe21499e1/projective_curve.py |
for x in R(g((X,one,zero))).roots(multiplicities=False): points.append(self.point([x,one,zero])) | g10 = R(g(X,one,zero)) if g10.is_zero(): for x in K: yield(self.point([x,one,zero])) else: for x in g10.roots(multiplicities=False): yield(self.point([x,one,zero])) | def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration. | 60fa2d7563b965c543d95334de8974fbe21499e1 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/60fa2d7563b965c543d95334de8974fbe21499e1/projective_curve.py |
for x in R(g((X,y,one))).roots(multiplicities=False): points.append(self.point([x,y,one])) | gy1 = R(g(X,y,one)) if gy1.is_zero(): for x in K: yield(self.point([x,y,one])) else: for x in gy1.roots(multiplicities=False): yield(self.point([x,y,one])) def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration. INPUT: - ``algorithm`` (string, ... | def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration. | 60fa2d7563b965c543d95334de8974fbe21499e1 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/60fa2d7563b965c543d95334de8974fbe21499e1/projective_curve.py |
def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration. | 60fa2d7563b965c543d95334de8974fbe21499e1 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/60fa2d7563b965c543d95334de8974fbe21499e1/projective_curve.py | ||
def tamagawa_product(self): | def tamagawa_product_bsd(self): | def tamagawa_product(self): r""" Given an elliptic curve `E` over a number field `K`, this function returns the integer `C(E/K)` that appears in the Birch and Swinnerton-Dyer conjecture accounting for the local information at finite places. If the model is a global minimal model then `C(E/K)` is simply the product of t... | 3a6cb29f6359bec17483b90200da8ff0007f845f /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/3a6cb29f6359bec17483b90200da8ff0007f845f/ell_number_field.py |
sage: E.tamagawa_product() | sage: E.tamagawa_product_bsd() | def tamagawa_product(self): r""" Given an elliptic curve `E` over a number field `K`, this function returns the integer `C(E/K)` that appears in the Birch and Swinnerton-Dyer conjecture accounting for the local information at finite places. If the model is a global minimal model then `C(E/K)` is simply the product of t... | 3a6cb29f6359bec17483b90200da8ff0007f845f /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/3a6cb29f6359bec17483b90200da8ff0007f845f/ell_number_field.py |
sage: E.tamagawa_product() | sage: E.tamagawa_product_bsd() | def tamagawa_product(self): r""" Given an elliptic curve `E` over a number field `K`, this function returns the integer `C(E/K)` that appears in the Birch and Swinnerton-Dyer conjecture accounting for the local information at finite places. If the model is a global minimal model then `C(E/K)` is simply the product of t... | 3a6cb29f6359bec17483b90200da8ff0007f845f /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/3a6cb29f6359bec17483b90200da8ff0007f845f/ell_number_field.py |
sage: E.tamagawa_product() | sage: E.tamagawa_product_bsd() | def tamagawa_product(self): r""" Given an elliptic curve `E` over a number field `K`, this function returns the integer `C(E/K)` that appears in the Birch and Swinnerton-Dyer conjecture accounting for the local information at finite places. If the model is a global minimal model then `C(E/K)` is simply the product of t... | 3a6cb29f6359bec17483b90200da8ff0007f845f /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/3a6cb29f6359bec17483b90200da8ff0007f845f/ell_number_field.py |
sage: E2 Elliptic Curve defined by y^2 + a*x*y + (a+1)*y = x^3 + (a+1)*x^2 + (12289755603565800754*a-75759141535687466985)*x + (51556320144761417221790307379*a-317814501841918807353201512829) over Number Field in a with defining polynomial x^2 - 38 | sage: E2 Elliptic Curve defined by y^2 + a*x*y + (a+1)*y = x^3 + (a+1)*x^2 + (368258520200522046806318444*a-2270097978636731786720859345)*x + (8456608930173478039472018047583706316424*a-52130038506793883217874390501829588391299) over Number Field in a with defining polynomial x^2 - 38 | def global_minimal_model(self, proof = None): r""" Returns a model of self that is integral, minimal at all primes. .. note:: | 3a6cb29f6359bec17483b90200da8ff0007f845f /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/3a6cb29f6359bec17483b90200da8ff0007f845f/ell_number_field.py |
This function is an automatic generated pexpect wrapper around the Singular | This function is an automatically generated pexpect wrapper around the Singular | def _sage_doc_(self): """ EXAMPLES:: sage: 'groebner' in singular.groebner._sage_doc_() True """ if not nodes: generate_docstring_dictionary() | a11bd733b420bf5cc9fbf35ee640484f1eaff9f4 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/a11bd733b420bf5cc9fbf35ee640484f1eaff9f4/singular.py |
""" | r""" | def eigenvalues(self,extend=True): """ Returns a list with the eigenvalues of the endomorphism of vector spaces. If the option extend is set to True (default), then eigenvalues in extensions of the base field are considered. EXAMPLES: We compute the eigenvalues of an endomorphism of QQ^3:: sage: V=QQ^3 sage: H=V.en... | 894cddf7d59c87fdda637de94cd68aea251b5e24 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/894cddf7d59c87fdda637de94cd68aea251b5e24/free_module_morphism.py |
If the option extend is set to True (default), then eigenvalues in extensions of the base field are considered. | INPUT: - ``extend`` -- boolean (default: True) decides if base field extensions should be considered or not. | def eigenvalues(self,extend=True): """ Returns a list with the eigenvalues of the endomorphism of vector spaces. If the option extend is set to True (default), then eigenvalues in extensions of the base field are considered. EXAMPLES: We compute the eigenvalues of an endomorphism of QQ^3:: sage: V=QQ^3 sage: H=V.en... | 894cddf7d59c87fdda637de94cd68aea251b5e24 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/894cddf7d59c87fdda637de94cd68aea251b5e24/free_module_morphism.py |
We compute the eigenvalues of an endomorphism of QQ^3:: | We compute the eigenvalues of an endomorphism of `\QQ^3`:: | def eigenvalues(self,extend=True): """ Returns a list with the eigenvalues of the endomorphism of vector spaces. If the option extend is set to True (default), then eigenvalues in extensions of the base field are considered. EXAMPLES: We compute the eigenvalues of an endomorphism of QQ^3:: sage: V=QQ^3 sage: H=V.en... | 894cddf7d59c87fdda637de94cd68aea251b5e24 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/894cddf7d59c87fdda637de94cd68aea251b5e24/free_module_morphism.py |
Note the effect of the extend option:: | Note the effect of the ``extend`` option:: | def eigenvalues(self,extend=True): """ Returns a list with the eigenvalues of the endomorphism of vector spaces. If the option extend is set to True (default), then eigenvalues in extensions of the base field are considered. EXAMPLES: We compute the eigenvalues of an endomorphism of QQ^3:: sage: V=QQ^3 sage: H=V.en... | 894cddf7d59c87fdda637de94cd68aea251b5e24 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/894cddf7d59c87fdda637de94cd68aea251b5e24/free_module_morphism.py |
def eigenvalues(self,extend=True): """ Returns a list with the eigenvalues of the endomorphism of vector spaces. If the option extend is set to True (default), then eigenvalues in extensions of the base field are considered. EXAMPLES: We compute the eigenvalues of an endomorphism of QQ^3:: sage: V=QQ^3 sage: H=V.en... | 894cddf7d59c87fdda637de94cd68aea251b5e24 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/894cddf7d59c87fdda637de94cd68aea251b5e24/free_module_morphism.py | ||
- extend (True) decides if base field extensions should be considered or not. | - ``extend`` -- boolean (default: True) decides if base field extensions should be considered or not. | def eigenvectors(self,extend=True): """ Computes the subspace of eigenvectors of a given eigenvalue. INPUT: - extend (True) decides if base field extensions should be considered or not. OUTPUT: A sequence of tuples. Each tuple contains an eigenvalue, a list with a basis of the corresponding subspace of eigenvectors... | 894cddf7d59c87fdda637de94cd68aea251b5e24 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/894cddf7d59c87fdda637de94cd68aea251b5e24/free_module_morphism.py |
A sequence of tuples. Each tuple contains an eigenvalue, a list with a basis of the corresponding subspace of eigenvectors, and the algebraic multiplicity of the eigenvalue. | A sequence of tuples. Each tuple contains an eigenvalue, a sequence with a basis of the corresponding subspace of eigenvectors, and the algebraic multiplicity of the eigenvalue. | def eigenvectors(self,extend=True): """ Computes the subspace of eigenvectors of a given eigenvalue. INPUT: - extend (True) decides if base field extensions should be considered or not. OUTPUT: A sequence of tuples. Each tuple contains an eigenvalue, a list with a basis of the corresponding subspace of eigenvectors... | 894cddf7d59c87fdda637de94cd68aea251b5e24 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/894cddf7d59c87fdda637de94cd68aea251b5e24/free_module_morphism.py |
[(3, [(0, 0, 1, -6/7)], 1), (-1*I, [(1, 1*I, 0, -0.571428571428572? + 2.428571428571429?*I)], 1), (1*I, [(1, -1*I, 0, -0.571428571428572? - 2.428571428571429?*I)], 1)] | [(3, [ (0, 0, 1, -6/7) ], 1), (-1*I, [ (1, 1*I, 0, -0.571428571428572? + 2.428571428571429?*I) ], 1), (1*I, [ (1, -1*I, 0, -0.571428571428572? - 2.428571428571429?*I) ], 1)] | def eigenvectors(self,extend=True): """ Computes the subspace of eigenvectors of a given eigenvalue. INPUT: - extend (True) decides if base field extensions should be considered or not. OUTPUT: A sequence of tuples. Each tuple contains an eigenvalue, a list with a basis of the corresponding subspace of eigenvectors... | 894cddf7d59c87fdda637de94cd68aea251b5e24 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/894cddf7d59c87fdda637de94cd68aea251b5e24/free_module_morphism.py |
[(3, [(0, 0, 1, -6/7)], 1)] | [(3, [ (0, 0, 1, -6/7) ], 1)] | def eigenvectors(self,extend=True): """ Computes the subspace of eigenvectors of a given eigenvalue. INPUT: - extend (True) decides if base field extensions should be considered or not. OUTPUT: A sequence of tuples. Each tuple contains an eigenvalue, a list with a basis of the corresponding subspace of eigenvectors... | 894cddf7d59c87fdda637de94cd68aea251b5e24 /local1/tlutelli/issta_data/temp/all_python//python/2010_temp/2010/9890/894cddf7d59c87fdda637de94cd68aea251b5e24/free_module_morphism.py |
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