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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.
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.
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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.
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.
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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.
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.
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def rank(self): """ Returns the rank of this curve, computed using 2-descent.
def rank(self): """ Returns the rank of this curve, computed using 2-descent.
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def rank(self): """ Returns the rank of this curve, computed using 2-descent.
def rank(self): """ Returns the rank of this curve, computed using 2-descent.
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def rank(self): """ Returns the rank of this curve, computed using 2-descent.
def rank(self): """ Returns the rank of this curve, computed using 2-descent.
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def rank_bound(self): """ Returns an upper bound for the rank of this curve, computed using 2-descent.
def rank_bound(self): """ Returns an upper bound for the rank of this curve, computed using 2-descent.
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def rank_bound(self): """ Returns an upper bound for the rank of this curve, computed using 2-descent.
def rank_bound(self): """ Returns an upper bound for the rank of this curve, computed using 2-descent.
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def rank_bound(self): """ Returns an upper bound for the rank of this curve, computed using 2-descent.
def rank_bound(self): """ Returns an upper bound for the rank of this curve, computed using 2-descent.
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def selmer_rank(self): r""" Returns the rank of the 2-Selmer group of the curve. EXAMPLES:
def selmer_rank(self): r""" Returns the rank of the 2-Selmer group of the curve. EXAMPLES:
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def selmer_rank(self): r""" Returns the rank of the 2-Selmer group of the curve. EXAMPLES:
def selmer_rank(self): r""" Returns the rank of the 2-Selmer group of the curve. EXAMPLES:
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def saturate(self, bound=-1): """ Compute the saturation of the Mordell-Weil group at all primes up to bound.
def saturate(self, bound=-1): """ Compute the saturation of the Mordell-Weil group at all primes up to bound.
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def saturate(self, bound=-1): """ Compute the saturation of the Mordell-Weil group at all primes up to bound.
def saturate(self, bound=-1): """ Compute the saturation of the Mordell-Weil group at all primes up to bound.
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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.
def certain(self): r""" 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()` using the default parameters.
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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.
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.
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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.
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.
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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.
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.
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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 $P$.
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 $P$.
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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 $P$.
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 $P$.
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def __init__(self, curve, verbose=True, pp=1, maxr=999): r""" Constructor for the :class:`mwrank_MordellWeil` class.
def __init__(self, curve, verbose=True, pp=1, maxr=999): r""" Constructor for the :class:`mwrank_MordellWeil` class.
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def __init__(self, curve, verbose=True, pp=1, maxr=999): r""" Constructor for the :class:`mwrank_MordellWeil` class.
def __init__(self, curve, verbose=True, pp=1, maxr=999): r""" Constructor for the :class:`mwrank_MordellWeil` class.
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def __repr__(self): r""" String representation of this Mordell-Weil subgroup.
def __repr__(self): r""" String representation of this Mordell-Weil subgroup.
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def __repr__(self): r""" String representation of this Mordell-Weil subgroup.
def __repr__(self): r""" String representation of this Mordell-Weil subgroup.
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def process(self, v, sat=0): """ This function allows one to add points to a mwrank_MordellWeil object.
def process(self, v, sat=0): """ This function allows one to add points to a mwrank_MordellWeil object.
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def process(self, v, sat=0): """ This function allows one to add points to a mwrank_MordellWeil object.
def process(self, v, sat=0): """ This function allows one to add points to a mwrank_MordellWeil object.
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def process(self, v, sat=0): """ This function allows one to add points to a mwrank_MordellWeil object.
def process(self, v, sat=0): """ This function allows one to add points to a mwrank_MordellWeil object.
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def process(self, v, sat=0): """ This function allows one to add points to a mwrank_MordellWeil object.
def process(self, v, sat=0): """ This function allows one to add points to a mwrank_MordellWeil object.
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def process(self, v, sat=0): """ This function allows one to add points to a mwrank_MordellWeil object.
def process(self, v, sat=0): """ This function allows one to add points to a mwrank_MordellWeil object.
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def process(self, v, sat=0): """ This function allows one to add points to a mwrank_MordellWeil object.
def process(self, v, sat=0): """ This function allows one to add points to a mwrank_MordellWeil object.
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def regulator(self): """ Return the regulator of the points in this subgroup of the Mordell-Weil group.
def regulator(self): """ Return the regulator of the points in this subgroup of the Mordell-Weil group.
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def regulator(self): """ Return the regulator of the points in this subgroup of the Mordell-Weil group.
def regulator(self): """ Return the regulator of the points in this subgroup of the Mordell-Weil group.
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def rank(self): """ Return the rank of this subgroup of the Mordell-Weil group.
def rank(self): """ Return the rank of this subgroup of the Mordell-Weil group.
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def rank(self): """ Return the rank of this subgroup of the Mordell-Weil group.
def rank(self): """ Return the rank of this subgroup of the Mordell-Weil group.
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def rank(self): """ Return the rank of this subgroup of the Mordell-Weil group.
def rank(self): """ Return the rank of this subgroup of the Mordell-Weil group.
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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...
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` (the default), an upper bound is computed for the primes at which the subgroup may not be satu...
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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...
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...
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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...
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...
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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...
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...
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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...
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...
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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...
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...
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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...
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...
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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...
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...
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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...
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...
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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...
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...
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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...
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...
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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...
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...
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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...
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...
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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...
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...
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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...
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...
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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...
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...
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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...
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...
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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...
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...
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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...
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...
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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...
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...
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def search(self, height_limit=18, verbose=False): r""" Search for new points, and add them to this subgroup of the 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.
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def search(self, height_limit=18, verbose=False): r""" Search for new points, and add them to this subgroup of the 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.
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def search(self, height_limit=18, verbose=False): r""" Search for new points, and add them to this subgroup of the 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.
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def search(self, height_limit=18, verbose=False): r""" Search for new points, and add them to this subgroup of the 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.
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def search(self, height_limit=18, verbose=False): r""" Search for new points, and add them to this subgroup of the 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.
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def search(self, height_limit=18, verbose=False): r""" Search for new points, and add them to this subgroup of the 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.
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def points(self): """ Return a list of the generating points in this Mordell-Weil group.
def points(self): """ Return a list of the generating points in this Mordell-Weil group.
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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...
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...
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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...
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...
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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...
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...
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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...
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...
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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...
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...
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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...
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...
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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...
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...
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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...
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...
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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...
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...
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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...
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...
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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...
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...
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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...
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...
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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...
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...
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def plot(self, *args, **kwds): """ Plot the real points of an affine patch of this projective plane curve.
def plot(self, *args, **kwds): """ Plot the real points of an affine patch of this projective plane curve.
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def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration.
def rational_points_iterator(self): r""" Return the rational points on this curve computed via enumeration.
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def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration.
def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration.
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def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration.
def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration.
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def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration.
def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration.
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def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration.
def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration.
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def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration.
def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration.
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def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration.
def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration.
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def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration.
def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration.
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def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration.
def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration.
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def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration.
defrational_points(self,algorithm="enum",sort=True):r"""Returntherationalpointsonthiscurvecomputedviaenumeration.
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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...
def tamagawa_product_bsd(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 ...
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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...
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...
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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...
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...
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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...
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...
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def global_minimal_model(self, proof = None): r""" Returns a model of self that is integral, minimal at all primes. .. note::
def global_minimal_model(self, proof = None): r""" Returns a model of self that is integral, minimal at all primes. .. note::
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def _sage_doc_(self): """ EXAMPLES:: sage: 'groebner' in singular.groebner._sage_doc_() True """ if not nodes: generate_docstring_dictionary()
def _sage_doc_(self): """ EXAMPLES:: sage: 'groebner' in singular.groebner._sage_doc_() True """ if not nodes: generate_docstring_dictionary()
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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...
def eigenvalues(self,extend=True): r""" 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.e...
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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...
def eigenvalues(self,extend=True): """ Returns a list with the eigenvalues of the endomorphism of vector spaces. INPUT: - ``extend`` -- boolean (default: True) decides if base field extensions should be considered or not. EXAMPLES: We compute the eigenvalues of an endomorphism of QQ^3:: sage: V=QQ^3 sage: H=V.endo...
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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...
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...
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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...
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...
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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...
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...
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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...
def eigenvectors(self,extend=True): """ Computes the subspace of eigenvectors of a given eigenvalue. INPUT: - ``extend`` -- boolean (default: 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 ...
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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...
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 sequence with a basis of the corresponding subspace of eigenvec...
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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...
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...
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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...
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...
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