14.1.1.2 mwrank_MordellWeil Objects

class mwrank_MordellWeil
The mwrank_MordellWeil class represents a subgroup of a Mordell-Weil group. Use this class to saturate a specified list of points on an mwrank_EllipticCurve, or to search for points up to some bound.
mwrank_MordellWeil( curve, [verbose=999], [pp=1], [maxr=True])

Create a mwrank_MordellWeil instance.

INPUT:
    curve -- \class{mwrank_EllipticCurve} instance
    verbose -- bool
    pp -- int
    maxr -- int

Instances of class mwrank_MordellWeil have the following methods (in addition to inherited methods and special methods):

points,$  $ process,$  $ rank,$  $ regulator,$  $ saturate,$  $ search

Further documentation:

points( )

Return a list of the generating points in this Mordell-Weil group.

process( v, [sat=0])

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), then saturate at all primes.

INPUT:

v - a point (3-tuple of ints), or a list of 3-tuples of integers, which define points on the curve.

sat - int, saturate at primes up to sat, or at all primes if sat=0.

rank( )

Return the rank of this subgroup of the Mordell-Weil group.

regulator( )

Return the regulator of the points in this subgroup of the Mordell-Weil group.

saturate( [max_prime=False], [odd_primes_only=-1])

Saturate this subgroup of the Mordell-Weil group.

INPUT:
    max_prime (int) -- (default: 97), saturation is performed
                       for all primes up to max_prime
                       
    odd_primes_only (bool) -- only do saturation at odd primes

OUTPUT:
    ok (bool) -- True if and only if the saturation
                 is provably correct at \emph{all} primes.
    index (int) -- The index of the group generated by
                   points in their saturation
    saturation (list) -- list of points that form
                         a basis for the saturation

Note: We emphasize that if this function returns True as the first return argument, then the points it found are saturated at all primes, i.e., saturating at the primes up to max_prime are sufficient to saturate at all primes. Note that the function might not have needed to saturate at all primes up to max_prime. It has worked out what prime you need to saturate up to, and that prime is $ \leq $ max_prime.

Note: Currently (July 2005), this does not remember the result of calling search. So calling search up to height 20 then calling saturate results in another search up to height 18.

search( [height_limit=False], [verbose=18])

Search for new points, and add them to this subgroup of the Mordell-Weil group.

INPUT:
    height_limit -- float (default: 18) search up to
                    this logarithmetic height.
           On 32-bit machines, h_lim MUST be < 21.48 else
           exp(h_lim)>2^31 and overflows.

Instances of class mwrank_MordellWeil also have the following special methods:

__repr__

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