#9342: rank method for elliptic curves over number fields
-------------------------------+--------------------------------------------
Reporter: ljpk | Owner: cremona
Type: enhancement | Status: new
Priority: minor | Milestone: sage-4.5
Component: elliptic curves | Keywords:
Author: ljpk | Upstream: N/A
Reviewer: | Merged:
Work_issues: |
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Here is a method to compute the rank of elliptic curves over number fields
using Simon 2-descent.
{{{
def rank(self,verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20,
limbigprime=30):
r"""
This computes the rank of elliptic curves over number fields using
Simon's algorithm for two-descent. If the upper and lower bounds given are
the same, then we return the rank, otherwise we return the upper and lower
bounds.
INPUT:
The parameters are those used by simon_two_descent, and are documented
there.
OUTPUT:
If the upper and lower bounds given by Simon two-descent are the same,
then the rank has been uniquely identified and we return this. Otherwise,
we return the upper and lower bounds with a warning that these are not the
same.
Note: For non-quadratic number fields, this code does return, but it
takes a long time.
EXAMPLES:
sage: K.<a> = NumberField(x^2 + 23, 'a')
sage: E = EllipticCurve(K, '37')
sage: E == loads(dumps(E))
True
sage: E.rank()
2
Here is a curve with two-torsion, so here the algorithm gives bounds on
the rank:
sage: Qrt5.<rt5>=NumberField(x^2-5)
sage: E=EllipticCurve([0,5-rt5,0,rt5,0])
sage: E.rank()
Lower and upper bounds differ!
Lower bound being returned
1
IMPLEMENTATION:
Uses Denis Simon's GP/PARI scripts from
\url{http://www.math.unicaen.fr/~simon/}.
"""
simon_output=self.simon_two_descent(verbose=verbose,lim1=lim1,lim3=lim3,limtriv=limtriv,maxprob=maxprob,limbigprime=limbigprime)
if simon_output[0]==simon_output[1]:
return simon_output[0]
print "Lower and upper bounds differ!"
print "Lower bound being returned"
return simon_output[0]
}}}
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/9342>
Sage <http://www.sagemath.org>
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