Opened 12 years ago

Last modified 9 years ago

#10735 closed defect

Simon 2-descent may not check for solubility at archimedean places. — at Initial Version

Reported by: Jamie Weigandt Owned by: John Cremona
Priority: minor Milestone: sage-6.2
Component: elliptic curves Keywords: simon_two_descent
Cc: John Cremona, William Stein, Robert Miller Merged in:
Authors: Reviewers:
Report Upstream: Reported upstream. No feedback yet. Work issues:
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Description

Given an elliptic curve E the method E.simon_two_descent() returns an ordered triple. This consists of a lower bound on the Mordell-Weil rank of E, an integer which is supposed to be the F_2 dimension of the 2-Selmer group of E, and list of points, generating the part of the Mordell-Weil group that has been found.

Sometimes the second entry is larger than the actual 2-Selmer rank as computed by mwrank, and predicted by BSD. The first curve I know of for which this happens is the elliptic curve '438e1' from Cremona's tables.

This curve definitely possesses 2-covers which are everywhere locally soluble EXCEPT at that infinite place. These probably explain the discrepancy.

sage: E=EllipticCurve('438e1')
sage: E.simon_two_descent()
(0, 3, [(13 : -7 : 1)])
sage: E.selmer_rank() #uses mwrank
1
sage: E.sha().an()
1

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