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 |
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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: | |

Branch: | Commit: | ||

Dependencies: | Stopgaps: |

### 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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