Opened 11 years ago

Last modified 7 years ago

## #11023 new enhancement

# analytic_rank() should set rank() and gens() when it returns 0 — at Initial Version

Reported by: | weigandt | Owned by: | weigandt |
---|---|---|---|

Priority: | major | Milestone: | sage-6.4 |

Component: | elliptic curves | Keywords: | rank, gens |

Cc: | cremona, was | Merged in: | |

Authors: | Reviewers: | ||

Report Upstream: | N/A | Work issues: | |

Branch: | Commit: | ||

Dependencies: | Stopgaps: |

### Description

If the analytic rank of an elliptic curve over QQ is computed to be 0, it is a theorem that the algebraic rank is 0, and we can set the generators to be []. This is not done.

sage: E=EllipticCurve([1, 0, 0, -1319539461660, -159402536950172400]) sage: E.analytic_rank() 0 sage: E.rank() Unable to compute the rank with certainty (lower bound=0). This could be because Sha(E/Q)[2] is nontrivial. Try calling something like two_descent(second_limit=13) on the curve then trying this command again. You could also try rank with only_use_mwrank=False. --------------------------------------------------------------------------- BOOM! RuntimeError: Rank not provably correct.

We could also set the rank to 1 if the analytic rank is provably 1, but it seems customary only to set the rank when we can also set the generators, so that shouldn't be done until improved Heegner point functionality is available.

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