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A naïve question. We definitely know an elliptic curve of rank $28$ or more exists by Elkies but no one knows exactly what the rank is for this curve (and for similar examples given previously).

Could we not get a (conjectural) answer assuming BSD? Is this feasible computationally?

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    $\begingroup$ Elkies curve has conductor $N\sim 3.5\times 10^{141}$, and to compute the $L$-function is not really feasable. $\endgroup$ May 28, 2014 at 11:24

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If we assume the Birch-Swinnerton-Dyer conjecture (BSD) and the Generalized Riemman Hypothesis (GRH), then we know that the rank $r$ of Elkies curve satisfies $r=28$ or $r= 30$. However, we still don't know the exact rank, even if we assume such strong conjectures. Usually, if we accept BSD, and the rank is smaller than $4$, then we are able to use the $L$-function to get information about the rank. When the rank is larger than this, though, currently the best one can do is determine that the first $r$ derivatives of the $L$-function are very close to $0$, and the $(r + 1)$-st is not, which will provide a very good guess for the rank and a rigorous upper bound, assuming BSD. For references see the paper of Bober.

Edit: Keith Conrad has linked a paper by Zev Klagsbrun, Travis Sherman, James Weigandt, which shows the following for the Elkies curve $E_{28}$. The second part again refers to Bober:

Theorem: Assuming GRH, the Mordell-Weil group $E_{28}(\mathbb{Q})$ is isomorphic to $\mathbb{Z}^{28}$, and the analytic rank of $E_{28}$ over $\mathbb{Q}$ is at most $28$.

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  • $\begingroup$ So is the obstruction entirely computational? Could we expect a definite answer with better computing power or are the algorithms not guaranteed to give us answers to a high enough level of accuracy? $\endgroup$
    – fretty
    May 28, 2014 at 23:05
  • $\begingroup$ Well, good question. With the present ideas it is computationally not feasible (see the comment above). With a better idea this might be different. $\endgroup$ May 29, 2014 at 10:04
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    $\begingroup$ A result from 2016 reduces what has to be assumed in order to conclude the rank should be 28: arxiv.org/abs/1606.07178. $\endgroup$
    – KCd
    Apr 14, 2018 at 7:22

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