# Exact rank of Elkies curve

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?

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

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 $28$.