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What is the current record (and reference) for the highest analytic rank of an elliptic curve over $\mathbb{Q}$?

The highest algebraic rank is the Elkies curve with rank at least 28, but I cannot determine whether its analytic rank has been computed with any certainty.

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    $\begingroup$ Only rank $3$ after Gross--Zagier. Anything higher would have nice consequences for lower bounds for class numbers (improving the power of log in the effective bound). $\endgroup$
    – Lucia
    Commented Nov 9, 2018 at 17:49
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    $\begingroup$ The problem is to prove that a higher derivative of the $L$-function, evaluated at 1, indeed vanishes, if it appears to do so numerically. This can be done for the value itself and the first derivative, but so far not for higher ones. $\endgroup$
    – Michael Stoll
    Commented Nov 9, 2018 at 20:29
  • $\begingroup$ @MichaelStoll Ah, I hadn't realized that one could prove that the first derivative vanishes, but not the second. (I had thought one couldn't even prove the first derivative vanished). Is there an intuitive description of the block? Stated differently, I'd thought that our lack of ability to understand Sha would have made many generic approaches fail. $\endgroup$
    – davidlowryduda
    Commented Nov 9, 2018 at 22:16
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    $\begingroup$ As stated, the record is unbounded, if you allow (number) fields other than Q... $\endgroup$
    – literature-searcher
    Commented Nov 10, 2018 at 0:07
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    $\begingroup$ @literature-searcher Do you have a reference for that? Unlike algebraic rank, it is not clear to me how passing to a curve over a nunber field would make analytic rank easier to bound. $\endgroup$
    – Wojowu
    Commented Nov 10, 2018 at 10:08

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