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A recent post on Math.SE discusses an upper bound on the rank in terms of the number of distinct prime divisors of $A$ and $B$, namely, \begin{align} r \leqslant \omega(A^{2} - 4 B) + \omega(B) - 1, \end{align} where $\omega(k) = \sum_{p \mid k} 1$ is the number of distinct prime divisor (arithmetic) function. (A proof of this fact can be found within links posted in the comments and answers.)

Is there a similar, non-trivial lower bound? That is, are there (possibly arithmetic) functions of $A$ and $B$ which provide a lower bound for the rank of the elliptic curve in question?

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I believe that such a lower bound is not known. Or if known, not effectively computable. Otherwise, I would think that more curves of high rank should be known than what is currently the case.

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