Lower bound on the rank of the elliptic curve $y^{2} = x^{3} + A x^{2} + B x$

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?

-