Ranks of elliptic curves and the class number problem

Goldfeld [MR 56 #8529] showed that if there is an elliptic curve $E$ whose $L$-function vanishes to order $g\ge 2$ at the critical point, then we can obtain a lower bound for the class number:

$$h(D)\gg_E (\log D)^{g-2} .$$

At present, $g=3$ is the largest value for which this has been shown to hold. See the paper by Oesterlé [ MR 86k:11064] for a readable account with the explicit calculation of the relevant constants, and the survey by Goldfeld [ MR 86k:11065] for background information.

In the article on the maximal rank of an elliptic curve as a function of its conductor it is suggested that there are elliptic curves $E$ with rank as large as $\log N/\log\log N$, where $N$ is the conductor of $E$. Might this lead to the lower bound

$$h(D)\gg \frac{\sqrt D}{\log D} ?$$

Is it possible to use a family of elliptic curves of high rank to improve Goldfeld's result?

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