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

At present, 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 with rank as
large as
, where is the conductor of .
Might this lead to the lower bound

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

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