  Given an elliptic curve $E$ over $\mathbb{Q}$, a celebrated conjecture of
Goldfeld asserts that a positive proportion of its quadratic twists should have
analytic rank 0 (resp. 1). We show this conjecture holds whenever $E$ has a
rational 3-isogeny. We also prove the analogous result for the sextic twists of
$j$-invariant 0 curves (Mordell curves). To prove these results, we establish a
general criterion for the non-triviality of the $p$-adic logarithm of Heegner
points at an Eisenstein prime $p$, in terms of the relative $p$-class numbers
of certain number fields and then apply this criterion to the special case
$p=3$. As a by-product, we also prove the 3-part of the Birch and
Swinnerton-Dyer conjecture for many elliptic curves of $j$-invariant 0.
