= Solution
For degree $3$ there are either three real embeddings or one real embedding and one complex pair. The respective <Minkowski bound for ideal classes> are
$$
\frac{2}{9}\sqrt{|\Delta^2(L)|},\qquad
\frac{8}{9\pi}\sqrt{|\Delta^2(L)|}.
$$
Under the given <discriminant> bound they are at most $\frac29\sqrt{45}<2$ and $\frac{8}{9\pi}\sqrt{45}<2$. Hence every <ideal class> contains an <integral ideal> with positive <integer> <norm> strictly below $2$, so with <norm> $1$. The only such <ideal> is $\mathcal O_L$. Therefore
$$
\boxed{h_L=1.}
$$
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