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We study infinite families of quadratic and cubic twists of the ellipticcurve $E = X_0(27)$. For the family of quadratic twists, we establish a lowerbound for the $2$-adic valuation of the algebraic part of the value of thecomplex $L$-series at $s=1$, and, for the family of cubic twists, we establisha lower bound for the $3$-adic valuation of the algebraic part of the same$L$-value. We show that our lower bounds are precisely those predicted by thecelebrated conjecture of Birch and Swinnerton-Dyer.

On the $p$-part of the Birch-Swinnerton-Dyer conjecture for elliptic  curves with complex multiplication by the ring of integers of  $\mathbb{Q}(\sqrt{-3})$