  We study infinite families of quadratic and cubic twists of the elliptic
curve $E = X_0(27)$. For the family of quadratic twists, we establish a lower
bound for the $2$-adic valuation of the algebraic part of the value of the
complex $L$-series at $s=1$, and, for the family of cubic twists, we establish
a 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 the
celebrated conjecture of Birch and Swinnerton-Dyer.
