  We study the Mahler measures of the polynomial family $Q_k(x,y) =
x^3+y^3+1-kxy$ using the method previously developed by the authors. An
algorithm is implemented to search for CM points with class numbers $\leqslant
3$, we employ these points to derive interesting formulas that link the Mahler
measures of $Q_k(x,y)$ to $L$-values of modular forms. As by-products, some
conjectural identities of Samart are confirmed, one of them involves the
modified Mahler measure $\tilde{n}(k)$ introduced by Samart recently. For
$k=\sqrt[3]{729\pm405\sqrt{3}}$, we also prove an equality that expresses a
$2\times 2$ determinant with entries the Mahler measures of $Q_k(x,y)$ as some
multiple of the $L$-value of two isogenous elliptic curves over
$\mathbb{Q}(\sqrt{3})$.
