We study the Mahler measures of the polynomial family
Q
k
(
x
,
y
)
=
x
3
+
y
3
+
1
−
k
x
y
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 complex multiplication points with class numbers
⩽
3
\leqslant 3
, we employ these points to derive interesting formulas that link the Mahler measures of
Q
k
(
x
,
y
)
Q_k(x,y)
to
L
L
-values of modular forms. As by-products, some conjectural identities of Samart are confirmed, one of them involves the modified Mahler measure
n
~
(
k
)
\tilde {n}(k)
introduced by Samart recently. For
k
=
729
±
405
3
3
k=\sqrt [3]{729\pm 405\sqrt {3}}
, we also prove an equality that expresses a
2
×
2
2\times 2
determinant with entries the Mahler measures of
Q
k
(
x
,
y
)
Q_k(x,y)
as some multiple of the
L
L
-value of two isogenous elliptic curves over
Q
(
3
)
\mathbb {Q}(\sqrt {3})
.