CM POINTS, CLASS NUMBERS, AND THE MAHLER MEASURES OF x3 + y3+1- kxy

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 complex multiplication points with class numbers 5 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 byproducts, some conjectural identities of Samart are confirmed, one of them involves the modified Mahler measure n(k) introduced by Samart recently. For k = (3)root 729 +/- 405 root 3, we also prove an equality that expresses a 2 x 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 Q(root 3).