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We study the Mahler measures of the polynomial family Q k (x, y) = x 3 + y 3 + 1 − k x y Qₖ (x, y) = x³+y³+1-kxy using the method previously developed by the authors. An algorithm is implemented to search for complex multiplication points with class numbers ⩽ 3 3, we employ these points to derive interesting formulas that link the Mahler measures of Q k (x, y) Qₖ (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) n (k) introduced by Samart recently. For k = 729 ± 405 3 3 k= 3729 405 {3}, we also prove an equality that expresses a 2 × 2 2 2 determinant with entries the Mahler measures of Q k (x, y) Qₖ (x, y) as some multiple of the L L -value of two isogenous elliptic curves over Q (3) Q (3).
Tao et al. (Sat,) studied this question.