Mahler measure of a non-reciprocal family of elliptic curves

Abstract In this article, we study the logarithmic Mahler measure of one-parameter family $$Q_\alpha=y^2+(x^2-\alpha x)y+x,$$ denoted by $\mathrm{m}(Q_\alpha)$. The zero loci Qα generically define elliptic curves Eα, which are 3-isogenous to Hessian curves. We particularly interested in case $\alpha\in (-1,3)$, has not been considered literature due certain subtleties. For α interval, establish a hypergeometric formula for (modified) Qα, $\tilde{n}(\alpha).$ This coincides, up constant factor, with known $\mathrm{m}(Q_\alpha)$ $|\alpha|$ sufficiently large. addition, verify numerically that if α3 is an integer, then $\tilde{n}(\alpha)$ rational multiple $L^{\prime}(E_\alpha,0)$. A proof identity = 2, corresponds curve conductor 19, given.