Mahler measure of a non-reciprocal family of elliptic curves

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