Mahler measure of a non-reciprocal family of elliptic curves

Abstract

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 $\tilde{n}(\alpha).$ This formula coincides, up to a constant factor, with the known formula for $m(Q_\alpha)$ with $|\alpha|$ sufficiently large. In addition, we verify numerically that if $\alpha^3$ is an integer, then $\tilde{n}(\alpha)$ is a rational multiple of $L'(E_\alpha,0)$. A proof of this identity for $\alpha=2$, which is corresponding to an elliptic curve of conductor $19$, is given.