Mahler measure of a non-reciprocal family of elliptic curves

Abstract

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 $\mathrm{m}(Q_\alpha)$. The zero loci of Qα generically define elliptic curves Eα, 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 α in this interval, we establish a hypergeometric formula for the (modified) Mahler measure of Qα, denoted by $\tilde{n}(\alpha).$ This formula coincides, up to a constant factor, with the known formula for $\mathrm{m}(Q_\alpha)$ with $|\alpha|$ sufficiently large. In addition, we verify numerically that if α3 is an integer, then $\tilde{n}(\alpha)$ is a rational multiple of $L^{\prime}(E_\alpha,0)$. A proof of this identity for α = 2, which corresponds to an elliptic curve of conductor 19, is given.