MAHLER MEASURE OF A NON-RECIPROCAL FAMILY OF ELLIPTIC CURVES

In this article, we study the logarithmic Mahler measure of the one-parameter family [GRAPHICS] 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 is an element of(-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 (n) over tilde(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 (n) over tilde(alpha) is a rational multiple of L '(E-alpha, 0). A proof of this identity for alpha = 2, which corresponds to an elliptic curve of conductor 19, is given.