Minimal Mahler measure in cubic number fields

The minimal integral Mahler measure of a number field K, M(K), is the minimal Mahler measure of an integral generator of K. Upper and lower bounds, which depend on the discriminant and degree of K, are known. We show that for three natural families of cubics, the lower bounds are sharp with respect to its growth as a function of discriminant. We construct an algorithm to compute M(K) for all cubics with absolute value of the discriminant bounded by N and show the resulting data for N = 10, 000.