Summary: We investigate the Mordell constant of certain families of lattices, in particular, of lattices arising from totally real fields. We define the almost sure value \(\kappa_{\mu}\) of the Mordell constant with respect to certain homogeneous measures on the space of lattices, and establish a strict inequality \(\kappa_{\mu_{1}}<\kappa_{\mu_{2}}\) when the \(\mu_{i}\) are finite and \(\mathrm {supp}(\mu_1) \varsubsetneq \mathrm {supp} (\mu_2)\). In combination with known results regarding the dynamics of the diagonal group we obtain isolation results as well as information regarding accumulation points of the Mordell-Gruber spectrum, extending previous work of P. M. Gruber and G. Ramharter [Acta Math. Acad. Sci. Hung. 39, 135–141 (1982; Zbl 0488.10028)]. One of the main tools we develop is the associated algebra, an algebraic invariant attached to the orbit of a lattice under a block group, which can be used to characterize closed and finite volume orbits.