This paper provides a comprehensive study of the dimer model on infinite minimal graphs with Fock's elliptic weights [Foc15]. Specific instances of such models were studied in [BdTR17, BdTR18, dT17]; we now handle the general genus 1 case, thus proving a non-trivial extension of the genus 0 results of [Ken02, KO06] on isora-dial critical models. We give an explicit local expression for a two-parameter family of inverses of the Kasteleyn operator with no periodicity assumption on the underlying graph. When the minimal graph satisfies a natural condition, we construct a family of dimer Gibbs measures from these inverses, and describe the phase diagram of the model by deriving asymptotics of correlations in each phase. In the Z 2-periodic case, this gives an alternative description of the full set of ergodic Gibbs measures constructed in [KOS06]. We also establish a correspondence between elliptic dimer models on periodic minimal graphs and Harnack curves of genus 1. Finally, we show that a bipartite dimer model is invariant under the shrinking/expanding of 2-valent vertices and spider moves if and only if the associated Kasteleyn coefficients are antisymmetric and satisfy Fay's trisecant identity.