We study the parametrized family of elliptic functions of the form F_Λ,b(z)=℘_Λ(z)+b for b∈ C, Λ a lattice, and ℘_Λ the Weierstrass elliptic ℘ function with period lattice Λ. We show that the dynamics depend on b as b varies within one fundamental region of C/Λ, and on the lattice Λ. We analyze properties of the Julia sets, and bifurcations of F_Λ,b, focussing on real rectangular lattices; in particular the dynamical properties are more diverse than those coming from the family ℘_Λ with Λ varying.