The flux surface topology of a toroidal plasma bounded by a magnetic separatrix allows edge moments of the toroidal current density profile to be identified in an MHD equilibrium reconstruction code using only external magnetic measurements. This is demonstrated analytically for simple plasma shapes and applied to experimental data on the ASDEX Upgrade tokamak where CLISTE reconstructions from magnetic data are shown to be consistent with those obtained from a more complete set of diagnostic data. An independent demonstration of edge current profile recoverability is obtained by analysing the reconstruction errors for a database of Monte Carlo-generated equilibria.