  Using the results of arXiv:0804.0009, where all timelike supersymmetric
backgrounds of N=2, D=4 matter-coupled supergravity with Fayet-Iliopoulos
gauging were classified, we construct genuine nut-charged BPS black holes in
AdS_4 with nonconstant moduli. The calculations are exemplified for the
SU(1,1)/U(1) model with prepotential F=-iX^0X^1. The resulting supersymmetric
black holes have a hyperbolic horizon and carry two electric, two magnetic and
one nut charge, which are however not all independent, but are given in terms
of three free parameters. We find that turning on a nut charge lifts the flat
directions in the effective black hole potential, such that the horizon values
of the scalars are completely fixed by the charges. We also oxidize the
solutions to eleven dimensions, and find that they generalize the geometry
found in hep-th/0105250 corresponding to membranes wrapping holomorphic curves
in a Calabi-Yau five-fold. Finally, a class of nut-charged Nernst branes is
constructed as well, but these have curvature singularities at the horizon.
