  We consider three-dimensional gravity with negative cosmological constant in
the presence of a scalar and an Abelian gauge field. Both fields are
conformally coupled to gravity, the scalar field through a nonminimal coupling
with the curvature and the gauge field by means of a Lagrangian given by a
power of the Maxwell one. A sixth-power self-interaction potential, which does
not spoil conformal invariance is also included in the action. Using a
circularly symmetric ansatz, we obtain black hole solutions dressed with the
scalar and gauge fields, which are regular on and outside the event horizon.
These charged hairy black holes are asymptotically anti-de Sitter spacetimes.
The mass and the electric charge are computed by using the Regge-Teitelboim
Hamiltonian approach. If both leading and subleading terms of the asymptotic
condition of the scalar field are present, a boundary condition that
functionally relates them is required for determining the mass. Since the
asymptotic form of the scalar field solution is defined by two integration
constants, the boundary condition may or may not respect the asymptotic
conformal symmetry. An analysis of the temperature and entropy of these black
holes is presented. The temperature is a monotonically increasing function of
the horizon radius as expected for asymptotically anti-de Sitter black holes.
However, restrictions on the parameters describing the black holes are found by
requiring the entropy to be positive, which, given the nonminimal coupling
considered here, does not follow the area law. Remarkably, the same conditions
ensure that the conformally related solutions become black holes in the
Einstein frame.
