The supersymmetry constraints of minimal gauged supergravity are analysed with the purpose of finding the most general asymptotically $\operatorname{AdS}_5$ black holes with a topologically spherical horizon. We show that under general arguments, such black holes are essentially determined by a two dimensional space, whose curvature obeys a 4th order differential equation. We recast the problem of finding solutions of such equation as a variational problem, and show that an infinite class of solutions exists, with each solution being characterised by the number and strength of conical singularities as well as the mean squared curvature of the 2D space.