  Magnetic monopoles in hyperbolic space are in correspondence with certain
algebraic curves in mini-twistor space, known as spectral curves, which are in
turn in correspondence with rational maps between Riemann spheres. Hyperbolic
monopoles correspond to circle-invariant Yang-Mills instantons, with an
identification of the monopole and instanton numbers, providing the curvature
of hyperbolic space is tuned to a value specified by the asymptotic magnitude
of the Higgs field. In previous work, constraints on ADHM instanton data have
been identified that provide a non-canonical realization of the circle symmetry
that preserves the standard action of rotations in the ball model of hyperbolic
space. Here formulae are presented for the spectral curve and the rational map
of a hyperbolic monopole in terms of its constrained ADHM matrix. This extends
earlier results that apply only to the subclass of instantons of JNR type. The
formulae are applied to obtain new explicit examples of spectral curves that
are beyond the JNR class.
