  We classify all possible charge-3 monopole spectral curves with non-trivial
automorphism group and within these identify those with elliptic quotients. By
focussing on elliptic quotients the transcendental constraints for a monopole
spectral curve become ones regarding periods of elliptic functions. We
construct the Nahm data and new monopole spectral curves with $D_6$ and $V_4$
symmetry, the latter based on an integrable complexification of Euler's
equations, and for which energy density isosurfaces are plotted. Extensions of
our approach to higher charge and hyperbolic monopoles are discussed.
