Aims. The aim of the study is to constrain the physics of gamma-ray bursts (GRBs) by analysing the multi-wavelength afterglow data set of GRB 121024A that covers the full range from radio to X-rays. Methods. Using multi-epoch broad-band observations of the GRB 121024A afterglow, we measured the three characteristic break frequencies of the synchrotron spectrum. We used six epochs of combined XRT and GROND data to constrain the temporal slopes, the dust extinction, the X-ray absorption, and the spectral slope with high accuracy. Two more epochs of combined data from XRT, GROND, APEX, CARMA, and EVLA were used to set constraints on the break frequencies and therefore on the micro-physical and dynamical parameters. Results. The XRT and GROND light curves show a simultaneous and achromatic break at around 49 ks. As a result, the crossing of the synchrotron cooling break is no suitable explanation for the break in the light curve. The multi wavelength data allow us to test two plausible scenarios explaining the break: a jet break, and the end of energy injection. The jet-break scenario requires a hard electron spectrum, a very low cooling break frequency, and a non-spreading jet. The energy injection avoids these problems, but requires $\epsilon_e > 1 (k = 2)$, spherical outflow, and $\epsilon_B < 10^{-9}$. Conclusions. In light of the extreme microphysical parameters required by the energy-injection model, we favour a jet-break scenario where $\nu_m < \nu_{sa}$ to explain the observations. This scenario gives physically meaningful microphysical parameters, and it also naturally explains the reported detection of linear and circular polarisation.
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