Towards two-loop QCD corrections to $ pp \to t \bar{t} j$

I discuss the status of the computation of the two-loop QCD corrections to top-quark pair production associated with a jet at hadron colliders. This amplitude is a missing ingredient for next-to-next-to-leading order (NNLO) QCD predictions. I briefly present computational techniques to tackle the algebraic and analytic complexities of two-loop multi-scale amplitudes, in particular where massive propagators give rise to elliptic Feynman integrals. I then describe how a special function basis for the helicity amplitudes is obtained and present first numerical evaluations for the finite remainders of the gg\to t\bar{t}g g g → t t ‾ g channel, after the infrared and ultraviolet poles have been identified analytically.