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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>g</mml:mi> <mml:mi>g</mml:mi> <mml:mo>→</mml:mo> <mml:mi>t</mml:mi> <mml:mover> <mml:mi>t</mml:mi> <mml:mo accent="true">‾</mml:mo> </mml:mover> <mml:mi>g</mml:mi> </mml:mrow> </mml:math> channel, after the infrared and ultraviolet poles have been identified analytically.