Two-loop Octagons, Algebraic Letters and $\bar{Q}$ Equations

Abstract: We compute the symbol of the first two-loop amplitudes in planar ${\cal N}=4$ SYM with algebraic letters, the eight-point NMHV amplitude (or the dual octagon Wilson loops). We show how to apply $\bar{Q}$ equations for computing the differential of two-loop $n$-point NMHV amplitudes and present the result for n=8 explicitly. The symbol alphabet for octagon consists of 180 independent rational letters and 18 algebraic ones involving Gram-determinant square roots. We comment on all-loop predictions for final entries and aspects of the result valid for all multiplicities.