Two-loop octagons, algebraic letters and Q ¯ equations

We compute the symbol of the first two-loop amplitudes in planar N =4 SYM with algebraic letters, the eight-point Next to Maximally Helicity Violating (NMHV) amplitude (or the dual octagon Wilson loops). We show how to apply Q ¯ equations of [S. Caron-Huot and S. He, J. High Energy Phys. 07 (2012) 174, 10.1007/JHEP07(2012)174] 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.