We study a symplectic variant of algebraic K-theory of the integers, which comes equipped with a canonical action of the absolute Galois group of Q. We compute this action explicitly. The representations we see are extensions of Tate twists Z_p(2 k- 1) by a trivial representation, and we characterize them by a universal property among such extensions. The key tool in the proof is the theory of complex multiplication for abelian varieties.