Let l_p = BP<1>_p be the p-complete connective Adams summand of topological K-theory, and let V(1) be the Smith-Toda complex. For p>3 we explicitly compute the V(1)-homotopy of the algebraic K-theory spectrum of l_p. In particular we find that it is a free finitely generated module over the polynomial algebra P(v_2), except for a sporadic class in degree 2p-3. Thus also in this case algebraic K-theory increases chromatic complexity by one. The proof uses the cyclotomic trace map from algebraic K-theory to topological cyclic homology, and the calculation is actually made in the V(1)-homotopy of the topological cyclic homology of l_p.