The Voevodsky motive of a rank one semiabelian variety

Over a perfect field k, let G be an extension of an abelian variety by the multiplicative group ${\text{G}}_m$. We compute the motive of G in Voevodsky’s category of etale motivic complexes with rational coefficients. The result is a decomposition similar to the classical decomposition of the Chow motive of an abelian variety obtained by Deninger/Murre, Kuennemann, Shermenev and Scholl, and other authors.