  Since the 1970s, the complete classification (up to isogeny) of abelian
varieties over finite fields with trivial group of rational points has been
known from results of Madan--Pal and Robinson; with two exceptions these are
all defined over $\mathbb{F}_2$. We determine the decomposition of these
varieties into simple factors over an algebraic closure of $\mathbb{F}_2$; this
requires solving a polynomial equation in three roots of unity.
