  A curve over the field of two elements with completely decomposable Jacobian
is shown to have at most six rational points and genus at most 26. The bounds
are sharp. The previous upper bound for the genus was 145. We also show that a
curve over the field of $q$ elements with more than $q^{m/2}+1$ rational points
has at least one Frobenius angle in the open interval $(\pi/m,3\pi/m)$. The
proofs make use of the explicit formula method.
