  We calculate the $\eta$-localization of the motivic stable homotopy ring over
the complex numbers, confirming a conjecture of Guillou and Isaksen. Our
approach is via the motivic Adams-Novikov spectral sequence. In fact, work of
Hu, Kriz, and Ormsby implies that it suffices to compute the corresponding
localization of the classical Adams-Novikov $E_2$-term, and this is what we do.
Guillou and Isaksen also proposed a pattern of differentials in the localized
motivic classical Adams spectral sequence, which we verify using a method first
explored by Novikov.
