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Let L 2 denote the Bousfield localization functor with respect to the Johnson-Wilson spectrum E(2). A spectrum L 2 X is called invertible if there is a spectrum Y such that L 2 X A Y = L 2 S 0 . Hovey and Sadofsky, Invertible spectra in the E(n)-local stable homotopy category, showed that every invertible spectrum is homotopy equivalent to a suspension of the E(2)-local sphere L 2 S 0 at a prime p > 3. At the prime 3, it is shown, A relation between the Picard group of the E(n)-local homotopy category and E(n)-based Adams spectral sequence, that there exists an invertible spectrum X that is not homotopy equivalent to a suspension of L 2 S 0 . In this paper, we show the homotopy equivalence v 3 2 : Σ 48 L 2 V(1) ≃ V(1) A X for the Smith-Toda spectrum V(1). In the same manner as this, we also show the existence of the self-map β: Σ 144 L 2 V(1) → L 2 V(1) that induces v 9 2 on the E(2)*-homology.

𝐸(2)-invertible spectra smashing with the Smith-Toda spectrum 𝑉(1) at the prime 3