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

Let $L_2$ denote the Bousfield localization functor with respect to the Johnson-Wilson spectrum $E(2)$. A spectrum $L_2X$ is called invertible if there is a spectrum $Y$ such that $L_2X\wedge Y=L_2S^0$. 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 $L_2S^0$ 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 $L_2S^0$. In this paper, we show the homotopy equivalence $v_2^3\colon \Sigma^{48}L_2V(1)\simeq V(1)\wedge X$ for the Smith-Toda spectrum $V(1)$. In the same manner as this, we also show the existence of the self-map $\beta\colon \Sigma^{144}L_2V(1)\to L_2V(1)$ that induces $v_2^9$ on the $E(2)_*$-homology.