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We study an inflationary $E$ model with the Gauss-Bonnet coupling, which canenhance the curvature perturbation at small scales and thus produce asignificant abundance of primordial black holes (PBHs) and detectable scalarinduced gravitational waves (SIGWs). PBHs from the $E$ model with mass$30M_{\odot}$, $10^{-5}M_{\odot}$, and $10^{-12}M_{\odot}$ can explain theLIGO-Virgo events, the ultrashort-timescale microlensing events in the OGLEdata, and all dark matter, respectively. SIGWs produced by the $E$ model canaccount for the recent NANOGrav signal. We also compute the primordialnon-Gaussianity and discuss its impact on PBHs and SIGWs. The probabilitydistribution of density contrast is modified to be right-tailed, which we findprompts the formation of PBHs, so that the abundance of PBHs is underestimatedwith Gaussian approximation. On the contrary, the fractional energy density ofSIGWs is hardly affected.

Primordial black holes and scalar induced gravitational waves from the $E$ model with a Gauss-Bonnet term