  We study an inflationary $E$ model with the Gauss-Bonnet coupling, which can
enhance the curvature perturbation at small scales and thus produce a
significant abundance of primordial black holes (PBHs) and detectable scalar
induced gravitational waves (SIGWs). PBHs from the $E$ model with mass
$30M_{\odot}$, $10^{-5}M_{\odot}$, and $10^{-12}M_{\odot}$ can explain the
LIGO-Virgo events, the ultrashort-timescale microlensing events in the OGLE
data, and all dark matter, respectively. SIGWs produced by the $E$ model can
account for the recent NANOGrav signal. We also compute the primordial
non-Gaussianity and discuss its impact on PBHs and SIGWs. The probability
distribution of density contrast is modified to be right-tailed, which we find
prompts the formation of PBHs, so that the abundance of PBHs is underestimated
with Gaussian approximation. On the contrary, the fractional energy density of
SIGWs is hardly affected.

General Relativity and Quantum Cosmology