  We have obtained 1 and 3 mm spectral scans of ASXDF1100.053.1 using NOEMA.
ASXDF1100.053.1 is an unlensed optically dark millimetre-bright SMG with
$K_{\rm AB}>25.7$ ($2\sigma$), which was expected to lie at $z=$5-7 based on
its radio-submm photo-$z$. Our data detected line emission due to
$^{12}$CO($J=$5-4) and ($J=$6-5), providing a $z_{\rm CO}= 5.2383\pm0.0005$.
Energy-coupled SED modelling indicates properties of $L_{\rm
IR}=8.3^{+1.5}_{-1.4}\times10^{12}$ L$_{\odot}$, SFR $=630^{+260}_{-380}$
M$_{\odot}$ yr$^{-1}$, $M_{\rm dust}=4.4^{+0.4}_{-0.3}\times10^{8}$
M$_{\odot}$, $M_{\rm stellar}=3.5^{+3.6}_{-1.4}\times10^{11}$ M$_{\odot}$, and
$T_{\rm dust}=37.4^{+2.3}_{-1.8}$ K. The CO luminosity allows us to estimate a
gas mass $M_{\rm gas}=3.1\pm0.3\times10^{10}$ M$_{\odot}$, suggesting a
gas-to-dust mass ratio of around 70, fairly typical for $z\sim2$ SMGs.
ASXDF1100.053.1 has $R_{\rm e, mm}=1.0^{+0.2}_{-0.1}$ kpc, so its surface
$L_{\rm IR}$ density $\Sigma_{\rm IR}$ is $1.2^{+0.1}_{-0.2}\times10^{12}$
L$_{\odot}$ kpc$^{-2}$. These properties indicate that ASXDF1100.053.1 is a
massive dusty star-forming (SF) galaxy with an unusually compact starburst. It
lies close to the SF main sequence at $z\sim5$, with low $M_{\rm gas}$/$M_{\rm
stellar}=0.09$, SFR/SFR$_{\rm MS} (R_{\rm SB})=0.6$, and a gas-depletion time
$\tau_{\rm dep}$ of $\approx 50$ Myr, modulo assumptions about the stellar
initial mass function in such objects. ASXDF1100.053.1 has extreme values of
$M_{\rm gas}/M_{\rm stellar}$, $R_{\rm SB}$, and $\tau_{\rm dep}$ compared to
SMGs at $z\sim$2-4, and those of ASXDF1100.053.1 are the smallest among SMGs at
$z>5$. ASXDF1100.053.1 is likely a late-stage dusty starburst prior to
passivisation. The number of $z=$5.1-5.3 unlensed SMGs now suggests a number
density $dN/dz=30.4\pm19.0$ deg$^{-2}$, barely consistent with the latest
cosmological simulations.
