= Solution
In the nonrelativistic dilute regime, $E(p)=mc^2+p^2/(2m)+\cdots$ and the <Fermi-Dirac distribution> reduces to the <Maxwell-Boltzmann distribution>. Hence
$$
n\simeq\frac{4\pi}{h^3}e^{(\mu-mc^2)/(kT)}
\int_0^\infty p^2e^{-p^2/(2mkT)}dp.
$$
Differentiating the <Gaussian integral> with respect to its coefficient gives $\int_0^\infty p^2e^{-ap^2}dp=\sqrt\pi/(4a^{3/2})$. Therefore
$$
\boxed{n\simeq\left(\frac{2\pi mkT}{h^2}\right)^{3/2}e^{(\mu-mc^2)/(kT)}}.
$$
The degeneracy factor here is the one in the supplied integral, namely one. Substituting $kT=mc^2/\alpha$, $\mu=0$ and dividing by the supplied <photon> density gives
$$
\boxed{\frac n{n_\gamma}=\frac{\sqrt{2\pi}}{8\zeta(3)}\alpha^{3/2}e^{-\alpha}}.
$$
At $\alpha=20$ this is about $4.8\times10^{-8}$. The relic is already nonrelativistic at decoupling and becomes still colder as its momenta redshift, making it a possible <cold dark matter> component if it is stable and interacts sufficiently weakly with light.
For orientation, write $\eta_b=n_b/n_\gamma$. Ignoring subsequent <entropy> transfer, its density relative to baryons would be $\rho_X/\rho_b=(1/20)(n/n_\gamma)/\eta_b$. With the cosmological baryon-to-photon ratio of order $6\times10^{-10}$, this is of order four, demonstrating the right cosmological scale. The measured dark-to-baryonic density ratio is likewise of order five; an observational reference is https://arxiv.org/abs/1807.06209 . This is a candidate abundance argument, not an exact relic-density calculation: <photon> heating after decoupling changes $n/n_\gamma$, and stability and interaction properties were not specified. The supplied mass and freeze-out parameter alone therefore do not establish a precise fit.
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