On the Faltings height of the curve $y^2=x^n-1$

We compute the stable Faltings height of the hyperelliptic curve $X_n\text{\hspace{0.17em}⁣}:y^2=x^n-1$ for every odd integer $n\ge 3$ in terms of special values of Euler’s gamma function. In particular, we prove the bounds $$-0.975n<h_{\mathrm{F}\mathrm{a}\mathrm{l}}(X_n)-\frac{n}{8}\log n<\frac{9}{64}n\log \log n-0.263n\mathrm{.}$$ As an application, we bound the Faltings height of any abelian variety with complex multiplication by the canonical CM-type of the $n$-th cyclotomic field by $\frac{n}{8}\log n+\frac{9}{64}n\log \log n-0.136n$.