Prime-ideal exponents of composite-conductor type D cyclotomic units

Derive a conceptual formula for the prime-ideal exponents of the cyclotomic \(S\)-unit \(2\Sigma_r\) for general composite conductors \(M=r(r-1)-1\) arising in type \(D_r\).

Background

The type-$D_r$ finite product establishes that $2\Sigma_r$ is a cyclotomic $S$-unit. For prime conductors, the computed ranks exhibit a simple parity pattern in the norm of this unit.

The first composite conductor occurs at $r=8$, where $M=55$ and the norm of $2\Sigma_8$ in the maximal real cyclotomic field is $1/11^2$. The paper leaves open the task of explaining the prime-ideal exponents uniformly for all composite values of $M$.