For k=5 the Dris index equation factors into the two cyclotomic terms

For every natural number $p$, $2 \cdot (p^2+p+1) \cdot \left(\frac{p+1}{2}\right)(p^2-p+1) = \sigma(p^5)$. This is the exact factorisation of the first Dris equation on the $k=5$ branch: writing $U = p^2+p+1$ and $V = \frac{p+1}{2}(p^2-p+1)$ gives $\sigma(p^5) = 2UV$, so that equation reads $m^2 = d^2 \cdot (qUV)$ once $m^2 = d^2 x$ is cancelled. The identity is a polynomial one, since $\sigma(p^5) = 1+p+\dots+p^5 = (1+p+p^2)(1+p^3)$; no oddness assumption on $p$ is required, and the division by $2$ is exact for every $p$ because $(p+1)(p^2-p+1) = p^3+1$.