Cohn polynomial irreducibility and degree conjecture for Lamé spectra

Let $J$ be the Klein invariant, and let the Type I and Type II Cohn polynomials be the polynomials in $J$ whose roots parametrize coincident values of the transformed energy $B$ for the corresponding Lamé spectral polynomials. Let $N_\ell^{\rm I}$ and $N_\ell^{\rm II}$ denote the degrees of the Type I and Type II spectral polynomials.

Cohn polynomial conjecture. As a polynomial in $J$ with integer coefficients, no Cohn polynomial has a nontrivial factor, except for the Type I Cohn polynomials with $\ell\equiv2\pmod3$, each of which is divisible by $J$. Moreover, the Type I and Type II Cohn polynomials have degrees

$$\left\lfloor \frac{(N_\ell^{\rm I})^2-N_\ell^{\rm I}+4}{6}\right\rfloor$$

and

$$\frac{N_\ell^{\rm II}(N_\ell^{\rm II}-1)}{2},$$

respectively.

The conjecture predicts the number of points in elliptic moduli space, labelled by $J$, at which the $2\ell+1$ algebraic energies reduce to $2\ell$ distinct energies. It is based on computations through $\ell\le25$; the degree formulas and the asserted irreducibility remain to be established in general.