I have several elliptic polynomials $P_i(u)$ in Jacobi elliptic functions $sn(u|k)$, $cn(u|k)$ and $dn(u|k)$ with standard definitions $sn^2(u|k)+cn(u|k)^2=1$, $dn^2(u|k)+k^2sn^2(u|k)=1$.
Coefficients of $P_i(u)$ depend rationally on the elliptic modulus $k$ but involve a square root $X=\sqrt{1+k+k^2}$. This square root is very unnatural and very hard to work with. In particular, it is very hard to factorize expressions over the field $F(k,X)$, $X^2=1+k+k^2$.
Ideally I would like to find a rational transformation $\tau=f(\tilde\tau)$ such that $\sqrt{1+k(\tau)+k^2(\tau)}=R(k(\tilde\tau),k'(\tilde\tau)))$, i.e. it becomes a rational function of $k(\tilde\tau), k'(\tilde\tau))$, $k^2+k'^2=1$. I expect that it has to be the third order transformation. However, formulas for third order transofmations are very complicated, say, $k^2(\tau)$ and $k^2(3\tau)$ are constrained by a fourth order relation.
Has anyone seen this before and know how to approach this problem of uniformization ? Thanks.