This is not a complete answer. The book "Elliptic Functions" by Serge Lang has some formulae (called the "modular equation") which are related. The idea is this. Any holomorphic function on the upper half plane which is $SL(2,{\mathbb Z})$ invariant and is meromorphic at the cusp at infinity is a rational function in $J$. Given $g\in SL(2,{\mathbb Q})$, the function $J(g\tau)$ is invariant under some congruence subgroup $\Delta $ (of finite index ) of $SL(2,{\mathbb Z})$; therefore, if $S$ denotes a full set of representatives of $SL(2,{\mathbb Z}/\Delta$ then the coefficients of the polynomial $$\prod _{\gamma \in S} (X-J(g\gamma \tau)=\sum a_i(J)X^i$$ are rational functions in $J(\tau)$. Taking '$g=\begin{pmatrix} 2 & 0 \cr 0 & 1 \end{pmatrix}$' we get that $J(2\tau)$ is a root of such a polynomial, and hence get "duplication formulae". I think the other formulae are similar.

This is not a complete answer. The book "Elliptic Functions" by Serge Lang has some formulae (called the "modular equation") which are related. The idea is this. Any holomorphic function on the upper half plane which is $SL(2,{\mathbb Z})$ invariant and is meromorphic at the cusp at infinity is a rational function in $J$. Given $g\in SL(2,{\mathbb Q})$, the function $J(g\tau)$ is invariant under some congruence subgroup $\Delta $ (of finite index ) of $SL(2,{\mathbb Z})$; therefore, if $S$ denotes a full set of representatives of $SL(2,{\mathbb Z})/\Delta$ then the coefficients of the polynomial $$\prod _{\gamma \in S} (X-J(g\gamma \tau))=\sum a_i(J)X^i$$ are rational functions in $J(\tau)$. Taking '$g=\begin{pmatrix} 2 & 0 \cr 0 & 1 \end{pmatrix}$' we get that $J(2\tau)$ is a root of such a polynomial, and hence get the "duplication formula". That is, a root of a cubic equation may be expressed in terms of radicals involving the coefficients $a_i(J)$. I think the other formulae are similar but I not sure.

This is not a complete answer. The book "Elliptic Functions" by Serge Lang has some formulae (called the "modular equation") which are related. The idea is this. Any holomorphic function on the upper half plane which is $SL(2,{\mathbb Z})$ invariant and is meromorphic at the cusp at infinity is a rational function in $J$. Given $g\in SL(2,{\mathbb Q})$, the function $J(g\tau)$ is invariant under some congruence subgroup $\Delta $ (of finite index ) of $SL(2,{\mathbb Z})$; therefore, if $S$ denotes a full set of representatives of $SL(2,{\mathbb Z}/\Delta$ then the coefficients of the polynomial $$\prod _{\gamma \in S} (X-J(g\gamma \tau)=\sum a_i(J)X^i$$ are rational functions in $J(\tau)$. Taking $g=\begin{pmatrix} 2 & 0 \cr 0 & 1 \end{pmatrix}$ we get that $J(2\tau)$ is a root of such a polynomial, and hence get "duplication formulae". I think the other formulae are similar.

This is not a complete answer. The book "Elliptic Functions" by Serge Lang has some formulae (called the "modular equation") which are related. The idea is this. Any holomorphic function on the upper half plane which is $SL(2,{\mathbb Z})$ invariant and is meromorphic at the cusp at infinity is a rational function in $J$. Given $g\in SL(2,{\mathbb Q})$, the function $J(g\tau)$ is invariant under some congruence subgroup $\Delta $ (of finite index ) of $SL(2,{\mathbb Z})$; therefore, if $S$ denotes a full set of representatives of $SL(2,{\mathbb Z}/\Delta$ then the coefficients of the polynomial $$\prod _{\gamma \in S} (X-J(g\gamma \tau)=\sum a_i(J)X^i$$ are rational functions in $J(\tau)$. Taking '$g=\begin{pmatrix} 2 & 0 \cr 0 & 1 \end{pmatrix}$' we get that $J(2\tau)$ is a root of such a polynomial, and hence get "duplication formulae". I think the other formulae are similar.