Let $(U_n)_{n \in \mathbb{N}}$ be a Lucas sequences given by $$U_0 = 0,\quad U_1 = 1,\quad U_n = P U_{n - 1} - Q U_{n-2},$$ where $P,Q$ are integers with $P^2 - 4Q \neq 0$. It is well known that the following product formula holds $$U_n = \prod_{d \mid n} \Phi_d(\alpha, \beta) ,$$ where $\Phi_d(\alpha, \beta) \in \mathbb{Z}$, $$\Phi_k(X,Y) := \prod_{\zeta \,\text{ $k$th primitive root of $1$}}(X - \zeta Y)$$ denotes the $k$th homogenous cyclotomic polynomial, and $\alpha,\beta$ are the two roots of $X^2 - PX + Q = 0$.

Let $(D_n)_{n \in \mathbb{N}}$ be an elliptic divisibility sequence, that is, there exists an elliptic curve $E$ over the rationals with a point $P$ of infinite order, and $D_n$ is determined by $$nP = \left(\frac{A_n}{D_n^2}, \frac{B_n}{D_n^3}\right) ,$$ where $A_n, B_n$ are integers with $\gcd(A_n, D_n) = \gcd(B_n, D_n) = 1$.

As far as I understand, elliptic divisibility sequences have many properties in common with Lucas sequences. For example, like Lucas sequences, they are strong divisibility sequences and (under certain condition) satisfy a primitive divisor theorem.

My question is if there exists a counterpart of cyclotomic polynomials for elliptic divisibility sequences, that is, some quantities $\Psi_d \in \mathbb{Z}$ such that a product formula $$D_n = \prod_{d \mid n} \Psi_d ,$$ holds; and, if so, what is known about $\Psi_d$.