Finite field mapping to elliptic curves of j-invariant 1728

In the report it is proposed new deterministic finite field mapping $\mathbb{F}_{\!q} \to E(\mathbb{F}_{\!q})$ in the case of any elliptic $\mathbb{F}_{\!q}$-curve $E$ of $j$-invariant 1728. For this purpose we will construct a rational $\mathbb{F}_{\!q}$-curve $C$ (and an efficiently computable birational $\mathbb{F}_{\!q}$-morphism $\mathbb{P}^1 \to C$) on the Kummer surface $K$ associated with the direct product $E \!\times\! E^\prime$, where $E^\prime$ is the quadratic $\mathbb{F}_{\!q}$-twist of $E$. More precisely, the curve $C$ is one of two absolutely irreducible $\mathbb{F}_{\!q}$-components of the inverse image $pr^{{-}1}(C_8)$ for some rational $\mathbb{F}_{\!q}$-curve $C_8$ of bidegree $(8,8)$ with 42 singular points, where $pr\!: K \to \mathbb{P}^1 \!\times\! \mathbb{P}^1$ is the two-sheeted projection to $x$-coordinates of $E$ and $E^\prime$.