In the report we will generalize the simplified Shallue–van de Woestijne–Ulas (SWU) method of a deterministic finite field mapping  F_q->E(F_q) to the case of any elliptic (F_q)-curve E of j-invariant 1728. More precisely, we will obtain a rational F_q-curve (and its explicit quite simple proper F_q-parametrization) on the Kummer surface K’ associated with the direct product ExE’, where E ‘ is the quadratic (F_q)-twist of E.  The simplified SWU method consists in computing the direct image of the parametrization and a subsequent inverse image (P,Q) of the natural two-sheeted covering ExE’->K’. Denoting by s:E’ -> E the corresponding  (F_{q^2})-isomorphism, it is easily seen that: P is in E(F_q) or s(Q)  belongs to  E(F_q).

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