  This article considers an F_q-conic contained in an F_q-subplane of
PG(2,q^3), and shows that it corresponds to a normal rational curve in the
Bruck-Bose representation in PG(6,q). This article then characterises which
normal rational curves of PG(6,q) correspond via the Bruck-Bose representation
to F_q-conics of PG(2,q^3). The normal rational curves of interest are called
3-special, which relates to how the extension of the normal rational curve
meets the transversal lines of the regular 2-spread of the Bruck-Bose
representation. This article uses geometric arguments that exploit the
interaction between the Bruck-Bose representation of PG(2,q^3) in PG(6,q), and
the Bose representation of PG(2,q^3) in PG(8,q).
