(2q+1)-arcs in PG(3,q3) stabilized by a Sylow p-subgroup of PSL(2,q)

We construct arcs K of cardinality 2q+1 in the projective space PG(3,q3), q=ph, p>3 prime, from a cubic curve C. By construction, K is stabilized by a Sylow p-subgroup of the projectivities preserving C and it is contained in no twisted cubic of PG(3,q3).