We consider the cyclic presentation of \(\textrm{PG}(3, q )\) whose points are in the finite field \(\mathbb {F}_{q^4}\) and describe the known ovoids therein. We revisit the set \(\mathcal {O}\) , consisting of \((q^2+1)\) th roots of unity in \(\mathbb {F}_{q^4}\) , and prove that it forms an elliptic quadric within the cyclic presentation of \(\textrm{PG}(3, q )\) . Additionally, following the work of Glauberman on Suzuki groups, we offer a new description of Suzuki–Tits ovoids in the cyclic presentation of \(\textrm{PG}(3, q )\) , characterizing them as the zeroes of a polynomial over \(\mathbb {F}_{q^4}\) .
