Abstract We consider the cyclic presentation of $$\textrm{PG}(3, q )$$ PG ( 3 , q ) whose points are in the finite field $$\mathbb {F}_{q^4}$$ F q 4 and describe the known ovoids therein. We revisit the set $$\mathcal {O}$$ O , consisting of $$(q^2+1)$$ ( q 2 + 1 ) th roots of unity in $$\mathbb {F}_{q^4}$$ F q 4 , and prove that it forms an elliptic quadric within the cyclic presentation of $$\textrm{PG}(3, q )$$ 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 $$\textrm{PG}(3, q )$$ PG ( 3 , q ) , characterizing them as the zeroes of a polynomial over $$\mathbb {F}_{q^4}$$ F q 4 .