Ovoids in the cyclic presentation of PG(3,q)

We consider the cyclic presentation of $PG(3,q)$ 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 $PG(3,q)$. Additionally, following the work of Glauberman on Suzuki groups, we offer a new description of Suzuki-Tits ovoids in the cyclic presentation of $PG(3,q)$, characterizing them as the zeroes of a polynomial over ${\mathbb{F}}_{q^4}$.