Binary forms with covariant points close to the real axis

For a real binary form $F(X, Z)$, Stoll and Cremona have defined a reduction theory using the action of the modular group $SL_2(\mathbb{Z})$, and associated to each binary form a covariant point $z(F)$ located in the upper half plane. When the point $z(F)$ is close to the real axis, then at least half of the roots will be on a circle of small radius $r$. Conversely, we find conditions depending on the radius $r$ such that the covariant point $z(F)$ to be close to the real axis. The results have further applications to improving the reduction algorithm for binary forms of Stoll and Cremona.