Consider an elliptic curve $E: y^2 = f(x) := x^3 + ax + b$ over a finite field $\mathbb{F}_q$ of characteristic $> 3$. Obviously, the projection to $x$ is a quadratic $\mathbb{F}_q$-cover of the line with at least one ramification $\mathbb{F}_q$-point, namely the infinity point. Are there (simple) necessary or sufficient conditions for $E$ to have a quadratic $\mathbb{F}_q$-cover of the line without ramification $\mathbb{F}_q$-points? In other words, is there or not a quartic $\mathbb{F}_q$-polynomial $g(x)$ without $\mathbb{F}_q$-roots such that $E$ is birationally $\mathbb{F}_q$-isomorphic to the curve $y^2 = g(x)$? Thank you!