Conic bundles and Mordell--Weil ranks of elliptic surfaces

Let $k$ be a number field and $\mathcal{E}$ an elliptic curve defined over the function field $k(T)$ given by an equation of the form $y^2=a_3{x}^3+a_2{x}^2+a_{1}x+a_0$, where $a_i\in k[T]$ and $deg(a_i)\le 2$. We explore the conic bundle structure over the $x$-line to obtain lower and upper bounds for the Mordell–Weil rank of $\mathcal{E}(k(T))$.