I think I am stuck at a calculation problem. Suppose there is an elliptic surface $X$ defined over $Q$, take $\overline{X}:=X\times_Q \overline{Q}$, and denote by $\varphi: G_{Q}\to Aut(H^2_{et}(\overline{X}, Q_{\ell}(1)))$ the natural representation. How do I know the Artin conductor of this representation once I am given the concrete elliptic fibration? For example, suppose $X$ is defined by the affine model $$ y^2=x(x^2+2x+t^4) $$ With $t$ a parameter. Then is there any way to tell the conductor of the representation space $H^2_{et}(\overline{X}, Q_{\ell}(1))$?
Thanks very much.