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Néron showed that an elliptic surface with rank $8$, and with base $B=P_1\mathbb{Q}$, and geometric genus $=0$, may be obtained by blowing up $9$ points in the plane. In this paper, we obtain parameterizations of the coefficients of the Weierstrass equations of such elliptic surfaces, in terms of the $9$ points. Manin also describes bases of the Mordell-Weil groups of these elliptic surfaces, in terms of the $9$ points ; we observe that, relative to the Weierstrass form of the equation, $$Y^2=X^3+A{X}^2+BX+C$$ (with $deg\left(A\right)\le 2,deg\left(B\right)\le 4$,...