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 $deg\left(A\right)\le 2,deg\left(B\right)\le 4$, and $deg\left(C\right)\le 6)$ a basis $\left\{(X_1,Y_1),\cdots ,(X_8,Y_8)\right\}$ can be found...

In this paper, the automorphism groups of relatively minimal rational elliptic surfaces with section which have constant J-maps are classified. The ground field is ℂ. The automorphism group of such a surface β: B → ℙ1, denoted by Au t(B), consists of all biholomorphic maps on the complex manifold B. The group Au t(B) is isomorphic to the semi-direct product MW(B) ⋊ Aut σ (B) of the Mordell-Weil groupMW(B) (the group of sections of B), and the subgroup Aut σ (B) of the automorphisms preserving a...