An elliptic surface of Mordell-Weil rank $8$ over the rational numbers

Schwartz, Charles F.

Journal de théorie des nombres de Bordeaux, Tome 6 (1994) no. 1, pp. 1-8

Résumé

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 with $X_i$ and $Y_i$ polynomial of degree $\le 2,\le 3$, respectively. One explicit example is computed, showing that for almost every elliptic surface given by a Weierstrass equation of the above form, a basis may be found with $X_i$ and $Y_i$ polynomial of degree $\le 2,\le 3$, respectively.