An elliptic surface of Mordell-Weil rank 8 over the rational numbers
Journal de théorie des nombres de Bordeaux, Tome 6 (1994) no. 1, pp. 1-8.

Néron showed that an elliptic surface with rank 8, and with base B=P 1 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 +AX 2 +BX+C
(with deg(A)≤2,deg(B)≤4, and deg(C)≤6) a basis (X 1 ,Y 1 ),⋯,(X 8 ,Y 8 ) can be found with X i and Y i polynomial of degree ≤2,≤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 ≤2,≤3, respectively.

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Schwartz, Charles F. An elliptic surface of Mordell-Weil rank $8$ over the rational numbers. Journal de théorie des nombres de Bordeaux, Tome 6 (1994) no. 1, pp. 1-8. http://www.numdam.org/item/JTNB_1994__6_1_1_0/

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