Geodesic
On Rational Elliptic Surfaces with Mordell-Weil Group of Rank Five
Bollettino della Unione matematica italiana, Série 9, Tome 3 (2010) no. 2, pp. 363-379.

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Let $E(K)$ be the Mordell-Weil group of a rational elliptic surface and let $r$ be its rank. In this note we classify all the rational elliptic surfaces with Mordell-Weil group of rank $r = 5$ over an algebraically closed field of arbitrary characteristic and using the theory of Mordell-Weil lattices, we find systems of generators for $E(K)$ in the coordinate-free situation. 
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Fusi, Davide; Tironi, Andrea Luigi. On Rational Elliptic Surfaces with Mordell-Weil Group of Rank Five. Bollettino della Unione matematica italiana, Série 9, Tome 3 (2010) no. 2, pp. 363-379. http://geodesic.mathdoc.fr/item/BUMI_2010_9_3_2_a7/

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