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. 
@article{BUMI_2010_9_3_2_a7,
     author = {Fusi, Davide and Tironi, Andrea Luigi},
     title = {On {Rational} {Elliptic} {Surfaces} with {Mordell-Weil} {Group} of {Rank} {Five}},
     journal = {Bollettino della Unione matematica italiana},
     pages = {363--379},
     publisher = {mathdoc},
     volume = {Ser. 9, 3},
     number = {2},
     year = {2010},
     zbl = {1200.14069},
     mrnumber = {2666364},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BUMI_2010_9_3_2_a7/}
}
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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/