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Published online by Cambridge University Press: 22 January 2016
It is well known [c.f. Kas] that every elliptic surface, with geometric genus 0, is given by a Weierstrass equation of the form
(relative to a suitable parameter, u, for the base) where the a’s and b’s are constants. For sufficiently general choices of a’s and b’s, the Mordell-Weil group (i.e., the group of solutions (x, y), with x and y rational functions of u) has rank 8.