We consider elliptic curves whose coefficients are degree 2 polynomials in avariable t. We prove that for infinitely many values of t the resultingelliptic curve has rank at least 1. All such curves together form an algebraicsurface which is birational to a conic bundle with 7 singular fibers. The mainstep of the proof is to show that such conic bundles are unirational. V.2: Maintheorem is corrected for small finite fields.
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