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