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We prove that the number of rational points of bounded height on certain delPezzo surfaces of degree 1 defined over Q grows linearly, as predicted byManin's conjecture. Along the way, we investigate the average number ofintegral points of small naive height on quadratic twists of a fixed ellipticcurve with full rational 2-torsion.

Integral points on quadratic twists and linear growth for certain elliptic fibrations