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We investigate the regulators of elliptic curves with rank 1 in some familiesof quadratic twists of a fixed elliptic curve. In particular, we formulate someconjectures on the average size of these regulators. We also describe anefficient algorithm to compute explicitly some of the invariants of an oddquadratic twist of an elliptic curve (regulator, order of the Tate-Shafarevichgroup, etc.) and we discuss the numerical data that we obtain and compare itwith our predictions.

Regulators of rank one quadratic twists