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