Orders of Tate–Shafarevich groups for the cubic twists of $X_0(27)$
This paper continues the authors’ previous investigations concerning orders of Tate–Shafarevich groups in quadratic twists of a given elliptic curve, and for the family of the Neumann–Setzer type elliptic curves. Here we present the results of our search for the (analytic) orders of Tate–Shafarevich groups for the cubic twists of $X_0(27)$. Our calculations extend those given by Zagier and Kramarz (1987) and by Watkins (2007). Our main observations concern the asymptotic formula for the frequency of orders of Tate–Shafarevich groups. In the last section we propose a similar asymptotic formula for the class numbers of real quadratic fields.