Orders of Tate–Shafarevich groups for the cubic twists of $X_0(27)$

Abstract

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 [20] and by Watkins [19]. 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.

Elliptic curves with exceptionally large analytic order of the Tate-Shafarevich groups

“…In our earlier papers, we have investigated (see [11], [7], [8], [9], [10]) some numerical examples of E defined over Q for which L(E, 1) is non-zero and the order of X(E) is large.…”

Elliptic curves with exceptionally large analytic order of the Tate–Shafarevich groups

“…In earlier papers (see [11], [7], [8], [9], [10]), we have investigated some numerical examples of E defined over Q for which L(E, 1) is non-zero and the order of Ш(E) is large.…”