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

TL;DR

This paper investigates the distribution of Tate-Shafarevich group orders in cubic twists of the elliptic curve $X_0(27)$, extending previous computational results and proposing asymptotic formulas for their frequency.

Contribution

The authors provide new computational data on Tate-Shafarevich groups for cubic twists of $X_0(27)$ and suggest an asymptotic formula for their frequency, building on prior work.

Findings

Extended calculations of Tate-Shafarevich group orders for cubic twists.

Observed asymptotic behavior in the distribution of these orders.

Proposed a similar asymptotic formula for class numbers of real quadratic fields.

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 \cite{ZK} and by Watkins \cite{Wat}. 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.