Constant Tamagawa numbers of special elliptic curves

TL;DR

This paper proves that certain elliptic curves have Tamagawa numbers always equal to one or zero, depending on parameters, using finite field matrices, and introduces a method to identify curves with low Mordell-Weil rank.

Contribution

The paper establishes a novel result on Tamagawa numbers of specific elliptic curves and provides a new method for quickly identifying curves with Mordell-Weil rank zero or one.

Findings

Tamagawa numbers are always 0 or 1 for the studied curves.

The value depends on the parameter ta.

A new sieve method for low-rank elliptic curves is proposed.

Abstract

For the elliptic curves $E_{\sigma 2D} : y^2 = x^3 + \sigma 2Dx$ , which has 2-isogeny curve $E'_{\sigma 2D} : y^2 = x^3 -\sigma 8Dx$, $\sigma = \pm 1,\ D = p_1^{e_1}p_2^{e_2}\cdots p_n^{e_n}$, where $p_i$ are different odd prime numbers and $e_i = 1 \text{ or } 3$, we demonstrate that Tamagawa numbers of these elliptic curves are always one or zero by the use of matrix in finite field $\mathbb F_2$. The specific number depends on the value of $\sigma$. By our proofs of these results, we find a method to quickly sieve a part of the elliptic curves with Mordell-Weil rank zero or rank one in this form as an application.