Constant Tamagawa numbers of special elliptic curves

Abstract

For the elliptic curves $E_{\sigma 2D}:y^2=x^3+\sigma 2Dx$ , which has 2-isogeny curve $E_{\sigma 2D}^{\prime }:y^2=x^3-\sigma 8Dx$, $\sigma =\pm 1,D=p_1^{e_1}{p}_2^{e_2}\cdots p_n^{e_n}$, 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.