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 $E'_{\sigma 2D} : y^2 = x^3 -\sigma 8Dx$, $\sigma = \pm 1,\ D = p_1^{e_1}p_2^{e_2}\cdots p_n^{e_n}$, 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.