Constant Tamagawa numbers of special elliptic curves

Luying Li, Chang Lv

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…