On the Selmer group and rank of a family of elliptic curves and curves of genus one violating the Hasse principle

Eleni Agathocleous

We study an infinite family of $j$-invariant zero elliptic curves $E_D:y^2=x^3+16D$ and their $\lambda$-isogenous curves $E_{D^{\prime }}:y^2=x^3-27\cdot 16D$, where $D$ and $D^{\prime }=-3D$ a…