  We study an infinite family of $j$-invariant zero elliptic curves
$E_{D}:y^{2}=x^{3}+16D$ and their $\lambda$-isogenous curves
$E_{D'}:y^{2}=x^{3}-27\cdot16D$, where $D$ and $D' = -3D$ are fundamental
discriminants of a specific form, and $\lambda$ is an isogeny of degree $3$. A
result of Honda guarantees that for our discriminants $D$, the quadratic number
field $K_{D} = \mathbb{Q}(\sqrt{D})$ always has non-trivial 3-class group. We
prove a series of results related to the set of rational points
$E_{D'}(\mathbb{Q}) \setminus \lambda(E_{D}(\mathbb{Q}))$, and the
$SL(2,\mathbb{Z})$-equivalence classes of irreducible integral binary cubic
forms of discriminant $D$. By assuming finiteness of the Tate-Shafarevich
group, we derive a parity result between the rank of $E_{D}$ and the rank of
its $3$-Selmer group, and we establish lower and upper bounds for the rank of
our elliptic curves. Finally, we give explicit classes of genus-$1$ curves that
correspond to irreducible integral binary cubic forms of discriminant
$D=48035713$, and we show that every curve in these classes should violate the
Hasse Principle.
