Prescribed Abscissae on Congruent-Number Curves over Simplest Cubic Fields

Let $K_t=\mathbb{Q}(θ_t)$, where $θ_t$ is the largest root of $X^3-tX^2-(t+3)X-1$ and $t\geq-1$ is an integer. We classify the points in $E_n(K_t)$ with abscissa $θ_t-1$, for $E_n:y^2=x^3-n^2x$, when $n$ is a positive integer and $E_n(\mathbb{Q})$ has rank zero. The main step excludes every nonzero