Bounds for 2-Selmer ranks in terms of seminarrow class groups

Let $E$ be an elliptic curve over a number field $K$ defined by a monic irreducible cubic polynomial $F(x)$. When $E$ is $\mathit{\text{nice}}$ at all finite primes of $K$, we bound its $2$-Selmer rank in terms of the $2$-rank of a modified ideal class group of the field $L=K[x]\mathrm{/}(F(x))$, which we call the $\mathit{\text{semi-narrow}}\mathit{\text{class}}\mathit{\text{group}}$ of $L$. We then provide several sufficient conditions for $E$ being nice at a finite prime. As an application, when $K$ is a real quadratic field, $E\mathrm{/}K$ is semistable and the discriminant of $F$ is totally negative, then we frequently determine the $2$-Selmer rank of $E$ by computing the root number of $E$ and the $2$-rank of the narrow class group of $L$.