Elliptic curve associated to a rational point (awaiting review)

Let $X_H$ be a modular curve with $-1\in H$. The non-cuspidal $K$-rational points on a modular curve $X_H$ are in one-to-one correspondence with isomorphism classes of pairs $(E,[\iota]_H)$. where $E$ is an elliptic curve over $K$ and $[\iota]_H$ is a $K$-rational $H$-level structure on $E$.

The isomorphism class of $(E,[\iota]_H)$ includes pairs $(E',[\iota']_H)$ for each of the infinitely many quadratic twists $E'$ of $E$ (because $-1\in H$), but on modular curve pages and search results we list only the minimal quadratic twist of $E$.

A quadratic twist-minimal elliptic curve $E$ may admit multiple non-isomorphic rational $H$-level structures corresponding to multiple $K$-rational points on $X_H$, but in search results for low degreee points we list $E$ only once.