assertion: one comes from the study of finite flat group schemes over $\operatorname{Spec}(\Z)$. The hypotheses ($E$ being semistable and the discriminant being a square) imply by a theorem of Mazur (see Theorem \[thm:Mazur\] below) that $E[2]$ is a finite flat group scheme of type $(2,2)$. As explained in [@Mestre], there are only $4$ such group schemes, and only two of them satisfy that the determinant of the group scheme is isomorphic to the group scheme $\mu_2$; they both have an invariant subspace. A different approach comes from the study of the mod-$2$ Galois representation itself: if it is irreducible, then by Ribet’s level-lowering result, there would exist a modular form of level $1$ and weight $2$, while if the image were irreducible but not absolutely irreducible (i.e. it is the cyclic subgroup of order $3$ in $\operatorname{GL}_2(\F_2)$), then there would exist a cubic extension of $\Q$ unramified outside $2$; but such an extension does not exist.

Over an imaginary quadratic field $K$ of class number one, in four cases such an extension does not exist either, so the same result holds (for modular elliptic curves, assuming an analogous level-lowering conjecture). However for $K=\Q(\sqrt{-d})$ for $d\in\{11,19,43,67,163\}$ where $2$ is inert in 