$K$, there is no reason for such Galois representation not to exist. In particular, they supply new examples of irreducible finite flat group schemes of type $(2,2)$ over $\operatorname{Spec}({{\mathcal{O}}}_K)$ (the irreducibility comes from the fact that irreducible $2$-groups are only the additive and the multiplicative ones as proved in [@Oort-Tate Corollary page 21], which give rational $2$-torsion). One simple example of this phenomenon comes from the elliptic curves over $\Q$ of conductor $11$, which have minimal discriminant either $-11$ or $-11^5$; after base-change to $K=\Q(\sqrt{-11})$ the conductor is still prime but the minimal discriminants are all now square. For an example which is not a base-change, the curve [[](http://www.lmfdb.org/EllipticCurve/2.0.11.1/47.1/a/1)]{} (using its LMFDB label, see [@lmfdb]): $$E:\quad y^2+y=x^3+\alpha x^2-x$$ where $\alpha=(1+\sqrt{-11})/2$, has prime conductor $\p=(\pi)$ of norm $47$, with $\pi=7-2\alpha$, and discriminant $\pi^2$, and is alone in its isogeny class. This phenomenon is studied in detail in Section \[section:evenexponents\] below. We suspect that there are infinitely many examples like this over each of these five fields.

Another interesting problem that arose is that of classifying elliptic curves (up to $2$-isogenies and twists) with a rational $2$-torsion point. The prime $2$ is always hard to handle, and we develop tools (including a 