general, and can be applied to different situations. Using this characterization, we describe all curves of prime power conductor $\id{p}^r$ with a rational $2$-torsion point, according to the following possible cases: either a twist of the curve has good reduction at $\id{p}$, or all twists have additive reduction at $\id{p}$, or the curve has a twist of multiplicative reduction. We prove that in the first case all curves have CM and come in families, described explicitly in Theorem \[thm:goodtwist\], using the results from Section \[section CM\]. For the additive ones, we prove in Theorem \[thm:additivetwist\] that there are only some sporadic cases (also all CM), and for multiplicative ones, we prove that there are potentially (and probably) infinitely many, almost all belonging to a family analogous to the so-called *Setzer-Neumann* family (as in [@Setzer]) for curves defined over $\Q$. As in the classical case, all such curves have rank $0$.

The important consequence of this detailed case by case study for our main result is that if $E$ is an elliptic curve defined over $K$ of odd prime conductor with a rational $2$-torsion point, then either $E$ or a $2$-isogenous curve has odd discriminant valuation.

Finally, in the last section, 