$\operatorname{{\mathcal N}}({{\mathcal{D}}}_{\min}(E)) \le \normid{p}^{6}$, the twisted curve $E'$ satisfies $\operatorname{{\mathcal N}}({{\mathcal{D}}}_{\min}(E'))=\operatorname{{\mathcal N}}({{\mathcal{D}}}_{\min}(E)) \cdot \normid{p}^6$ and $\operatorname{cond}(E')=\id{p}^2$.

By Theorem \[thm:mainthm\] the result holds for some curve in the isogeny class. Furthermore, when the isogeny class of a semistable curve has at most $2$ elements, Theorem \[thm:absred2\] and Corollary \[coro:2torsion-odddiscriminant\] prove that the discriminant valuation is at most $2$.

When there is a $3$-isogeny, the isogenous curve has discriminant valuation $3$ or $6$ (depending on whether or not the conductor is ramified in $K$), and there is a unique semistable case with a $5$-isogeny (see Theorem \[thm:5-torsion\]), which is precisely the curve [[](http://www.lmfdb.org/EllipticCurve/Q/11/a/2)]{}.

We end this section with a result used in the proof of Theorem \[thm:mainthm\] above.

Let $K$ be an imaginary quadratic field of class number $1$ and $E/K$ be a semistable elliptic curve. Let $\ell \ge 3$ be a prime number such that the residual representation of $E$ at $\ell$ is not absolutely irreducible. Then either $E$, or a curve $\ell$-isogenous to $E$ over $K$, has a point of order $\ell$ defined over $K$. \[thm:Kraus\]

The residual representation associated to $E$ takes values in $\operatorname{GL}(2,\F_\ell)$, and may be either reducible (over $\F_{\ell}$), irreducible but absolutely reducible (i.e., reducible over 