Notation: Let $K/\mathbb{Q}$ be a quadratic number field; let $p\geq 3$ be a rational prime and let $\mathfrak{p}$ denote a prime lying above $p$; let $K_{\mathfrak{p}}$ denote the completion of $K$ with respect to the prime $\mathfrak{p}$; let $R$ denote the ring of integers of $K_{\mathfrak{p}}$.
Let $E/\mathbb{Q}$ be an elliptic curve define over $\mathbb{Q}$, however, we want to consider $E$ as an elliptic curve over $K$. Suppose that $v_{\mathfrak{p}}(j(E)) \geq 0$, which is equivalent to $j(E) \in R$. From Proposition VII.5.5 of Silverman's AEC, we know that $E$ has potential good reduction at $\mathfrak{p}$, meaning that there exists a finite extension $K'/K$ such that $E$ has good reduction modulo all primes $\mathfrak{p}_{K'}|\mathfrak{p}$ over $K'$.
Question: Can we pick $K'/K$ to be a normal extension such that $\gcd(p,[K':K]) = 1$?
Assumptions: I know that $E/\mathbb{Q}$ does not have any $\mathbb{Q}$-rational torsion of prime order, however, over the quadratic extension $K$, $E$ does gain a $K$-rational point of order 3,5 or 7. Also, one may assume that $K$ has class number 1, if that helps.
Example: In a paper of Frey, he constructs such extensions for elliptic curves defined over $\mathbb{Q}$ that have a $\mathbb{Q}$-rational point $P$ of order $p$ for $p\in \{5,7\}$, where $P$ is not contained in the kernel of reduction modulo $p$. In particular, he writes if $v_p(j(E)) \geq 0$, then $E$ has good reduction modulo all primes $\mathfrak{p}_N | p$ where $$N = \mathbb{Q}(\zeta_{12},\sqrt[12]{p}).$$ The case of $p = 3$ is dealt with separately.
Intial Idea: From the proof of Proposition VII.5.5, one can take $K'$ to be the extension of $K$ over which $E$ has a Weierstrass equation in Legendre form. From Proposition III.1.7 of Silverman's AEC, we have a procedure for writing $E$ in Legendre form using the defining cubic equation, however, it is not immediately clear to me how to use this construction to determine the degree of $K'$ over $K$.
Thank you in advance for you time!
