Let $E$ be an elliptic curve over a local field $K$.

If $E$ has non-integral $j$-invariant, under what conditions will there exist an isogenous curve with integral $j$-invariant?

Here, saying an element $a \in K$ is integral means that $a$ belongs to the ring of integers of $K$.

I feel that the following result may be relevant.

Let $E$ be an elliptic curve over a local field $K$. We say $E$ has potential good reduction over $K$ if there is a finite extension $K^{\prime}/K$ such that $E$ has good reduction over $K^{\prime}$. The $j$-invariant of $E$ is integral if and only if $E$ has potential good reduction over $K$.