On the $p$-isogenies of elliptic curves with multiplicative reduction over quadratic fields

Let $q>5$ be a prime and $K$ a quadratic number field. In this article we extend a previous result of Najman and the author and prove that if $E\mathrm{/}K$ is an elliptic curve with potentially multiplicative reduction at all primes $\mathfrak{q}\mid q$, then $E$ does not have prime isogenies of degree greater than $71$ and different from $q$. As an application to our main result, we present a variant of the asymptotic version of Fermat’s Last Theorem over quadratic imaginary fields of class number one.