absolutely irreducible. While working over the rationals, either the residual image is reducible, in which case the curve has a $\Q$-point (which does not happen for $\ell>7$) or it is absolutely irreducible. The reducible case for small primes can be discarded by a result of Fontaine ([@Fontaine Theorem B]).

One of the main results of this article is a generalization of Theorem \[thm:MO\] to $K$.

Let $K$ be an imaginary quadratic field of class number $1$. Let $E/K$ be a modular elliptic curve of prime conductor. Assume Conjecture \[conj:loweringlevel\] holds. Then there exists a curve isogenous to $E$ over $K$ with prime or prime square discriminant. \[thm:mainthm\]

Let $\id{p}$ be the conductor of $E$. For the case $\id{p}\mid 2$, A. Koutsianas used the methods of [@Angelos] to compute for us all elliptic curves with conductor a power of $\id{p}$, and found that over each of the nine fields there are none with conductor $\id{p}$. This is consistent with the tables of automorphic forms of [@CremonaTessellations] and [@Cremona-Whitley].

From now on we may assume that $\id{p} \nmid 2$. Suppose that ${{\mathcal{D}}}(E)$ is not prime, and let $\ell$ be a prime number dividing $v_{\id{p}}({{\mathcal{D}}}(E))$. Recall that by semistability and Mazur’s result, the 