with their LMFDB label, see [@lmfdb].

It is well-known that given an elliptic curve $E/K$, if the class number of $K$ equals $1$ then $E$ has a global minimal model. Such model might not be unique, hence there is in general no notion of a *minimal discriminant*. However, over an imaginary quadratic field, all units are annihilated by $12$, and hence the value of the minimal discriminant ${{\mathcal{D}}}(E)$ is well-defined.

**Acknowledgments:** we would like to thank Nicolas Vescovo for participating in some discussions of the present article, to Luis Dieulefait, for explaining to us some technicalities used in the proof of Theorem \[thm:Kraus\], to Angelos Koutsianas for his computations used in the proof of Theorem \[thm:mainthm\], and to Samir Siksek and Haluk [Ş]{}engün for many useful conversations. The second author would like to thank the University of Warwick for its hospitality during his visit as a Leverhulme Visiting Professor. Last but not least, we would like to thank the anonymous referee for many useful comments and corrections.

Bianchi modular forms and modularity of elliptic curves
=======================================================

For background on Bianchi modular forms and modularity of elliptic curves defined over imaginary quadratic fields, we refer to the survey article of [Ş]{}engün 