John Cremona
- John Cremona and Ariel Pacetti
title: On elliptic curves of prime power conductor over imaginary quadratic fields with class number one
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Introduction {#introduction .unnumbered}
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The theory of elliptic curves plays a crucial role in modern number theory. An important advance came with the systematic construction of tables, as done by the first author for elliptic curves defined over $\Q$ ([@CremonaAlgo]). The original purpose of the present article was to prove analogues over imaginary quadratic fields of class number one to well-known bounds relating the conductor and discriminant of an elliptic curve over $\Q$ with prime power conductor. A useful application of such a result is an effective algorithm to construct tables of elliptic curves of prime power conductor, via solving Thue equations. While succeeding in carrying out our original aim, we also found several examples of phenomena for curves over imaginary quadratic fields that do not occur over $\Q$, and were able to classify all elliptic curves with prime power conductor and nontrivial torsion over the fields in question, extending the results of Miyawaki [@Miyawaki] for elliptic curves over $\Q$ and Shumbusho [@Shumbusho] over imaginary quadratic fields.

Over $\Q$, there are only finitely many elliptic 