Conductor labels for elliptic curves (reviewed)

The LMFDB uses a systematic labelling of conductors of elliptic curves defined over each number field $K$. The conductors are integral ideals of $K$. For an ideal of norm $N$ the label has the form $N.i$ where $i$ is the index (starting from $1$) of the ideal in an ordered list of all integral ideals of norm $N$.

• over imaginary quadratic fields, we use the standard labelling scheme for ideals;
• over totally real fields, we use the same labels for ideals as are used for levels of Hilbert modular forms; the order of these is recorded in the database;
• over other fields, the ordering of the ideals of fixed norm is recorded in the database.

In future it is planned to move to using the standard labelling scheme for ideals over all fields.