Ideal labels (reviewed)

In the LMFDB ideals in rings of integers of number fields are identified using the labeling system developed by John Cremona, Aurel Page and Andrew Sutherland [arXiv:2005.09491].

In a number field $K$, each nonzero ideal $I$ of its ring of integers $\mathcal{O}_K$ is assigned an ideal label of the form $\texttt{N.i}$, where $N$ and $i$ are positive integers, in which $N:=[\mathcal{O}_K:I]$ is the norm of the ideal and $i$ is the index of the ideal in a sorted list of all ideals of norm $N$. Once an integral primitive element $\alpha$ for the field $K$ is fixed, the ordering of ideals of the same norm is defined in a deterministic fashion (involving no arbitrary choices).

In the LMFDB we always represent number fields as $K = \mathbb{Q}[X]/(g(X))$ where $g$ is the unique monic integral polynomial which satisfies the polredabs condition. In this representation the image of $X$ under the quotient map $\mathbb{Q}[X]\rightarrow\mathbb{Q}[X]/(g(X))$ is a canonical integral primitive element $\alpha$ for $K$. Fixing this element determines a unique ordering of all $\mathcal{O}_K$-ideals of the same norm.