Labels for number fields

Each (global) number field has a unique label of the form d.r.D.i where

• $d$ is the degree;
• $r$ is the real signature; the full signature is therefore $[r,(d-r)/2]$;
• $D$ is the absolute value of the discriminant;
• $i$ is the index, counting from 1. This is in case there is more than one field with the same signature and absolute value of the discriminant: for example 4.0.1008.1 and 4.0.1008.2.

The discriminant portion of the label can take the form $a_1$ e $\epsilon_1$ _ $a_2$ e $\epsilon_2$ _ $\cdots$ _ $a_k$ e $\epsilon_k$ to mean the absolute value of the discriminant equals $a_1^{\epsilon_1}a_2^{\epsilon_2}\cdots a_k^{\epsilon_k}$. The separators are the letter e and the underscore symbol.