1Function: bnrdisclist
2Section: number_fields
3C-Name: bnrdisclist0
4Prototype: GGDG
5Help: bnrdisclist(bnf,bound,{arch}): list of discriminants of
6 ray class fields of all conductors up to norm bound.
7 The ramified Archimedean places are given by arch; all possible values are
8 taken if arch is omitted. Supports the alternative syntax
9 bnrdisclist(bnf,list), where list is as output by ideallist or ideallistarch
10 (with units).
11Doc: $\var{bnf}$ being as output by \kbd{bnfinit} (with units), computes a
12 list of discriminants of Abelian extensions of the number field by increasing
13 modulus norm up to bound \var{bound}. The ramified Archimedean places are
14 given by \var{arch}; all possible values are taken if \var{arch} is omitted.
15
16 The alternative syntax $\kbd{bnrdisclist}(\var{bnf},\var{list})$ is
17 supported, where \var{list} is as output by \kbd{ideallist} or
18 \kbd{ideallistarch} (with units), in which case \var{arch} is disregarded.
19
20 The output $v$ is a vector, where $v[k]$ is itself a vector $w$, whose length
21 is the number of ideals of norm $k$.
22
23 \item We consider first the case where \var{arch} was specified. Each
24 component of $w$ corresponds to an ideal $m$ of norm $k$, and
25 gives invariants attached to the ray class field $L$ of $\var{bnf}$ of
26 conductor $[m, \var{arch}]$. Namely, each contains a vector $[m,d,r,D]$ with
27 the following meaning: $m$ is the prime ideal factorization of the modulus,
28 $d = [L:\Q]$ is the absolute degree of $L$, $r$ is the number of real places
29 of $L$, and $D$ is the factorization of its absolute discriminant. We set $d
30 = r = D = 0$ if $m$ is not the finite part of a conductor.
31
32 \item If \var{arch} was omitted, all $t = 2^{r_1}$ possible values are taken
33 and a component of $w$ has the form
34 $[m, [[d_1,r_1,D_1], \dots, [d_t,r_t,D_t]]]$,
35 where $m$ is the finite part of the conductor as above, and
36 $[d_i,r_i,D_i]$ are the invariants of the ray class field of conductor
37 $[m,v_i]$, where $v_i$ is the $i$-th Archimedean component, ordered by
38 inverse lexicographic order; so $v_1 = [0,\dots,0]$, $v_2 = [1,0\dots,0]$,
39 etc. Again, we set $d_i = r_i = D_i = 0$ if $[m,v_i]$ is not a conductor.
40
41 Finally, each prime ideal $pr = [p,\alpha,e,f,\beta]$ in the prime
42 factorization $m$ is coded as the integer $p\cdot n^2+(f-1)\cdot n+(j-1)$,
43 where $n$ is the degree of the base field and $j$ is such that
44
45 \kbd{pr = idealprimedec(\var{nf},p)[j]}.
46
47 \noindent $m$ can be decoded using \tet{bnfdecodemodule}.
48
49 Note that to compute such data for a single field, either \tet{bnrclassno}
50 or \tet{bnrdisc} are (much) more efficient.
51