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