\input amstex
\documentstyle{amsppt}
\magnification=\magstep1
\font\petit=cmr6
\font\moyen=cmr8
\def\size{\operatorname{size}}
\def\disc{\operatorname{disc}}
\hbox{}
\vskip 3cm
\parskip 0pt
\topmatter
\title Description and Use of the Tables
\endtitle
\endtopmatter
\document
\vskip 3truecm
For the number fields belonging to tables of reasonable
length available to us, we have computed the signature, the Galois group
of the Galois closure of the field, the discriminant of the number field and the index of
$\Bbb Z[x]$ in the ring of integers, the class number, the structure of the class
group as a product of cyclic groups, an ideal in the class for each class generating these
cyclic groups, the regulator, the number of roots
of unity in the field, a generator of the torsion part of the unit
group and a system of fundamental units.
The computations have been done using the PARI package, which assumes GRH.
All these tables are available by anonymous ftp from
\medskip
\centerline{{\tt megrez.math.u-bordeaux.fr} (147.210.16.17)}
\bigskip
This directory contains subdirectories with files corresponding to tables of
number fields which have been compiled by different authors. The tables under
consideration are tables of degrees 3, 4, 5, 6 and 7 for
all possible signatures.
The length of the tables is rather small for degrees 6 and 7.
Each directory contains indexes corresponding to each signature,
so that it is easy to determine in which table a given number
field belongs (there may be an ambiguity if the same discriminant appears
in two consecutive tables).
For the tables considered, the size and the name of the authors of the computations are as follows:
\vfill\eject
\noindent\newline
\qquad**\quad degree$=3$\quad signature$=(1,1)$\quad $|d_K|\le 1\,000\,000$\quad by M. Olivier\newline
\qquad**\quad degree$=3$\quad signature$=(3,0)$\quad $|d_K|\le 2\,000\,000$\quad by M. Olivier\newline
\qquad**\quad degree$=4$\quad signature$=(0,2)$\quad $|d_K|\le 1\,000\,000$\quad by J. Buchmann, D. Ford, M. Pohst\newline
\qquad**\quad degree$=4$\quad signature$=(2,1)$\quad $|d_K|\le 1\,000\,000$\quad by J. Buchmann, D. Ford, M. Pohst\newline
\qquad**\quad degree$=4$\quad signature$=(4,0)$\quad $|d_K|\le 1\,000\,000$\quad by J. Buchmann, D. Ford, M. Pohst\newline
\qquad**\quad degree$=5$\quad signature$=(1,2)$\quad $|d_K|\le 1\,000\,000$\quad by F. Diaz y Diaz, M. Pohst, A. Schwarz\newline
\qquad**\quad degree$=5$\quad signature$=(3,1)$\quad $|d_K|\le 1\,000\,000$\quad by F. Diaz y Diaz, M. Pohst, A. Schwarz\newline
\qquad**\quad degree$=5$\quad signature$=(5,0)$\quad $|d_K|\le 20\,000\,000$\quad by F. Diaz y Diaz, M. Pohst, A. Schwarz\newline
\qquad**\quad degree$=6$\quad signature$=(0,3)$\quad $|d_K|\le 200\,000$\quad by M. Olivier\newline
\qquad**\quad degree$=6$\quad signature$=(2,2)$\quad $|d_K|\le 400\,000$\quad by M. Olivier\newline
\qquad**\quad degree$=6$\quad signature$=(4,1)$\quad $|d_K|\le 1\,000\,000$\quad by M. Olivier\newline
\qquad**\quad degree$=6$\quad signature$=(6,0)$\quad $|d_K|\le 10\,000\,000$\quad by M. Olivier\newline
\qquad**\quad degree$=7$\quad signature$=(1,3)$\quad $|d_K|\le 600\,000$\quad by P. L\'etard\newline
\qquad**\quad degree$=7$\quad signature$=(3,2)$\quad $|d_K|\le 1\,800\,000$\quad by P. L\'etard\newline
\qquad**\quad degree$=7$\quad signature$=(5,1)$\quad $|d_K|\le 12\,000\,000$\quad by P. L\'etard\newline
\qquad**\quad degree$=7$\quad signature$=(7,0)$\quad $|d_K|\le 150\,000\,000$\quad by P. L\'etard
\vskip 1truecm
The files containing the arithmetic information
are denoted by $txy.zzz$ where $x$ is a digit that indicates the degree
of the number field in the table, $y$ is a digit corresponding to the
number of real places of the number field in the table
and $zzz$ is a three digits number that denotes
the order number of the file in the $txy.$-table
when the corresponding table contains more than 1000 discriminants.
Hence, the length of $txy.zzz$ is exactly equal to 1000 except for the largest
value of $zzz$ in a given signature. In this last case the length is always less than 1000.
\bigskip
Each file contains a single Pari vector having 1000 components (except
for the last which may contain less), each component corresponding to
a single number field. After being gunzipped, they are human-readable, but
are also made to be read by GP. Since the file size is much larger than
the default GP buffer (30000), and the necessary stack size is also
insufficient, one should use the following command to launch GP:
\medskip
\centerline{\tt gp -s 10000000 -b 500000}
\medskip
and then use the usual GP command {\tt $\backslash$r filename} to read in the
1000 (or less) number fields as a single vector {\tt v}.
\bigskip
The information contained in each one of these components appears as a new vector
having 9 components and the following structure :
\vfill\eject
Let $k,\ 1\le k\le m$ be fixed. Then we have
$$\matrix
v[k][1]=P(x)&\text{the polynomial}\\
&\text{generating the field (1)}\\
&\\
&\\
v[k][2]=[r_1,r_2]&\text{the signature of the field (2)}\\
&\\
&\\
v[k][3]=[\# G,s,m]&\text{the Galois group $G$ of $P(x)$ (3)}\\
&\\
&\\
v[k][4]=[d,a]&\text{the discriminant}\\
&\text{of the field and the index of}\\
&[\Bbb Z_K:\Bbb Z[x]]\\
&\\
&\\
v[k][5]=[1,w_2,\dots,w_n]&\text{an integral basis given}\\
&\text{in HNF on the power basis}\\
&\\
&\\
v[k][6]=[h,[c_1,\dots,c_t],[\frak a_1,\dots,\frak a_t],[\alpha_1,\dots,\alpha_t]]&\text{the class number, the}\\
&\text{structure of the class group }\\
&\text{as a product of cyclic groups}\\
&\text{of orders $c_1,\dots,c_t$,}\\
&\text{an ideal generating these cyclic }\\
&\text{groups and the generator of the}\\
&\text{principal ideal $\frak a_i^{c_i}$ (4)}\\
&\\
&\\
v[k][7]=R&\text{the regulator}\\
&\\
&\\
v[k][8]=[w,\zeta]&\text{the number of roots of}\\
&\text{unity in the field and a generator}\\
&\text{of this cyclic group (5)}\\
&\\
&\\
v[k][9]=[u_1,\dots,u_r]&\text{a system of fundamental units (6)}
\endmatrix
$$
\bigskip
(1)\ \ The polynomial used to define a number field is not completely
canonical,
but can be obtained via a completely deterministic process:
If $P(X)=a_nX^n+\cdots+a_0$, we set $\size(P)=\sum_i|\theta_i|^2$, where
the $\theta_i$ are the complex roots of $P$. Then the polynomial $P$
which is chosen is a {\it monic\/} polynomial which minimizes for the
lexicographic order the vector
$v(P)=(\size(P),|\disc(P)|,|a_n|,\dots,|a_0|)$
and such that the non-zero monomial of largest degree $d$ such that
$d\not\equiv n\pmod2$, is one exists, is negative.
It is possible that this still does not determine the polynomial $P$
completely, but in the range of our tables it does.
Note that the choice of $P$ is slightly different from the one made in
[Cohen] (p. 170-171), in particular because of the exchange of the
$\size(P)$ and $|\disc(P)|$ components. Indeed, making a complete list of
polynomials $P$ having the smallest possible $\size(P)$ is a straightforward
backtracking procedure, and in this small finite list one can then choose
the smallest lexicographic polynomial. On the other hand, finding the smallest
possible discriminants involve finding all the solutions of index norm
equations, and this is a difficult process.
\bigskip
(2)\ \ As usual, $r_1$ denotes the number of real places of $K$ and $r_2$
the number of pairs of complex places.
\bigskip
(3)\ \ The Galois group $G$ of $P(x)$ is given as a three-component vector.
The first component gives the order of the group, the second component denotes
the group signature (hence, $s=1$ if $G\subset A_n$ and $s=-1$ otherwise) and
the third component denotes the number of the groups corresponding to the same
pairs $(\#G,s)$. For the tables considered here, one has $m=1$ except in two
cases in degree six:\newline
$C_6=[6,-1,1],\ S_3=[6,-1,2],\ S^-_4=[24,-1,1],\ A_4\times C_2=[24,-1,2]$.
\bigskip
(4)\ \ The first component of this vector denotes the class number. When one
has $h=1$, the other three components are empty vectors. The second component is a vector
$[c_1,\dots,c_t]$ giving the structure of the class group as a product of
cyclic groups:
$$\Cal H_K\thickapprox \Bbb Z/c_1\Bbb Z\times\cdots\times\Bbb Z/c_t\Bbb Z$$
with $c_i|c_{i-1}$. The third component is an other vector having $t$
components each one of them gives an ideal in
a class generating the corresponding cyclic group. Finally,
the last component is a vector whose $i$th-component is a generator of
the principal ideal $\frak a_i^{c_i}$.
Note that the ideals used as generators of the cyclic factors are
not canonical, and we are aware that it is easy to find much
smaller ideals in many cases.
\bigskip
(5)\ \ The first component of this vector denotes the order of the group
of roots of unity into the field and the second component gives a
generator of this cyclic group.
\bigskip
(6)\ \ As usual, one has $r=r_1+r_2-1$ for the rank of the unit group.
All the elements of the field are given as vectors in the integral basis. All
the ideals are given by their HNF matrix in the integral basis.
\bigskip
{\sl Evidently we welcome corrections to the tables. We also welcome additional
contributions. However, in that case we ask you to use exactly the same
format for submitting your tables. If you want to submit only the equations
for the number fields, we can compute the necessary invariants. Note that
additional data is most welcome in tables where only a small number of fields
is available (e.g. not in degree 3 or 4).}
\enddocument