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These files present 
\roster
\item"" the relative class numbers of the imaginary abelian number fields
of prime power conductors $<10000$.
\endroster
Files are divided into two types:
\roster
\item"" The file ``rcn\_a\_b'' presents
for each prime number $p$ with $a\leqq p<b$,
(normalizations of) absolute norms of
$-\frac{\,1\,}{\,2\,}\,B_{1,\,\chi}$
and their factorization (not necessarily complete)
for odd Dirichlet characters $\chi$ of
$p$-power conductors $<10000$.
\item"" The file ``rcnf\_a\_b'' presents
the relative class numbers of the imaginary abelian number fields
of $p$-power conductors $<10000$
for each prime number $p$ with $a\leqq p<b$.
\endroster

We use the following notations.
$\chi_4$ denotes the unique primitive Dirichlet character of conductor $4$.
For odd prime numbers $p$, $\chi_p$ denotes
a primitive Dirichlet character of conductor $p$ and order $p-1$.
For prime powers $q=p^m$ ($\ne 4$), $\psi_q$ denotes
an even primitive Dirichlet character of conductor $q$ and
order $p^{m-1}$ or $2^{m-2}$ according as $p$ is odd or $p=2$.
For a prime power $q=p^m$ and an even integer $e$,
$h_{e}(q)$ denotes the relative class number of conductor $q$ and degree $e$,
that is, $h_{e}(q)$ is the normalization of the absolute norm of
$-\,\frac{\,1\,}{\,2\,}\,B_{1,\,\chi}$
for (odd) primitive Dirichlet character of conductor $q$ and order $e$:
If $p=2$, then $h_{2}(4)=4N(-B_{1,\chi}/2)=4(-B_{1,\chi}/2)$ and
$$
h_{e}(q)=2\ds \prod_{\ord(\chi)=e}
\left(-\,\frac{\,1\,}{\,2\,}\,B_{1,\,\chi}\right)
$$
for $q\geqq 8$.
If $p$ is odd and not a Fermat prime,
$$
h_{e}(q)=p\ds \prod_{\ord(\chi)=e}
\left(-\,\frac{\,1\,}{\,2\,}\,B_{1,\,\chi}\right)
$$
for $e=\varphi(q)$,
$$
h_{e}(p)=2\ds \prod_{\ord(\chi)=e}
\left(-\,\frac{\,1\,}{\,2\,}\,B_{1,\,\chi}\right)
$$
for $e$ the exact $2$-power dividing $p-1$, and
$$
h_{e}(q)=\ds \prod_{\ord(\chi)=e}
\left(-\,\frac{\,1\,}{\,2\,}\,B_{1,\,\chi}\right)
$$
otherwise.
If $p$ is an odd Fermat prime,
necessarily we have $e=\varphi(q)$ and,
$$
h_{p-1}(p)=2p\ds \prod_{\ord(\chi)=e}
\left(-\,\frac{\,1\,}{\,2\,}\,B_{1,\,\chi}\right),
$$
and
$$
h_{e}(q)=p\ds \prod_{\ord(\chi)=e}
\left(-\,\frac{\,1\,}{\,2\,}\,B_{1,\,\chi}\right)
$$
for $q>p$.

$K_{f,\,d}$ denotes the imaginary abelian number field
of prime power conductor $f$ and degree $d$,
and $h^-(K_{f,\,d})$ its relative class number.
Then we have
$$
h^-(K_{f,\,d})=\ds \prod_{q\mid f,\ e\mid d}h_{e}(q).
$$

In the File ``rcn\_a\_b'',
for each prime number $p$ with $a\leqq p<b$,
the values of the relative class numbers $h_{e}(q)$ of degrees $e$
are given for $q$, $p$-powers $<10000$.
For each $p$,
data are ordered by $q$ and $e$.
For each $q$ and $e$,
the data are character $\chi$, its conductor $f(\chi)$,
its order $\ord(\chi)$, and $h_{e}(q)$.
We tried to factor $h_{e}(q)$ as possible.
(Factorizations are complete for numbers of at most $130$ digits.)
p(n) means the number is a prime with n (decimal) digits and
c(n) means the number is a composite with n digits.
c means the factor is a composite.
For large probable primes,
we certified the primality by `Titanix' and `Primo'.
We do not execute the primality proving
for the following titanic probable primes:
\roster
\item"" $h_{8760}(8761)$ : $2709$ decimal digit probable prime.
\item"" $h_{6580}(6581)$ : $2458$ decimal digit probable prime.
\endroster

In the File ``rcnf\_a\_b'',
for each prime number $p$ with $a\leqq p<b$.
the relative class numbers of the imaginary abelian number fields
of $p$-powers $<10000$ are given.
For each $p$,
data are ordered by conductor and degree.
Each relative class number is expressed as the product of $h_{e}(q)$.
But we do not give information on factorization.

The computation was done on a NeXT computer and PC.
We used both the vector manipulation method and
the ring homomorphism -- modular arithmetic method.
(See \cite{1,2,3,5,6}.)
It seems that the former is neglected.
However, when $\varphi(f)$ ($f$ conductor) has only small prime factors,
we can compute absolute norms of integers in cyclotomic fields
several times faster by the former than by the latter
by taking relative norms repeatedly.

Remark.
M. A. Shokrollahi \cite{5} has already computed
the relative class numbers of imaginary abelian number fields
of prime conductor $<10000$,
and his data are available from amin\@shokrollahi.com upon request.
However, his data are not kind for others in the following points:
First, each number has radix $2^{32}$ and
each digit is hexadecimal.
Each relative class number is factored in some sense,
however the meaning of factors is not clear.
Some error can be found.
Data are excluded for Fermat primes.

Complete factorizations of the relative class numbers of
$p$\/th cyclotomic fields for $p\leqq 509$
are given in R. Schoof \cite{4}.
We have extended this to $p\leqq 631$.

I will appreciate any information on further factorizations.

Acknowledgement.
I am indebted for factorization and primality proving to the following:
\NoBlackBoxes
\roster
\item"" factorize.c (included in GMP) due to GNU;
\item"" GMP-ECM due to P. Zimmermann et al.;
%\item"" FACTOR.EXE due to Shamus Software Ltd.;
\item"" PPMPQS.EXE due to S. Tomabechi
(available at\newline
http://www.asahi-net.or.jp/\~{}KC2H-MSM/mathland/matha1/index.htm);
\item"" msieve.EXE due to J. S. Papadopoulos\newline
(available at
%http://www.boo.net/\~{}jasonp/qs.html);
http://sourceforge.net/projects/msieve/);
\item"" ggnfs due to C. Monico;
\item"" Titanix and Primo due to M. Martin
(available
%\footnote{Now, the use is forbidden in Canada, Japan and USA
%because of Patent Office Policy.}
at http://www.ellipsa.net/).
\endroster
I appreciate the programmers and contributors.
\newpage

\Refs

\ref \no 1 \by G. Fung, A. Granville, and H. C. Williams
\paper Computation of the fisrt factor of the class number of
cyclotomic fields
\jour J. Number Theory
\vol 42 \yr 1992 \issue 3 \pages 297--312
\moreref MR 93k:11097
\endref

\ref \no 2 \by G. Fee and A. Granville
\paper The prime factors of Wendt's binomial circulant determinant
\jour Math. Comp.
\vol 57 \yr 1991 \issue 196 \pages 839--848
\moreref MR 92f:11183
\endref

\ref \no 3 \by H. Hasse
\book \"Uber die Klassenzahl abelscher Zahlk\"orper
\publ Akademie-Verlag \publaddr Berlin \yr 1952
\moreref MR {\bf 14}, 141
\moreref Nachdruck der ersten Aufgabe.
Mit einer Einleitung von J. Martinet 
\publ Springer-Verlag \publaddr New York and Berlin \yr 1985
\moreref MR 87j:11122a
\endref

\ref \no 4 \by R. Schoof
\paper Minus class groups of the fields of the $l$-th roots of unity
\jour Math. Comp. \vol 67 \yr 1998 \issue 223 \pages 1225--1245
\moreref MR 98j:11085
\endref

\ref \no 5 \by M. A. Shokrollahi
\paper Relative class number of imaginary abelian fields
of prime conductor below $10000$
\jour Math. Comp. \vol 68 \yr 1999 \issue 228 \pages 1717--1728
\moreref MR 99m:11147
\endref

\ref \no 6 \by R. Spira
\paper Calculation of the first factor of the cyclotomic class number
\inbook Computers in number theory
\bookinfo (Proc. Sci. Res. Council Atlas Sympos. No. 2, Oxford, 1969)
\ed A. O. L. Atkin and B. J. Birch
\publ Academic Press \publaddr London and New York
\yr 1971 \pages 149--151
\endref

\endRefs

\enddocument
