In future we present the files giving the relative class numbers of the imaginary abelian number fields of non prime power conductors < 10000.

Files are divided into two types:

The file “nb a b” presents for each non prime power conductor f with a ≦ f < b, absolute norms of − \frac{1}{2} B_{1, χ} and their factorization (not necessarily complete) for odd Dirichlet characters χ of non prime power conductors f < 10000. Regrettably, different from the prime power conductor case, values are not uniquely determined by conductor and degree of χ. Therefore, for specification of values, we need additional information. The file “rcnf a b” presents the relative class numbers of the imaginary abelian number fields of non prime power conductors f < 10000 with a ≦ f < b.

We use the following notations. χ_{4} denotes the unique primitive Dirichlet character of conductor 4. For odd prime numbers p, χ_{p} denotes a primitive Dirichlet character of conductor p and order p − 1. For prime powers q = p^{m} (≠ 4), ψ_{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.

We tried to factor N(− \frac{1}{2} B_{1, χ}) as possible. 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’.

In the File “rcnf a b”, for each non prime power conductor f with a ≦ f < b. the relative class numbers of the imaginary abelian number fields of non prime power conductors f < 10000 are given. For each f, data are ordered by conductor and degree. Regrettably, different from the prime power conductor case, fields are not uniquely determined by its conductor and degree. Therefore, for specification of fields, we need additional information. We do not give information on factorization.

The computation was done on a PC. We used the vector manipulation method. (See [2].)

I will appreciate any information on further factorizations. Acknowledgement. I am indebted for factorization and primality proving to the

following:

factorize.c (included in GMP) due to GNU; GMP-ECM due to P. Zimmermann et al.; PPMPQS.EXE due to S. Tomabechi (available at http://www.asahi-net.or.jp/˜ KC2H-MSM/mathland/matha1/index.htm); msieve.EXE due to J. S. Papadopoulos (available at http://www.boo.net/˜ jasonp/qs.html); Titanix and Primo due to M. Martin (available^{1} at http://www.ellipsa.net/).

I appreciate the programmers and contributors.

1Now, the use is forbidden in Canada, Japan and USA because of Patent Office Policy.

References

1. H. Hasse, Über die Klassenzahl abelscher Zahlkörper, Akademie-Verlag, Berlin, 1952; MR 14, 141; Nachdruck der ersten Aufgabe. Mit einer Einleitung von J. Martinet, Springer-Verlag, New York and Berlin, 1985; MR 87j:11122a.

2. R. Spira, Calculation of the first factor of the cyclotomic class number, Computers in number theory (A. O. L. Atkin and B. J. Birch, ed.), (Proc. Sci. Res. Council Atlas Sympos. No. 2, Oxford, 1969), Academic Press, London and New York, 1971, pp. 149–151.