These files present the relative class numbers of the imaginary abelian number fields of prime power conductors < 10000.

Files are divided into two types:

The file “rcn a b” presents for each prime number p with a ≦ p < b, (normalizations of) absolute norms of − \frac{1}{2} B_{1, χ} and their factorization (not necessarily complete) for odd Dirichlet characters χ of p-power conductors < 10000. 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 ≦ p < 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. 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, χ} for (odd) primitive Dirichlet character of conductor q and order e: If p = 2, then h_{2}(4) = 4N(−B_{1,χ}/2) = 4(−B_{1,χ}/2) and

h_{e}(q) = 2 ∏_{ord(χ)=e} (− \frac{1}{2} B_{1, χ})

for q ≧ 8. If p is odd and not a Fermat prime,

h_{e}(q) = p ∏_{ord(χ)=e} (− \frac{1}{2} B_{1, χ})

for e = φ(q),

h_{e}(p) = 2 ∏_{ord(χ)=e} (− \frac{1}{2} B_{1, χ})

for e the exact 2-power dividing p − 1, and

h_{e}(q) = ∏_{ord(χ)=e} (− \frac{1}{2} B_{1, χ})

otherwise. If p is an odd Fermat prime, necessarily we have e = φ(q) and,

h_{p−1}(p) = 2p ∏_{ord(χ)=e} (− \frac{1}{2} B_{1, χ}) ,

and

h_{e}(q) = p ∏_{ord(χ)=e} (− \frac{1}{2} B_{1, χ})

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}) = ∏_{q|f, e|d} h_{e}(q).

In the File “rcn a b”, for each prime number p with a ≦ 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 χ, its conductor f(χ), its order ord(χ), 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:

h_{8760}(8761) : 2709 decimal digit probable prime. h_{6580}(6581) : 2458 decimal digit probable prime.

In the File “rcnf a b”, for each prime number p with a ≦ 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 [1,2,3,5,6].) It seems that the former is neglected. However, when φ(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 [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 pth cyclotomic fields for p ≦ 509 are given in R. Schoof [4]. We have extended this to p ≦ 631.

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://sourceforge.net/projects/msieve/); ggnfs due to C. Monico; Titanix and Primo due to M. Martin (available at http://www.ellipsa.net/).

I appreciate the programmers and contributors.

References

1. G. Fung, A. Granville, and H. C. Williams, Computation of the fisrt factor of the class number of cyclotomic fields, J. Number Theory 42 (1992), no. 3, 297–312; MR 93k:11097.

2. G. Fee and A. Granville, The prime factors of Wendt’s binomial circulant determinant, Math. Comp. 57 (1991), no. 196, 839–848; MR 92f:11183.

3. 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.

4. R. Schoof, Minus class groups of the fields of the l-th roots of unity, Math. Comp. 67 (1998), no. 223, 1225–1245; MR 98j:11085.

5. M. A. Shokrollahi, Relative class number of imaginary abelian fields of prime conductor below 10000, Math. Comp. 68 (1999), no. 228, 1717–1728; MR 99m:11147.

6. 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.