Tables

1. Bernoulli computations
2. Relative class number computations
   1. Class number of subfields
   2. Partial factorization
   3. Statistics file I
   4. Statistics file II
   5. Quadratic subfields

The file Table.gz contains a (compressed) table of the irregular pairs (and Vandiver and cyclotomic residues) for all primes up to 8 million. The compressed file is about 6MB and the uncompressed file is about 13MB. Note that considerable space savings can be realized just by deleting all primes of index of irregularity equal to 0 (i.e., all regular primes).

A typical line in the table looks like

P k t1:v1,s11,s31 t2:v2,s12,s32 ... tk:vk,s1k,s3k

where k is the index of irregularity for a prime p and (P,t1) ... (P,tk) are the irregular pairs. The s_{ij} and the v_j are additional data for verifying Vandiver's conjecture and computing the lambda-invariant.

For additional information, consult the paper by J. Buhler, R. Crandall, R. Ernvall, and T. Metsänkylä, Irregular primes and cyclotomic invariants to four million. Math. Comp., 61, no. 203, 151--153 (1993), or the more recent paper by J. Buhler, R. Crandall, R. Ernvall, T. Metsänkylä, and M.A. Shokrollahi, Irregular primes below 8 million .

The file REL-SUBF.gz contains the relative class number of the cyclotomic field Q(zeta_p) and all its non-real subfields for all primes p, 3 < p < 10000, p not a Fermat prime .

A typical entry of the table is

1bb 4

1ba d

d46 6240f605 9ef38e12 6b5efe71 37498448 f5aab571 b7e0e4c3 8f94c29b 2cfeb850 0b85a3a0 47aa3987 b0889612 360714c9

22 1

6da68769

1a 1

2b1d567

2 1

5

The first line is of the form P d , where P is the prime in hexadecimal and d is the number of odd divisor of p-1, also in hexadecimal. In the example above p=443 and d=4.

The next lines start with D L a_{L-1} ... a_0 . D is the degree of the subfield over the rationals, L is the number of 32-bit integers to follow, a_{L-1},...,a_0 is the 2^32-ary representation of the relative class number of that subfield.

In the example above, the relative class number of Q(zeta_{443}) equals

133903139692160215294582251745437893243000261183001775228876376028383380685059813631891614281926522017244649796657419465

that of the subfield of degree 34 equals 1839630185, that of the subfield of degree 26 equals 45208935, and that of the quadratic subfield equals 5.

The uncompressed file is about 4 M, and the compressed file is about 2.3 M.

The file REL-FACT.gz contains a partial factorization of the relative class number h^-(p) of the p-th cyclotomic field for all primes p, 3 < p < 10000, p not a Fermat prime .

A typical entry of the table is

1bb:

:: 398204 f2867c14 ee921850 239969e3 46af836e 741f9e04 13d74fb5 d1e4f37a f5547008 528e44ac 29dcd4d3

::15ee1b15

:: 89f77b

:: 5

The entries are in hexadecimal representation. The first line corresponds to the prime, in this case 443. The next lines begin with a double colon :: and contain different factors of the class number in 2^32-ary notation. Hence, the first entry corresponds to the number

8050186860070466537284940331284584002110165270872096566461673720566704214999469121068754923269188801747

REL-FACT.gz is 1.7MB large. The uncompressed version is about 3.1MB.

The file REL-STATI.gz contains statistics for divisibility properties of relative class numbers of imaginary abelian fields of prime conductor below 10000 by primes below 100. A typical entry in this file is of the form

P D

d_1 n2 n3 n5 n7 .... n97

d_2 n2 n3 n5 n7 .... n97

...

Here P is the conductor of the field, D is the number of imaginary subfields of the cyclotomic field of conductor P, d_i is the degree of the subfield over the rationals, and n_q is the exact power of the prime q dividing the relative class number of the Abelian field in question.

For example, an entry in the file is

The uncompressed file is 550 KB, the compressed version is 39 KB large.

The file REL-STATII.gz contains some other statistics for the above primes. A typical line in that file is of the form

2957 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0.911446

The first entry corresponds to the prime. The next one is the exact power of 2 dividing the relative class number. Entries 3-27 correspond to divisibility by the first 24 odd primes, a 1 indicating divisibility. Entry 28 is 1 iff the prime is irregular. Entry 29 is the ratio of h^-(p) and the function G(p), see the paper by Fung et al., Journal of Number Theory, 42, 297-312 (1992).

The uncompressed file is 83KB large. The compressed file is 13KB large.

The file REL-QUAD.gz contains the relative class number of imaginary quadratic subfields of Q(zeta_p) for all primes p between 3 and 10000, p congruent to 3 mod 4. It was only used for program checking.

The uncompressed file is 13KB, the compressed file is 2.3KB large.