Sextic Number Fields with Discriminant - j 2 a 3 b

Complete lists of number fields, of given degree n and unramified outside a given finite set S of primes, are both of intrinsic interest and useful in some applications. For degrees n ≤ 5 and S = {∞, 2, 3}, the complete lists have appeared previously; there are in total 85 such fields. Here we give the complete list for n = 6 and S = {∞, 2, 3}, finding in particular exactly 398 such fields. We use a three-pronged approach to obtain this classification: an exhaustive computer search, sextic twinning, and class field theory. Also we completely identify the 2-adic and 3-adic completions of all these degree ≤ 6 fields, this information being one of the focal points of interest and essential in applications. There is a considerable literature on the classification of number fields by means of their discriminants. At present for n = 3, 4, 5, 6, 7 there are large tables, available at [B], giving all number fields of degree n and absolute discriminant less than certain bounds. It would also be interesting to have complete tables of number fields unramified outside a given finite set S of primes. For a given degree n, these sets of number fields are finite, by a classical theorem of Hermite. For a recent survey of the general subject of number fields with prescribed ramification, we recommend [H]; there it is made clear that very little is known about the set of non-solvable number fields unramified outside a given S. In this paper, we focus on the set S = {∞, 2, 3}. We present tables for degrees ≤ 6, the following chart giving an overview the situation. Degree n Sn An Hn Upper Bound for |D| Method 2 7 – – 23 = 24 Trivial 3 8 1 – 23 = 1,944 Old tables or CFT 4 22 1 39 23 = 497,664 Old tables or CFT 5 5 0 1 23 = 1,492,992 Old tables 6 54 8 336 23 = 2,902,376,448 New here 7 23 = 2,902,376,448 8 23 ≈ 1.1× 10 9 23 ≈ 5.5× 10 Here the Sn and An columns give the number of fields with the given Galois group. Similarly, the Hn entry is the number of fields with smaller Galois group. 1991 Mathematics Subject Classification. 11Y40. This is the final form of the paper. c ©1998 American Mathematical Society 141 142 JOHN W. JONES AND DAVID P. ROBERTS For n ≤ 4, the maximal possible absolute discriminant is well within the range of the existing tables; alternatively, one can use class field theory, as all fields here are solvable. The case n = 5 is just barely within the range of the existing tables. For n = 6, the existing tables for arbitrary signature fields cover absolute discriminants up to 200,000; the maximum possible absolute discriminant required here is well beyond the range of these tables. The case n = 7 may also be in the range of our current techniques, while the cases n ≥ 8 certainly are not. In general, we would like to advocate the point of view that number fields with discriminants divisible by only a few small primes should be regarded as “jewels”. On the one hand, they occur very rarely on complete tables of number fields ordered by absolute discriminant. On the other hand, they are exactly the fields which appear most often in certain other contexts. For example, they appear as fields of definition of three-point covers of the projective line, see e.g. [Mal]. Similarly, they arise naturally in the study of motives. Suppose M is a motive with bad reduction only in S and ` is a finite prime in S; then the fields which correspond to the `-adic representations associated to M are ramified only in S. We have already investigated some simple cases from quite varied geometric situations with S = {∞, 2, 3}. The fields we tabulate here appear repeatedly. Our tables thus play an important role in the analysis of certain motives. This application served as our principal motivation for writing the present paper. To obtain the complete classification of sextics, we proceed as follows. There are sixteen conjugacy classes of transitive subgroups of S6. We divide them into three classes: • Two “new” non-solvable groups: A6 and S6 • Two “new” solvable groups: C 3 . C4 and C 3 . D4 • Twelve “old” groups. We have made a large specialized computer search for fields of discriminant −j2a3b, in particular guaranteed to find all the new non-solvable fields. The search, described in Section 2, found 62 new non-solvable fields, 54 new solvable fields and 282 old fields. In Section 3.2, we use sextic twinning to directly construct all old fields from the list of degree ≤ 5 fields; this proves that there are indeed 282 of them. In Section 3.4, we apply class field theory to the list of C4 and D4 quartics, proving directly that there are indeed 54 new solvable fields. The fields we consider here being “jewels”, it is worth studying them in some detail. We tabulate essentially all such fields of degree ≤ 5 and all of the new sextic fields. These tables provide common invariants for each field, and also complete descriptions of their 2-adic and 3-adic completions.