Enumeration of Totally Real Number Fields of Bounded Root Discriminant

We enumerate all totally real number fields F with root discriminant δF ≤ 14. There are 1229 such fields, each with degree [F : Q] ≤ 9. In this article, we consider the following problem. Problem 1. Given B ∈ R>0, enumerate the set NF (B) of totally real number fields F with root discriminant δF ≤ B, up to isomorphism. To solve Problem 1, for each n ∈ Z>0 we enumerate the set NF (n,B) = {F ∈ NF (B) : [F : Q] = n} which is finite (a result originally due to Minkowski). If F is a totally real field of degree n = [F : Q], then by the Odlyzko bounds [27], we have δF ≥ 4πe − O(n−2/3) where γ is Euler’s constant; thus for B < 4πe < 60.840, we have NF (n,B) = ∅ for n sufficiently large and so the set NF (B) is finite. Assuming the generalized Riemann hypothesis (GRH), we have the improvement δF ≥ 8πe − O(log−2 n) and hence NF (B) is conjecturally finite for all B < 8πe < 215.333. On the other hand, for B sufficiently large, the set NF (B) is infinite: Martin [23] has constructed an infinite tower of totally real fields with root discriminant δ ≈ 913.493 (a long-standing previous record was held by Martinet [25] with δ ≈ 1058.56). The value lim inf n→∞ min{δF : F ∈ NF (n,B)} is presently unknown. If B is such that #NF (B) =∞, then to solve Problem 1 we enumerate the set NF (B) = ⋃ nNF (n,B) by increasing degree. Our restriction to the case of totally real fields is not necessary: one may place alternative constraints on the signature of the fields F under consideration (or even analogous p-adic conditions). However, we believe that Problem 1 remains one of particular interest. First of all, it is a natural boundary case: by comparison, Hajir-Maire [14, 15] have constructed an unramified tower of totally complex number fields with root discriminant ≈ 82.100, which comes within a factor 2 of the GRH-conditional Odlyzko bound of 8πe ≈ 44.763. Secondly, in studying certain problems in arithmetic geometry and number theory—for example, in the enumeration of arithmetic Fuchsian groups [21] and the computational investigation of the Stark conjecture and its generalizations—provably complete and extensive tables of totally real fields are useful, if not outright essential. Indeed, existing strategies for finding towers with small root discriminant as above often start by finding a good candidate base field selected from existing tables. The main result of this note is the following theorem, which solves Problem 1 for δ = 14. Theorem 2. We have #NF (14) = 1229. The complete list of these fields is available online [35]; the octic and nonic fields (n = 8, 9) are recorded in Tables 3–4 in §4, and there are no dectic fields (NF (14, 10) = ∅). For a comparison of this theorem with existing results, see §1.2. The note is organized as follows. In §1, we set up the notation and background. In §2, we describe the computation of primitive fields F ∈ NF (14); we compare well-known methods and provide some improvements. In §3, we discuss the extension of these ideas to imprimitive fields, and we report timing details on the computation. Finally, in §4 we tabulate the fields F . The author wishes to thank: Jürgen Klüners, Noam Elkies, Claus Fieker, Kiran Kedlaya, Gunter Malle, and David Dummit for useful discussions; William Stein, Robert Bradshaw, Craig Citro, Yi Qiang, and the rest of the Sage development team for computational support (NSF Grant No. 0555776); and Larry Kost and Helen Read for their technical assistance.