Artin Lfunctions of small conductor
 John W. Jones^{1} and
 David P. Roberts^{2}Email authorView ORCID ID profile
Received: 19 December 2016
Accepted: 21 March 2017
Published: 10 July 2017
Abstract
We study the problem of finding the Artin Lfunctions with the smallest conductor for a given Galois type. We adapt standard analytic techniques to our novel situation of fixed Galois type and obtain much improved lower bounds on the smallest conductor. For small Galois types we use complete tables of number fields to determine the actual smallest conductor.
Keywords
Artin representation Lfunction Number field Conductor1 Overview
Artin Lfunctions \(L(\mathcal {X},s)\) are remarkable analytic objects built from number fields. Let \(\overline{\mathbf {Q}}\) be the algebraic closure of the rational number field \(\mathbf {Q}\) inside the field of complex numbers \(\mathbf {C}\). Then Artin Lfunctions are indexed by continuous complex characters \(\mathcal {X}\) of the absolute Galois group \({\mathbb {G}}= {{\mathrm{Gal}}}(\overline{\mathbf {Q}}/\mathbf {Q})\), with the unital character 1 giving the Riemann zeta function \(L(1,s) = \zeta (s)\). An important problem in modern number theory is to obtain a fuller understanding of these higher analogs of the Riemann zeta function. The analogy is expected to be very tight: all Artin Lfunctions are expected by the Artin conjecture to be entire except perhaps for a pole at \(s=1\); they are all expected to satisfy the Riemann hypothesis that all zeros with \(\text{ Re }(s) \in (0,1)\) satisfy \(\text{ Re }(s)=1/2\).
The two most basic invariants of an Artin Lfunction \(L(\mathcal {X},s)\) are defined via the two explicit elements of \({\mathbb {G}}\), the identity e and the complex conjugation element \(\sigma \). These invariants are the degree \(n = \mathcal {X}(e)\) and the signature \(r = \mathcal {X}(\sigma )\) respectively. A measure of the complexity of \(L(\mathcal {X},s)\) is its conductor \(D \in \mathbf {Z}_{\ge 1}\), which can be computed from the discriminants of related number fields. It is best for purposes such as ours to focus instead on the root conductor \(\delta = D^{1/n}\).
 1::

Use known and the above conjectured properties of Lfunctions to obtain a lower bound \(\mathfrak {d}(G,c,\chi )\) on the root conductors of Lfunctions in \(\mathcal {L}(G,c,\chi )\).
 2::

Explicitly identify the sets \(\mathcal {L}(G,c,\chi ;B)\) with B as large as possible.
There is a large literature on a special case of the situation we study. Namely let \((G,c,\phi )\) be a Galois type where \(\phi \) is the character of a transitive permutation representation of G. Then the set \(\mathcal {L}(G,c,\phi ;B)\) is exactly the set of Dedekind zeta functions \(\zeta (K,s)\) arising from a corresponding set \(\mathcal {K}(G,c,\phi ;B)\) of arithmetic equivalence classes of number fields. In this context, root conductors are just root discriminants, and lower bounds date back to Minkowski’s work on the geometry of numbers. Use of Dedekind zeta functions as in 1 above began with work of Odlyzko [17–19], Serre [29], Poitou [26, 27], and Martinet [16]. Extensive responses to 2 came shortly thereafter, with papers often focusing on a single degree \(n=\phi (e)\). Early work for quartics, quintics, sextics, and septics include respectively [2–4, 7, 13, 21–23, 25, 31]. Further results towards 2 in higher degrees are extractable from the websites associated to [10, 12, 14].
The full situation that we are studying here was identified clearly by Odlyzko in [20], who responded to 1 with a general lower bound. However this more general case of Artin Lfunctions has almost no subsequent presence in the literature. A noticeable exception is a recent paper of PizarroMadariaga [24], who improved on Odlyzko’s results on 1. A novelty of our paper is the separation into Galois types. For many Galois types this separation allows us to go considerably further on 1. This paper is also the first systematic study of 2 beyond the case of number fields.
Sections 2 and 3 review background on Artin Lfunctions and tools used to bound their conductors. Sections 4, 5 and 6 form the new material on the lower bound problem 1, while Sects. 7, 8, and 9 focus on the tabulation problem 2. Finally, Sect. 10 returns to 1 and considers asymptotic lower bounds for root conductors of Artin Lfunctions in certain families. In regard to 1, Fig. 3 and Corollary 10.1 give a quick indication of how our typebased lower bounds compare with the earlier degreebased lower bounds. In regard to both 1 and 2, Tables 3, 4, 5, 6 and 7 show how the new lower bounds compare with actual first conductors for many types.
Artin Lfunctions have recently become much more computationally accessible through a package implemented in Magma [1] by Tim Dokchitser. Thousands are now collected in a section on the LMFDB [14]. The present work increases our understanding of all this information in several ways, including by providing completeness certificates for certain ranges.
2 Artin Lfunctions
In this section we provide some background. An important point is that our problems allow us to restrict consideration to Artin characters which take rational values only. In this setting, Artin Lfunctions can be expressed as products and quotients of roots of Dedekind zeta functions, minimizing the background needed. General references on Artin Lfunctions include [15, 28].
2.1 Number fields
A number field K has many invariants relevant for our study. First of all, there is the degree \(n = [K{:}\mathbf {Q}]\). The other invariants we need are local in that they are associated with a place v of \(\mathbf {Q}\) and can be read off from the corresponding completed algebra \(K_v = K \otimes \mathbf {Q}_v\), but not from other completions. For \(v=\infty \), the complete invariant is the signature r, defined by \(K_\infty \cong \mathbf {R}^{r} \times \mathbf {C}^{(nr)/2}\). It is more convenient sometimes to work with the eigenspace dimensions for complex conjugation, \(a = (n+r)/2\) and \(b = (nr)/2\). For an ultrametric place \(v=p\), the full list of invariants is complicated. The most basic one is the positive integer \(D_p = p^{c_p}\) generating the discriminant ideal of \(K_p/\mathbf {Q}_p\). We package the \(D_p\) into the single invariant \(D = \prod _p D_p \in \mathbf {Z}_{\ge 1}\), the absolute discriminant of K.
2.2 Dedekind zeta functions
2.3 Analytic properties of Dedekind zeta functions
2.4 Permutation characters
We recall from the introduction that throughout this paper we are taking \(\overline{\mathbf {Q}}\) to be the algebraic closure of \(\mathbf {Q}\) in \(\mathbf {C}\) and \({\mathbb {G}}= {{\mathrm{Gal}}}(\overline{\mathbf {Q}}/\mathbf {Q})\) its absolute Galois group. A degree n number field K then corresponds to the transitive nelement \({\mathbb {G}}\)set \(X = \text{ Hom }(K,\overline{\mathbf {Q}})\). A number field thus has a permutation character \(\Phi = \Phi _K = \Phi _X\) with \(\Phi (e)=n\). Also signature has the charactertheoretic interpretation \(\Phi (\sigma ) = r\), where \(\sigma \) as before is the complex conjugation element.
2.5 General characters and Artin Lfunctions
For a finite set of primes S, let \(\overline{\mathbf {Q}}_S\) be the compositum of all number fields in \(\overline{\mathbf {Q}}\) with discriminant divisible only by primes in S. Let \({\mathbb {G}}_S = {{\mathrm{Gal}}}(\overline{\mathbf {Q}}_S/\mathbf {Q})\) be the corresponding quotient of \({\mathbb {G}}\). Then for primes \(p \not \in S\) one has a welldefined Frobenius conjugacy class \(\text{ Fr }_p\) in \({\mathbb {G}}_S\). The local factor \(P_p(x)\) in (2.1) is the characteristic polynomial \(\det (1  \rho (\text{ Fr }_p) x)\), where \(\rho \) is a representation with character \(\mathcal {X}\).
2.6 Relations with other objects
Artin Lfunctions of degree 1 are exactly Dirichlet Lfunctions, so that \(\mathcal {X}\) can be identified with a faithful character of the quotient group \((\mathbf {Z}/D\mathbf {Z})^\times \) of \({\mathbb {G}}\), with D the conductor of \(\mathcal {X}\). Artin Lfunctions coming from irreducible degree 2 characters and conductor D are expected to come from cuspidal modular forms on \(\Gamma _1(D)\), holomorphic if \(r=0\) and nonholomorphic otherwise. This expectation is proved in all cases, except for those with \(r = \pm 2\) and projective image the nonsolvable group \(A_5\). In general, to understand how an Artin Lfunction \(L(\mathcal {X},s)\) qualitatively relates to other objects, one needs to understand its Galois theory, including the placement of complex conjugation; in other words, one needs to identify its Galois type. To be more quantitative, one brings in the conductor.
2.7 Analytic properties of Artin Lfunctions
An Artin Lfunction has a meromorphic continuation and functional equation, although each with an extra complication in comparison with the special case of Dedekind zeta functions. For the meromorphic continuation, the behavior at \(s=1\) is known: the pole order is the multiplicity \((1,\mathcal {X})\) of 1 in \(\mathcal {X}\). The complication is that one has poor control over other possible poles. The Artin conjecture for \(\mathcal {X}\) says however that there are no poles other than \(s=1\).
Recall from the introduction that an Artin Lfunction is said to satisfy the Riemann hypothesis if all its zeros in the critical strip \(0<\text{ Re }(s)<1\) are actually on the critical line \(\text{ Re }(s)= 1/2\). We will be using the Riemann hypothesis through Lemma 3.1. If we replaced the function (3.1) with the appropriately scaled version of (5.17) from [24], then our lower bounds would be only conditional on the Artin conjecture, which is completely known for some Galois types \((G,c,\chi )\). However the bounds obtained would be much smaller, and the comparison with first conductors as presented in Tables 3, 4, 5, 6 and 7 below would be less interesting.
2.8 Rational characters and rational Artin Lfunctions
3 Signaturebased analytic lower bounds
Here and in the next section we aim to be brief, with the main point being to explain how typebased lower bounds are usually much larger than signaturebased lower bounds. We employ the standard framework for establishing lower bounds for conductors and discriminants, namely Weil’s explicit formula. General references for the material in this section are [20, 24].
3.1 Basic quantities
3.2 The quantity M(n, r, u)
3.3 Lower bounds for root discriminants
Lemma 3.1
If \(\Phi \) is a permutation character, then the nonnegativity condition (3.3) is automatically satisfied. This makes the application of the analytic theory to lower bounds of root discriminants of fields relatively straightforward.
3.4 Lower bounds for general Artin conductors
To pass from nonnegative characters to general characters, the classical method uses the following lemma.
Lemma 3.2
A proof of this lemma from first principles is given in [20].
Combining Lemma 3.1 with Lemma 3.2 one gets the following immediate consequence
Theorem 3.3
4 Typebased analytic lower bounds
In this section we establish Theorem 4.2, which is a family of lower bounds on the root conductor \(\delta _\mathcal {X}\) of a given Artin character, dependent on the choice of an auxiliary character \(\phi \).
4.1 Conductor relations
Let G be a finite group, c an involution in G, \(\chi \) a faithful character of G, and \(\phi \) a nonzero realvalued character of G. Say that a pair of Artin characters (\(\mathcal {X}\),\(\Phi \)) has joint type \((G,c,\chi ,\phi )\) if there is a surjection \(h : {\mathbb {G}}\rightarrow G\) with \(h(\sigma )=c\), \(\mathcal {X}= \chi \circ h\), and \(\Phi = \phi \circ h\).
Because (4.1)–(4.3) are only correct for \(\psi \) rational, when we apply them to characters \(\chi \) and \(\phi \) of interest, we are always assuming that \(\chi \) and \(\phi \) are rational. As explained in Sect. 2.8, the restriction to rational characters still allows obtaining general lower bounds. Also, as will be illustrated by an example in Sect. 5.5, focusing on rational characters does not reduce the quality of these lower bounds.
4.2 Root conductor relations
Our discussion establishes the following lemma.
Lemma 4.1
4.3 Bounds via an auxiliary Artin character \(\Phi \)
Theorem 4.2
Computing the right side of (4.10) is difficult because the base in (4.9) requires evaluating the maximum of a complicated function, while the exponent \(b(G,\chi ,\phi )\) involves an exhaustive study of wild ramification. Almost always in the sequel, \(\chi \) and \(\phi \) are rationalvalued and we replace \(b(G,\chi ,\phi )\) by \(\widehat{\underline{\alpha }}(G,\chi ,\phi )\); in the common case that all three quantities of (4.7) are equal, this is no loss.
5 Four choices for \(\phi \)
This section fixes a type \((G,c,\chi )\) where the faithful character \(\chi \) is rationalvalued and uses the notation \((n,r) = (\chi (e),\chi (c))\). The section introduces four nonnegative characters \(\phi _i\) built from \((G,\chi )\). For the first character \(\phi _L\), it makes \(m(G,c,\chi ,\phi _L,b)\), the lower bound appearing in Theorem 4.2, more explicit. For the remaining three characters \(\phi _i\), it makes the perhaps smaller quantity \(m(G,c,\chi ,\phi _i,\widehat{\underline{\alpha }})\) more explicit.
Two simple quantities enter into the constructions as follows. Let X be the set of values of \(\chi \), so that \(X \subset \mathbf {Z}\) by our rationality assumption. Let be the least element of X. The greatest element of X is of course \(\chi (e)=n\), and we let \(\widehat{\chi }\) be the second greatest element. Thus, .
5.1 Linear auxiliary character
In [24, §3] the auxiliary character \(\chi +n\) is used, which has the advantage of being nonnegative for any rational character \(\chi \). Odlyzko also uses \(\chi +n\) in [20], and suggests using the auxiliary character since it gives a better bound whenever . This strict inequality holds exactly when the center of G has odd order.
5.2 Square auxiliary character
The character \(\phi _S\) is used prominently in [20, 24]. When \(\widehat{\chi }=n2\), the lower bound \(m(G,c,\chi ,\phi _S,\widehat{\underline{\alpha }})\) coincides with that of Lemma 3.2. Thus for \(\widehat{\chi }=n2\), Theorem 4.2 with \(\phi = \phi _S\) gives the same bound as Theorem 3.3. On the other hand, as soon as \(\widehat{\chi }<n2\), Theorem 4.2 with \(\phi = \phi _S\) is stronger. The remaining case \(\widehat{\chi }=n1\) occurs only three times among the 195 characters we consider in Sect. 8. In these three cases, the bound in Theorem 3.3 is stronger because the exponent is larger. However, in each of these cases, the tamewild principle applies [11] and we can use exponent \(m(G,c,\chi ,\phi _S,{\underline{\alpha }})\), which gives the same bound as Theorem 3.3 in two cases, and a better bound in the third.
5.3 Quadratic auxiliary character
5.4 Galois auxiliary character
5.5 Spectral bounds and rationality
To get large lower bounds on root conductors, one wants to be small for (5.1) or \(\widehat{\chi }/n\) to be small for (5.2)–(5.4). The analogous quantities and \(\widehat{\chi }/n_1\) are welldefined for a general real character \(\chi _1\), and replacing \(\chi _1\) by the sum \(\chi \) of its conjugates can substantially reduce them.
6 Other choices for \(\phi \)
To apply Theorem 4.2 for a given Galois type \((G,c,\chi )\), one needs to choose an auxiliary character \(\phi \). We presented four choices in Sect. 5. We discuss all possible choices here, using \(G=A_4\) and \(G=A_5\) as illustrative examples.
6.1 Rational character tables
Rational character tables for \(A_4\) and \(A_5\)

Two examples are given in Table 1. We index characters by their degree, with \(I = \{1,2,3\}\) for \(A_4\) and \(I = \{1,4,5,6\}\) for \(A_5\). All characters are absolutely irreducible except for \(\chi _2\) and \(\chi _{6}\), which each break as a sum of two conjugate irreducible complex characters. We likewise index powerconjugacy classes by the order of a representing element, always adding letters as is traditional. Thus \(J = \{1A,2A,3AB\}\) for \(A_4\) and \(J = \{1A,2A,3A,5AB\}\) for \(A_5\), with 3AB and 5AB each consisting of two conjugacy classes.
6.2 The polytope \(P_G\) of normalized nonnegative characters
A general realvalued function \(\phi \in \mathbf {R}(G^\sharp )\) has an expansion \(\sum x_i \chi _i\) with \(x_i \in \mathbf {R}\). The coefficients are recovered via inner products, \(x_i = (\phi ,\chi _i)/(\chi _i,\chi _i)\). Alternative coordinates are given by \(y_j = \phi (C_j)\). The \(\phi \) allowed for Theorem 4.2 are the nonzero \(\phi \) with the \(x_i\) and \(y_j\) nonnegative integers.
An allowed \(\phi \) gives the same lower bound in Theorem 4.2 as any of its positive multiples \(m \phi \). Without getting any new bounds, we can therefore give ourselves the convenience of allowing the \(x_i\) and \(y_j\) to be nonnegative rational numbers. Similarly, we can extend by continuity to allow the \(x_i\) and \(y_j\) to be nonnegative real numbers. The allowed \(\phi \) then become the cone in kdimensional Euclidean space given by \(x_i \ge 0\) and \(y_j \ge 0\), excluding the tip of the cone at the origin.
Figure 2 continues our two examples. On the left, \(P_{A_4}\) is drawn in the \(x_2\)–\(x_3\) plane. The character faces give the coordinate axes and are dashed. The class faces are calculated from columns in the rational character table and are solid. On the right, a view of \(P_{A_5}\) is given in \(x_{4}\)–\(x_{5}\)–\(x_{6}\) space. The three pairwise intersections of character faces give coordinate axes and are dashed, while all other edges are solid. In this view, the point \(\phi _G = \phi _{60} = (4,5,3)\) should be considered as closest to the reader, with the solid lines visible and the dashed lines hidden by the polytope. Note that \(P_{A_4}\) has the combinatorics of a square and \(P_{A_5}\) has the combinatorics of a cube. While the general \(P_G\) is the intersection of an orthant with tip \(\phi _1\) and an orthant with tip \(\phi _G\), its combinatorics are typically more complicated than \([0,1]^{(k1)}\). For example, the groups \(G=A_6\), \(S_5\), \(A_7\), and \(S_6\), have \(k=6\), 7, 8, and 11 respectively; but instead of having 32, 64, 128 and 1024 vertices, their polytopes \(P_G\) have 28, 40, 115, and 596 vertices respectively.
6.3 Points in \(P_G\)
In the previous subsection, we have mentioned already the distinguished vertices \(\phi _1\) and \(\phi _G\). For every rationally irreducible character, we also have , \(\phi _{\chi ,S} = \chi ^2\), and \(\phi _{\chi ,Q} = \chi \chi ^*\), as in Sect. 5.
For every subgroup H of G, another element of \(P_G\) is the permutation character \(\phi _{G/H}\). For \(H = G\), this character is just the \(\phi _1\) considered before, which is a vertex. Otherwise, a theorem of Jordan, discussed at length in [30], says that \(\phi _{G/H}(C_j)=0\) for at least one j; in other words, \(\phi _{G/H}\) is on at least one character face. For \(A_4\) and \(A_5\), there are respectively five and nine conjugacy classes of subgroups, distinguished by their orders. Figure 2 draws the corresponding points, labeled by \(\phi _{G/H}\). All four vertices of \(P_{A_4}\) and six of the eight vertices of \(P_{A_5}\) are of the form \(\phi _{N}\). The remaining one \(\phi _N\) in \(P_{A_4}\) is on an edge, while the remaining three \(\phi _N\) in \(P_{A_5}\) are on edges as well.
6.4 The best choice for \(\phi \)
Given \((G,c,\chi )\) and \(u \in \{\widehat{\underline{\alpha }},b,\underline{\alpha }\}\), let \(m(G,c,\chi ,u) = \max _{\phi \in P_G} m(G,c,\chi ,\phi ,u)\). Computing these maxima seems difficult. Instead we vary \(\phi \) over a modestly large finite set, denoting the largest bound appearing as \(\mathfrak {d}(G,c,\chi ,u)\). For most G, the set of \(\phi \) we inspect consists of all \(\phi _{\chi ,L}\), \(\phi _{\chi ,S}\), and \(\phi _{\chi ,Q}\), all \(\phi _{G/H}\) including the regular character \(\phi _G\), and all vertices. For some G, like \(S_7\), there are too many vertices and we exclude them from the list of \(\phi \) we try.
For each \((G,\chi )\), we work either with \(u=\widehat{\underline{\alpha }}\) or with \(u=\underline{\alpha }\), as explained in the “middle four columns” part of Sect. 8.2.2. We then report \(\mathfrak {d}(G,\chi ) = \min _c \mathfrak {d}(G,c,\chi ,u)\) in Sect. 8.
7 The case \(G=S_5\)
Our focus in the next two sections is on finding initial segments \(\mathcal {L}(G,\chi ; B)\) of complete lists of Artin Lfunctions, and in particular on finding the first root conductor \(\delta _1(G,\chi )\). It is a question of transferring completeness statements for number fields to completeness statements for Artin Lfunctions via conductor relations. In this section, we explain the process by presenting the case \(G=S_5\) in some detail.
7.1 Different orders on the same set of fields
Consider the set \(\mathcal {K}\) of isomorphism classes of quintic fields K over \(\mathbf {Q}\) with splitting field \(L/\mathbf {Q}\) having Galois group \({{\mathrm{Gal}}}(L/\mathbf {Q}) \cong S_5\). The group \(S_5\) has seven irreducible characters which we index by degree and an auxiliary label: \(\chi _{1a} = 1\), \(\chi _{1b}\), \(\chi _{4a}\), \(\chi _{4b}\), \(\chi _{5a}\), \(\chi _{5b}\), and \(\chi _{6a}\). For \(\phi \) a permutation character, let \(D_\phi (K) = D(K_\phi )\) be the absolute discriminant of the associated resolvent algebra \(K_\phi \) of K. Extending by multiplicativity, functions \(D_\chi : \mathcal {K}\rightarrow \mathbf {R}_{>0}\) are defined for general \(\chi = \sum m_n \chi _n\). They do not depend on the coefficient \(m_{1a}\). We follow our practice of often shifting attention to the corresponding root conductors \(\delta _\chi (K) = D_\chi (K)^{1/\chi (e)}\).
Let \(\mathcal {K}(\chi ; B) = \{K \in \mathcal {K}: \delta _\chi (K) \le B\}\). Suppose now all the \(m_{n}\) are nonnegative with at least one coefficient besides \(m_{1a}\) and \(m_{1b}\) positive. Then \(\delta _\chi \) is a height function in the sense that all the \(\mathcal {K}(\chi ; B)\) are finite. Suppressing the secondary phenomenon that ties among a finite number of fields can occur, we think of each \(\delta _\chi \) as giving an ordering on the set \(\mathcal {K}\).
The orderings coming from different \(\delta _\chi \) can be very different. For example, consider the field \(K \in \mathcal {K}\) defined by the polynomial \(x^5  2x^4 + 4x^3  4x^2 + 2x  4\). This field is the first field in \(\mathcal {K}\) when ordered by the regular character \(\phi _{120} = \sum _n \chi _n(n) \chi _n\). However it is the 22nd field when ordered by \(\phi _6 = 1 + \chi _{5b}\) only the 2298th field when ordered by \(\phi _5 = 1+ \chi _{4a}\).
This phenomenon of different orderings on the same set of number fields plays a prominent role in asymptotic studies [32]. Here we are interested instead in initial segments and how they depend on \(\chi \). Our formalism lets us treat any \(\chi \). Following the conventions for general G of the next section, we focus on the five irreducible \(\chi \) with \(\chi (e)>1\), thus \(\chi _n\) for \(n \in \{4a,4b,5a,5b,6a\}\).
7.2 Computing Artin conductors
For all the groups G considered in the next section, we proceeded similarly. Thus we started with rational character tables from Magma. We used linear algebra to express rationally irreducible characters in terms of permutation characters. We used Magma again to compute resolvents and then Pari to evaluate their discriminants. In this last step, we often confronted large degree polynomials with large coefficients. The discriminant computation was only feasible because we knew a priori the set of primes dividing the discriminant, and could then easily compute the pparts of the discriminants of these resolvent fields for relevant primes p using Pari/gp without fully factoring the discriminants of the resolvent polynomials.
Magma’s Artin representation package computes conductors of Artin representations in a different and more local manner. Presently, it does not compute all conductors in our range because some decomposition groups are too large.
7.3 Transferring completeness results
The final results are on Table 3. In particular for \(n = 4a\), 4b, 5a, 5b, 6a the unique minimizing fields are \(K^{103}\), \(K^{21}\), \(K^{14}\), \(K^{6}\), and \(K^{12}\), with root conductors approximately 6.33, 18.72, 17.78, 16.27, and 18.18. The lengths of the initial segments identified are 45, 15, 186, 592, and 110. Note that because of the relations \(\phi _5 = 1+ \chi _{4a}\) and \(\phi _6 = 1 + \chi _{5b}\), the results for 4a and 5b are just translation of known minima of discriminants of number fields with Galois groups 5T5 and 6T14 respectively. For 4b, 5b, 6a, and the majority of the characters in the tables of the next section, the first root conductor and the entire initial segment are new.
8 Tables for 84 groups G
In this section, we present our computational results for small Galois types. For simplicity, we focus on results coming from complete lists of Galois number fields. Summarizing statements are given in Sect. 8.1 and then many more details in Sect. 8.2.
8.1 Lower bounds and initial segments
We consider all groups with a faithful transitive permutation representation in some degree from two to nine, except we exclude the nonsolvable groups in degrees eight and nine. There are 84 such groups, and we consider all associated Galois types \((G,\chi )\) with \(\chi \) a rationally irreducible faithful character. Our first result gives conditional lower bounds:
Theorem 8.1
For each of the 195 Galois types \((G,\chi )\) listed in Tables 3, 4, 5, 6 and 7, the listed value \(\mathfrak {d}\) gives a lower bound for the root conductor of all Artin representations of type \((G,\chi )\), assuming the Artin conjecture and Riemann hypothesis for relevant Lfunctions.
The bounds in Tables 3, 4, 5, 6 and 7 are graphed with the best previously known bounds from [24] in Fig. 3. The horizontal axis represents the dimension \(n_1=\chi _1(e)\) of any irreducible constituent \(\chi _1\) of \(\chi \). The vertical axis corresponds to lower bounds on root conductors. The piecewiselinear curve connects bounds from [24], and there is one dot at height \(\mathfrak {d}(G,\chi )\) for each \((G,\chi )\) from Tables 3, 4, 5, 6 and 7 with \(\chi _1(e)\le 20\). Here we are freely passing back and forth between a rational character \(\chi \) and an irreducible constituent \(\chi _1\) via \(\delta _1(G,\chi ) = \delta _1(G,\chi _1)\), which is a direct consequence of (2.4).
Artin Lfunctions with small conductor from groups in degrees 2, 3, 4, and 5
\(\varvec{G}\)  \(\varvec{n_1}\)  \(\varvec{z}\)  \(\varvec{\mathfrak {d}}\)  \(\varvec{\delta _1}\)  \(\varvec{\Delta _1}\)  Pos’n  \(\varvec{\beta }\)  \(\varvec{B^\beta }\)  \(\varvec{\#}\)  

\(C_2\)  2  TW  2, 2  1.73  \(3^*\)  100  6086  
2T1  1  \([1, 1]\)  \(2.97_{\ell }\)  3.00  3  1  2.00  10,000  6086  
\(C_3\)  3  TW  2, 2  3.66  \(7^*\)  500  1772  
3T1  1  \(\sqrt{3}\)  \([1, 1]\)  \(6.93_{\ell }\)  7.00  7  1  1.50  11180.34  1772 
\(S_3\)  6  TW  3, 4  4.80  \(23^*\)  250  24,484  
3T2  2  \([1, 0]\)  \(4.74_{\ell }\)  4.80  23  1  1.00  250  13,329  
\(C_4\)  4  TW  3, 4  3.34  \(5^*\)  150  2668  
4T1  1  i  \([2, 0]\)  \(4.96_{S}\)  5.00  5  1  \(1.33_\bullet \)  796.99  489 
\(D_4\)  8  TW  5, 10  6.03  \(3^*7^*\)  150  31,742  
4T3  2  \([2, 0]\)  \(5.74_{q}\)  6.24  \(3 \cdot 13\)  2  1.00  150  9868  
\(A_4\)  12  TW  3, 4  10.35  \(2^*7^*\)  150  846  
4T4  3  \([1, 0]\)  \(7.60_{q}\)  14.64  \(2^{6} 7^{2}\)  1  1.00  150  270  
\(S_4\)  24  TW  5, 12  13.56  \(2^*11^*\)  150  14,587  
6T8  3  \([1, 1]\)  \(8.62_{G}\)  11.30  \(2^{2} 19^{2}\)  4  \(0.89_\bullet \)  85.96  779  
4T5  3  \([1, 1]\)  \(5.49_{p}\)  6.12  229  9  0.67  28.23  1603  
\(C_5\)  5  TW  2, 2  6.81  \(11^*\)  200  49  
5T1  1  \(\zeta _{5}\)  \([1, 1]\)  \(10.67_{\ell }\)  11.00  11  1  1.25  752.12  49 
\(D_5\)  10  TW  3, 4  6.86  \(47^*\)  200  3622  
5T2  2  \(\sqrt{5}\)  \([1, 0]\)  \(6.73_{q}\)  6.86  47  1  1.00  200  3219 
\(F_5\)  20  TW  4, 8  11.08  \(2^*5^*\)  200  3010  
5T3  4  \([1, 0]\)  \(10.28_{q}\)  13.69  \(2^{4} 13^{3}\)  2  1.00  200  2066  
\(A_5\)  60  TW  4, 8  18.70  \(2^*17^*\)  85  473  
5T4  4  \([1, 1]\)  \(8.18_{g}\)  11.66  \(2^{6} 17^{2}\)  1  0.75  27.99  46  
6T12  5  \([1, 1]\)  \(10.18_{p}\)  12.35  \(2^{6} 67^{2}\)  3  0.80  34.96  216  
12T33  3  \(\sqrt{5}\)  \([2, 1]\)  \(10.34_{g}\)  26.45  \(2^{6} 17^{2}\)  1  0.83  40.54  18 
\(S_5\)  120  TW  7, 40  24.18  \(2^*3^*5^*\)  85  2080  
5T5  4  \([1, 2]\)  \(6.28_{\ell }\)  6.33  1609  103  0.50  9.22  45  
10T12  4  \([2, 1]\)  \(10.28_{V}\)  18.72  \(5^{2} 17^{3}\)  21  0.75  27.99  15  
10T13  5  \([1, 1]\)  \(12.13_{V}\)  16.27  \(2^{4} 3^{2} 89^{2}\)  6  0.80  34.96  592  
6T14  5  \([1, 1]\)  \(11.09_{g}\)  17.78  \(2^{6} 3^{4} 7^{3}\)  14  0.80  34.96  186  
20T35  6  \([2, 1]\)  \(12.26_{g}\)  18.18  \(2^{4} 3^{3} 17^{4}\)  12  0.83  40.54  110 
Artin Lfunctions of small conductor from sextic groups
\(\varvec{G}\)  \(\varvec{n_1}\)  \(\varvec{z}\)  \(\varvec{\mathfrak {d}}\)  \(\varvec{\delta _1}\)  \(\varvec{\Delta _1}\)  Pos’n  \(\varvec{\beta }\)  \(\varvec{B^\beta }\)  \(\varvec{\#}\)  

\(C_6\)  6  TW  4, 6  5.06  \(7^*\)  200  9609  
6T1  1  \(\sqrt{3}\)  \([2, 1]\)  \(6.93_{P}\)  7  7  1  \(1.20_\bullet \)  577.08  617 
\(D_6\)  12  TW  6, 14  8.06  \(3^*5^*\)  150  46,197  
6T3  2  \([2, 1]\)  \(7.60_{G}\)  9.33  \(3 \cdot 29\)  6  \(1_\bullet \)  150  10,242  
\(S_3C_3\)  18  6, 17  10.06  \(2^*3^*7^*\)  200  9420  
6T5  2  \(\sqrt{3}\)  \([2, 1]\)  \(5.69_{q*}\)  7.21  \(2^{2} 13\)  4  0.75  53.18  503 
\(A_4C_2\)  24  6, 16  12.31  \(2^*7^*\)  150  6676  
6T6  3  \([3, 1]\)  \(7.60_{p}\)  8.60  \(7^{2} 13\)  3  0.67  28.23  98  
\(S_3^2\)  36  9, 69  15.53  \(2^*19^*\)  200  45,117  
6T9  4  \([2, 1]\)  \(7.98_{q*}\)  14.83  \(2^{4} 5^{2} 11^{2}\)  27  0.75  53.18  824  
\(C_3^2{\rtimes }C_4\)  36  TW  5, 16  23.57  \(3^*5^*\)  150  331  
6T10  \(4^{{{2}}}\)  \([2, 1]\)  \(7.98_{q*}\)  17.80  \(2^{11} 7^{2}\)  2  0.75  42.86  33  
\(S_4C_2\)  48  10, 96  16.13  \(2^*23^*\)  150  70,926  
6T11  \(3^{{{2}}}\)  \([3, 1]\)  \(6.14_{g}\)  6.92  \(2^{2} 83\)  7  0.67  28.23  3694  
\(C_3^2{\rtimes }D_4\)  72  TW  9, 105  21.76  \(3^*11^*\)  150  8536  
6T13  \(4^{{{2}}}\)  \([2, 2]\)  \(7.60_{p}\)  7.90  \(3^{2} 433\)  52  0.50  12.25  41  
12T36  \(4^{{{2}}}\)  \([2, 1]\)  \(11.29_{P}\)  23.36  \(3^{5} 5^{2} 7^{2}\)  18  0.75  42.86  106  
\(A_6\)  360  TW  6, 28  31.66  \(2^*3^*\)  60  26  
6T15  \(5^{{{2}}}\)  \([1, 2]\)  \(7.71_{\ell }\)  12.35  \(2^{6} 67^{2}\)  8  0.60  11.67  0  
10T26  9  \([1, 1]\)  \(17.69_{g}\)  28.20  \(2^{18} 3^{16}\)  1  0.89  38.07  7  
30T88  10  \([2, 1]\)  \(18.34_{g}\)  30.61  \(2^{24} 3^{16}\)  1  0.90  39.84  4  
36T555  8  \(\sqrt{5}\)  \([2, 1]\)  \(20.70_{g}\)  42.81  \(2^{18} 3^{16}\)  1  0.94  46.45  3 
\(S_6\)  720  11, 596  33.50  \(2^*3^*5^*\)  60  99  
12T183  \(5^{{{2}}}\)  \([3, 2]\)  \(8.21_{v}\)  11.53  \(11^{2} 41^{2}\)  6  0.60  11.67  1  
6T16  \(5^{{{2}}}\)  \([1, 3]\)  \(6.23_{\ell }\)  6.82  14731  53  0.40  5.14  0  
10T32  9  \([1, 3]\)  \(10.77_{v}\)  16.60  \(2^{15} 11^{3} 13^{3}\)  74  0.67  15.33  0  
20T145  9  \([3, 1]\)  \(19.33_{g}\)  31.25  \(2^{6} 5^{6} 73^{4}\)  16  0.89  38.07  4  
30T176  \(10^{{{2}}}\)  \([2, 2]\)  \(16.88_{v}\)  24.22  \(11^{4} 41^{6}\)  6  0.80  26.46  1  
36T1252  16  \([2, 1]\)  \(22.73_{g}\)  35.46  \(2^{36} 3^{8} 7^{12}\)  5  0.94  46.45  11 
Artin Lfunctions of small conductor from septic groups
\(\varvec{G}\)  \(\varvec{n_1}\)  \(\varvec{z}\)  \(\varvec{\mathfrak {d}}\)  \(\varvec{\delta _1}\)  \(\varvec{\Delta _1}\)  Pos’n  \(\varvec{\beta }\)  \(\varvec{B^\beta }\)  \(\varvec{\#}\)  

\(C_7\)  7  TW  2, 2  17.93  \(29^*\)  200  15  
7T1  1  \(\zeta _{7}\)  \([1, 1]\)  \(14.03_{\ell }\)  29  29  1  1.17  483.65  15 
\(D_7\)  14  TW  3, 4  8.43  \(71^*\)  200  2078  
7T2  2  \(\zeta _7^+\)  \([1, 0]\)  \(8.38_{q}\)  8.43  71  1  1  200  1948 
\(C_7{\rtimes }C_3\)  21  TW  3, 4  31.64  \(2^*73^*\)  100  11  
7T3  3  \(\sqrt{7}\)  \([1, 0]\)  \(25.50_{q}\)  34.93  \(2^{3} 73^{2}\)  1  1  100  8 
\(F_7\)  42  TW  5, 12  15.99  \(2^*7^*\)  75  342  
7T4  6  \([1, 0]\)  \(14.47_{q}\)  18.34  \(11^{3} 13^{4}\)  2  1  75  287  
\(\text {GL}_3(2)\)  168  TW  5, 14  32.25  \(2^*3^*11^*\)  45  19  
42T37  3  \(\sqrt{7}\)  \([2, 2]\)  \(15.55_{G}\)  26.06  \(7^{2} 19^{2}\)  7  \(0.89_\bullet \)  29.48  1 
7T5  6  \([1, 2]\)  \(9.36_{p}\)  11.23  \(13^{2} 109^{2}\)  4  0.67  12.65  1  
8T37  7  \([1, 1]\)  \(14.10_{g}\)  32.44  \(3^{8} 7^{8}\)  11  0.86  26.12  0  
21T14  8  \([1, 1]\)  \(14.90_{g}\)  23.16  \(2^{6} 3^{6} 11^{6}\)  1  0.88  27.96  1  
\(A_7\)  2520  8, 115  39.52  \(2^*3^*7^*\)  45  1  
7T6  6  \([1, 3]\)  \(9.13_{\ell }\)  12.54  \(3^{6} 73^{2}\)  26  0.50  6.71  0  
15T47  14  \([1, 2]\)  \(19.39_{g}\)  36.05  \(3^{24} 53^{6}\)  4  0.86  26.12  0  
21T33  14  \([1, 2]\)  \(19.39_{g}\)  31.07  \(3^{18} 17^{10}\)  2  0.86  26.12  0  
42T294  15  \([1, 3]\)  \(18.18_{v}\)  35.73  \(2^{12} 3^{20} 7^{12}\)  1  0.80  21.02  0  
70  10  \(\sqrt{7}\)  \([4, 2]\)  \(22.49_{g}\)  41.21  \(2^{9} 3^{14} 7^{8}\)  1  0.90  30.75  0 
42T299  21  \([3, 1]\)  \(26.95_{g}\)  38.33  \(2^{18} 3^{30} 7^{16}\)  1  0.95  37.54  0  
70  35  \([1, 1]\)  28.79 \(_{g}\)  41.28  \(2^{30} 3^{50} 7^{28}\)  1  \(0.97_\circ \)  40.36  0  
\(S_7\)  5040  15, –  40.49  \(2^*3^*5^*\)  35  0  
7T7  6  \([1, 4]\)  \(7.50_{\ell }\)  7.55  184607  0.33  3.27  0  
14T46  6  \([4, 3]\)  7.66 \(_{p}\)  17.02  \(2^{2} 7^{5} 19^{2}\)  194  0.50  5.92  0  
30T565  14  \([2, 4]\)  \(16.32_{p}\)  26.02  \(2^{20} 53^{8}\)  2  0.71  12.67  0  
30T565  14  \([4, 2]\)  \(20.24_{g}\)  30.98  \(2^{14} 71^{9}\)  46  0.86  21.06  0  
42T413  14  \([6, 2]\)  \(20.24_{g}\)  38.27  \(2^{20} 3^{12} 11^{10}\)  6  0.86  21.06  0  
21T38  14  \([1, 6]\)  \(13.12_{p}\)  22.02  \(2^{24} 3^{12} 29^{4}\)  170  0.57  7.63  0  
42T412  15  \([3, 5]\)  \(16.96_{p}\)  32.90  \(3^{12} 5^{5} 11^{13}\)  24  0.67  10.70  0  
42T411  15  \([5, 3]\)  \(16.56_{g}\)  29.92  \(2^{30} 3^{12} 17^{6}\)  3  0.80  17.19  0  
70  20  \([4, 2]\)  \(23.53_{g}\)  35.18  \(2^{34} 53^{12}\)  2  0.90  24.53  0  
42T418  21  \([3, 3]\)  \(20.24_{g}\)  33.42  \(2^{41} 3^{18} 17^{9}\)  3  0.86  21.06  0  
84  21  \([3, 1]\)  \(28.27_{g}\)  39.59  \(2^{38} 3^{18} 7^{16}\)  4  0.95  29.55  0  
70  35  \([1, 5]\)  25.92 \(_{p}\)  40.71  \(2^{61} 3^{30} 7^{28}\)  4  0.86  21.06  0  
126  35  \([5, 1]\)  30.23 \(_{g}\)  43.26  \(2^{54} 3^{42} 5^{30}\)  \(0.97_\circ \)  31.62  0 
Artin Lfunctions of small conductor from octic groups
\(\varvec{G}\)  \(\varvec{n_1}\)  \(\varvec{z}\)  \(\varvec{\mathfrak {d}}\)  \(\varvec{\delta _1}\)  \(\varvec{\Delta _1}\)  Pos’n  \(\varvec{\beta }\)  \(\varvec{B^\beta }\)  \(\varvec{\#}\)  

\(C_8\)  8  TW  4, 8  11.93  \(17^*\)  125  198  
8T1  1  \(\zeta _{8}\)  \([4, 0]\)  \(8.84_{S}\)  17  17  1  \(1.14_\bullet \)  249.15  41 
\(Q_8\)  8  TW  5, 10  18.24  \(2^*3^*\)  100  72  
8T5  2  \([2, 0]\)  \(26.29_{S}\)  48  \(2^{8} 3^{2}\)  2  \(1.33_\bullet \)  464.16  41  
\(D_8\)  16  TW  6, 20  9.75  \(5^*19^*\)  125  6049  
8T6  2  \(\sqrt{2}\)  \([4, 0]\)  \(9.07_{q}\)  9.75  \(5 \cdot 19\)  1  1  125  2296 
\(C_8{\rtimes }C_2\)  16  7, 24  9.32  \(3^*5^*\)  125  672  
8T7  2  i  \([4, 0]\)  \(9.07_{q}\)  15  \(3^{2} 5^{2}\)  1  1  125  75 
\(QD_{16}\)  16  6, 20  10.46  \(2^*3^*\)  125  1664  
8T8  2  \(\sqrt{2}\)  \([4, 0]\)  \(9.07_{q}\)  16.97  \(2^{5} 3^{2}\)  1  1  125  155 
\(Q_8{\rtimes }C_2\)  16  9, 32  9.80  \(2^*3^*\)  100  3366  
8T11  2  i  \([4, 0]\)  \(9.07_{q}\)  10.95  \(2^{3} 3 \cdot 5\)  3  1  100  825 
\(\text {SL}_2(3)\)  24  TW  5, 14  29.84  \(163^*\)  250  681  
24T7  2  \([2, 1]\)  \(65.51_{P}\)  163  \(163^{2}\)  1  \(1.20_\bullet \)  754.27  94  
8T12  2  \(\sqrt{3}\)  \([4, 1]\)  \(8.09_{p}\)  12.77  163  1  0.75  62.87  78 
32  11, 74  13.79  \(2^*5^*\)  125  11,886  
8T15  4  \([4, 0]\)  \(12.92_{q}\)  16.12  \(2^{4} 5^{2} 13^{2}\)  4  1  125  3464  
32  9, 58  13.56  \(5^*11^*\)  125  766  
8T16  4  \([4, 0]\)  \(12.92_{q}\)  16.58  \(5^{4} 11^{2}\)  1  1  125  129  
\(C_4\wr C_2\)  32  10, 90  13.37  \(2^*5^*\)  125  2748  
8T17  \(2^{{{2}}}\)  i  \([4, 2]\)  5.74 \(_{p}\)  8.25  \(2^{2} 17\)  6  \(0.50_\circ \)  11.18  3 
32  9, 58  14.05  \(2^*\)  125  2720  
8T19  4  \([4, 0]\)  \(12.92_{q}\)  19.03  \(2^{17}\)  1  1  125  1282  
32  17, 806  18.42  \(2^*3^*5^*\)  100  3284  
8T22  4  \([4, 0]\)  \(12.92_{q}\)  20.49  \(2^{4} 3^{2} 5^{2} 7^{2}\)  3  1  100  1162  
\(\text {GL}_2(3)\)  48  7, 41  16.52  \(2^*43^*\)  100  2437  
24T22  2  \(\sqrt{2}\)  \([4, 2]\)  5.74 \(_{v}\)  16.82  283  2  \(0.50_\circ \)  10  0 
8T23  4  \([4, 1]\)  \(9.07_{p}\)  9.95  \(3^{4} 11^{2}\)  4  0.75  31.62  99  
\(C_2^3{\rtimes }C_7\)  56  TW  3, 4  17.93  \(29^*\)  200  28  
8T25  7  \([1, 0]\)  \(16.10_{q}\)  17.93  \(29^{6}\)  1  1  200  27  
64  16, –  20.37  \(2^*5^*\)  125  10,317  
8T26  \(4^{{{2}}}\)  \([4, 2]\)  \(9.07_{p}\)  12.85  \(3^{2} 5^{2} 11^{2}\)  7  \(0.50_\circ \)  11.18  0  
\(C_2\wr C_4\)  64  11, 206  19.44  \(2^*\)  125  2482  
8T27  \(4^{{{2}}}\)  \([4, 2]\)  \(9.07_{p}\)  10.60  \(5^{3} 101\)  19  0.50  11.18  1  
\(C_2 \wr C_2^2\)  64  16, –  19.41  \(2^*7^*\)  125  11,685  
8T29  \(4^{{{2}}}\)  \([4, 2]\)  \(9.07_{p}\)  10.13  \(2^{4} 3^{2} 73\)  28  0.50  11.18  1  
64  11, 206  19.44  \(2^*\)  125  1217  
8T30  \(4^{{{2}}}\)  \([4, 2]\)  \(9.07_{p}\)  14.57  \(5^{3} 19^{2}\)  3  \(0.50_\circ \)  11.18  0  
96  9, 49  34.97  \(2^*5^*13^*\)  250  5520  
8T32  4  \([4, 1]\)  \(9.12_{g}\)  22.80  \(2^{6} 5^{2} 13^{2}\)  2  0.75  62.87  180  
24T97  4  \(\sqrt{3}\)  \([8, 1]\)  13.19 \(_{g}\)  43.30  \(2^{6} 5^{2} 13^{3}\)  2  \(0.88_\circ \)  125.37  112 
96  8, 44  30.01  \(2^*5^*7^*\)  150  791  
8T33  \(6^{{{2}}}\)  \([2, 2]\)  \(11.29_{p}\)  25.14  \(5^{3} 7^{4} 29^{2}\)  12  0.67  28.23  3  
96  10, 92  27.28  \(2^*3^*31^*\)  110  1915  
8T34  6  \([2, 2]\)  \(11.29_{p}\)  22.61  \(31^{3} 67^{2}\)  64  0.67  22.96  1  
\(C_2\wr D_4\)  128  20, –  22.91  \(2^*3^*13^*\)  125  14,369  
8T35  \(4^{{{4}}}\)  \([4, 2]\)  \(9.07_{p}\)  9.45  \(5^{2} 11 \cdot \ 29\)  110  0.50  11.18  9  
\(C_2^3{\rtimes }F_{21}\)  168  5, 14  31.64  \(2^*73^*\)  200  342  
8T36  7  \([1, 1]\)  \(16.06_{p}\)  21.03  \(2^{6} 73^{4}\)  1  0.86  93.82  120  
24T283  7  \(\sqrt{3}\)  \([2, 1]\)  20.23 \(_{p}\)  38.55  \(2^{6} 7^{11}\)  2  \(0.93_\circ \)  136.98  81 
\(C_2\wr A_4\)  192  12, 700  37.27  \(2^*5^*7^*\)  250  13,649  
8T38  \(4^{{{2}}}\)  \([4, 2]\)  \(8.56_{v}\)  15.20  \(2^{6} 7^{2} 17\)  11  0.50  15.81  1  
24T288  \(4^{{{2}}}\)  \(\sqrt{3}\)  \([8, 4]\)  10.84 \(_{p}\)  23.95  \(2^{6} 3 \cdot 5 \cdot 7^{3}\)  66  0.50  15.81  0 
192  13, 559  32.35  \(2^*23^*\)  100  1193  
8T39  \(4^{{{2}}}\)  \([4, 2]\)  \(5.74_{p}\)  8.71  \(2^{4} 359\)  49  0.50  10  8  
24T333  8  \([8, 1]\)  15.28 \(_{g}\)  39.94  \(2^{10} 43^{6}\)  2  \(0.88_\circ \)  56.23  16  
192  13, 559  29.71  \(2^*23^*\)  100  2001  
8T40  \(4^{{{2}}}\)  \([4, 2]\)  5.74 \(_{p}\)  13.04  \(2^{2} 5^{2} 17^{2}\)  9  \(0.50_\circ \)  10  0  
24T332  8  \([8, 1]\)  15.28 \(_{g}\)  29.71  \(2^{12} 23^{6}\)  1  \(0.88_\circ \)  56.23  47  
192  14, 1210  28.11  \(2^*11^*\)  100  4723  
12T108  \(6^{{{2}}}\)  \([2, 2]\)  11.29 \(_{p}\)  20.78  \(2^{12} 3^{9}\)  5  0.67  21.54  2  
8T41  \(6^{{{2}}}\)  \([2, 2]\)  \(11.29_{p}\)  13.01  \(5^{3} 197^{2}\)  13  0.67  21.54  40  
\(A_4\wr C_2\)  288  10, 178  32.18  \(2^*37^*\)  135  1362  
8T42  6  \([2, 3]\)  \(11.29_{p}\)  11.58  \(5^{3} 139^{2}\)  76  0.50  11.62  1  
18T112  9  \([3, 1]\)  17.10 \(_{g}\)  35.11  \(2^{24} 13^{6}\)  2  0.89  78.28  66  
12T128  9  \([3, 3]\)  \(13.59_{p}\)  22.52  \(7^{6} 233^{3}\)  56  0.67  26.32  6  
24T703  6  \(\sqrt{3}\)  \([4, 4]\)  13.79 \(_{p}\)  32.18  \(2^{4} 37^{5}\)  1  0.67  26.32  0 
\(C_2 \wr S_4\)  384  20, –  31.38  \(5^*197^*\)  100  6400  
8T44  \(4^{{{4}}}\)  \([4, 2]\)  \(5.74_{p}\)  7.53  \(5 \cdot 643\)  391  0.50  10  26  
24T708  \(8^{{{2}}}\)  \([8, 4]\)  13.19 \(_{p}\)  25.55  \(2^{12} 5^{2} 11^{6}\)  4  0.50  10  0  
576  16, –  29.35  \(2^*3^*\)  100  2664  
12T161  6  \([2, 3]\)  7.60 \(_{p}\)  19.04  \(2^{8} 3^{3} 83^{2}\)  1179  0.50  10  0  
8T45  6  \([2, 3]\)  \(7.60_{p}\)  15.48  \(2^{14} 29^{2}\)  110  0.50  10  0  
18T179  9  \([3, 1]\)  \(18.84_{g}\)  29.69  \(2^{12} 5^{6} 23^{4}\)  16  0.89  59.95  121  
12T165  9  \([3, 3]\)  \(10.60_{p}\)  17.20  \(2^{6} 19^{3} 67^{3}\)  161  0.67  21.54  12  
18T185  \(9^{{{2}}}\)  \([3, 3]\)  13.74 \(_{p}\)  22.69  \(2^{12} 3^{11} 13^{3}\)  6  0.67  21.54  0  
24T1504  12  \([4, 4]\)  16.40 \(_{p}\)  35.03  \(2^{8} 5^{6} 31^{8}\)  51  0.67  21.54  0  
576  11, 522  49.75  \(3^*5^*7^*\)  100  153  
8T46  6  \([2, 3]\)  \(9.66_{p}\)  19.51  \(3^{6} 5^{4} 11^{2}\)  42  0.50  10  0  
12T160  6  \([2, 3]\)  7.60 \(_{p}\)  27.27  \(2^{23} 7^{2}\)  11  0.50  10  0  
16T1030  9  \([3, 1]\)  \(18.84_{g}\)  35.40  \(2^{22} 3^{6} 13^{4}\)  6  0.89  59.95  50  
18T184  9  \([3, 1]\)  \(18.84_{g}\)  35.50  \(2^{12} 5^{7} 23^{4}\)  7  0.89  59.95  32  
24T1506  12  \([4, 4]\)  16.40 \(_{p}\)  55.16  \(2^{20} 3^{18} 5^{9}\)  4  0.67  21.54  0  
36T766  9  i  \([6, 2]\)  18.84 \(_{g}\)  58.55  \(2^{36} 7^{6}\)  3  0.89  59.95  2 
\(S_4\wr C_2\)  1152  20, –  35.05  \(2^*5^*41^*\)  150  23,694  
12T200  6  \([2, 4]\)  7.60 \(_{p}\)  15.62  \(5^{3} 11^{2} 31^{2}\)  20,668  \(0.33_\circ \)  5.31  0  
8T47  6  \([2, 4]\)  \(7.60_{p}\)  10.51  \(2^{9} 2633\)  20,566  0.33  5.31  0  
12T201  6  \([4, 3]\)  7.60 \(_{p}\)  19.19  \(3^{7} 151^{2}\)  21  0.50  12.25  0  
12T202  6  \([4, 3]\)  7.60 \(_{p}\)  16.51  \(3^{10} 7^{3}\)  12  0.50  12.25  0  
18T272  9  \([3, 3]\)  9.70 \(_{p}\)  19.73  \(3^{6} 853^{3}\)  450  0.67  28.23  44  
18T274  9  \([3, 3]\)  \(15.95_{p}\)  27.07  \(2^{12} 5^{7} 29^{3}\)  105  0.67  28.23  3  
18T273  9  \([3, 3]\)  12.88 \(_{p}\)  30.86  \(2^{16} 3^{18}\)  5  \(0.67_\circ \)  28.23  0  
16T1294  9  \([3, 3]\)  \(10.60_{p}\)  13.16  \(43^{3} 53^{3}\)  39  0.67  28.23  295  
36T1946  12  \([4, 4]\)  14.77 \(_{p}\)  32.80  \(2^{10} 13^{7} 17^{6}\)  16  0.67  28.23  0  
24T2821  12  \([4, 4]\)  16.03 \(_{p}\)  26.48  \(2^{16} 5^{6} 41^{5}\)  1  0.67  28.23  1  
36T1758  18  \([6, 2]\)  20.29 \(_{g}\)  36.08  \(2^{24} 5^{9} 41^{9}\)  1  0.89  85.96  1222 
Artin Lfunctions of small conductor from nonic groups
\(\varvec{G}\)  \(\varvec{n_1}\)  \(\varvec{z}\)  \(\varvec{\mathfrak {d}}\)  \(\varvec{\delta _1}\)  \(\varvec{\Delta _1}\)  Pos’n  \(\varvec{\beta }\)  \(\varvec{B^\beta }\)  \(\varvec{\#}\)  

\(C_9\)  9  TW  3, 4  13.70  \(19^*\)  200  48  
9T1  1  \(\zeta _{9}\)  \([3, 0]\)  \(17.02_{Q}\)  19.00  19  1  \(1.13_\bullet \)  387.85  26 
\(D_9\)  18  TW  4, 8  12.19  \(2^*59^*\)  200  699  
9T3  2  \(\zeta _9^+\)  \([3, 0]\)  \(9.70_{q}\)  14.11  199  3  1.00  200  638 
\(3^{1+2}_{}\)  27  6, 12  31.18  \(7^*13^*\)  200  32  
9T6  3  \(\sqrt{3}\)  \([3, 0]\)  \(30.32_{q}\)  38.70  \(7^{3} 13^{2}\)  1  1.00  200  14 
\(3^{1+2}_{+}\)  27  TW  6, 12  50.20  \(3^*19^*\)  200  16  
9T7  3  \(\sqrt{3}\)  \([3, 0]\)  \(30.32_{q}\)  64.08  \(3^{6} 19^{2}\)  1  1.00  200  12 
\(3^{1+2}.2\)  54  7, 34  17.01  \(2^*3^*\)  200  981  
9T10  6  \([3, 0]\)  \(15.90_{q}\)  17.49  \(31^{5}\)  2  1.00  200  741  
54  7, 34  16.83  \(3^*7^*\)  200  880  
9T11  6  \([3, 0]\)  \(15.90_{q}\)  19.01  \(3^{9} 7^{4}\)  1  1.00  200  805  
54  8, 42  16.72  \(2^*3^*5^*\)  200  2637  
9T12  3  \(\sqrt{3}\)  \([3, 2]\)  \(7.75_{p}\)  10.71  \(2^{2} 307\)  7  0.67  34.20  256 
18T24  3  \(\sqrt{3}\)  \([3, 1]\)  10.03 \(_{g}\)  20.08  \(2^{2} 3^{4} 5^{2}\)  1  \(0.83_\circ \)  82.70  77 
\(M_9\)  72  6, 20  29.72  \(2^*3^*\)  100  27  
9T14  8  \([1, 0]\)  \(17.50_{q}\)  31.59  \(2^{24} 3^{10}\)  1  1.00  100  26  
\(C_3^2{\rtimes }C_8\)  72  TW  5, 16  25.41  \(2^*3^*\)  100  19  
9T15  8  \([1, 0]\)  \(17.50_{q}\)  25.41  \(2^{31} 3^{4}\)  1  1.00  100  16  
\(C_3\wr C_3\)  81  9, 59  75.41  \(3^*19^*\)  500  131  
9T17  \(3^{{{3}}}\)  \(\sqrt{3}\)  \([3, 3]\)  \(12.92_{q}\)  30.14  \(7^{2} 13 \cdot 43\)  22  0.50  22.36  0 
\(C_3^2{\rtimes }D_6\)  108  11, 262  22.06  \(3^*23^*\)  150  12,002  
18T55  6  \([3, 1]\)  11.98 \(_{g}\)  24.38  \(2^{8} 3^{8} 5^{3}\)  4  \(0.83_\circ \)  65.07  229  
9T18  6  \([3, 2]\)  \(9.70_{p}\)  13.46  \(3^{3} 7^{2} 67^{2}\)  53  0.67  28.23  216  
\(C_3^2{\rtimes }QD_{16}\)  144  8, 62  23.41  \(3^*7^*\)  100  488  
9T19  8  \([1, 2]\)  12.13 \(_{v}\)  25.65  \(3^{13} 7^{6}\)  1  0.75  31.62  14  
18T68  8  \([2, 1]\)  14.44 \(_{g}\)  25.65  \(3^{13} 7^{6}\)  1  \(0.88_\circ \)  56.23  48  
\(C_3\wr S_3\)  162  13, 2004  29.89  \(3^*\)  200  1617  
9T20  \(3^{{{3}}}\)  \(\sqrt{3}\)  \([3, 3]\)  \(7.75_{p}\)  11.17  \(7 \cdot 199\)  23  0.50  14.14  20 
18T86  \(3^{{{3}}}\)  \(\sqrt{3}\)  \([3, 3]\)  8.23 \(_{v}\)  19.48  \(2^{2} 43^{2}\)  5  0.50  14.14  0 
162  11, 223  24.90  \(2^*3^*5^*\)  100  597  
9T21  \(6^{{{3}}}\)  \([3, 3]\)  \(9.70_{p}\)  15.58  \(5^{2} 83^{3}\)  2  0.50  14.14  0  
162  10, 205  26.46  \(3^*\)  100  180  
9T22  \(6^{{{3}}}\)  \([3, 3]\)  \(9.70_{p}\)  17.21  \(2^{6} 7^{4} 13^{2}\)  6  0.50  10  0  
\(C_3^2{\rtimes }\text {SL}_2(3)\)  216  7, 44  49.57  \(349^*\)  100  37  
9T23  8  \([1, 2]\)  \(11.29_{p}\)  23.39  \(547^{4}\)  10  0.75  31.62  3  
24T569  8  \(\sqrt{3}\)  \([2, 1]\)  \(18.99_{g}\)  38.84  \(349^{5}\)  1  0.94  74.99  20 
324  17, –  30.64  \(2^*3^*11^*\)  100  1816  
18T129  \(6^{{{3}}}\)  \([3, 3]\)  9.34 \(_{p}\)  22.88  \(2^{9} 23^{4}\)  16  0.50  14.14  0  
9T24  \(6^{{{3}}}\)  \([3, 3]\)  \(9.70_{p}\)  15.84  \(3^{7} 5^{2} 17^{2}\)  399  0.50  14.14  0  
324  9, 116  29.96  \(2^*3^*\)  100  107  
9T25  6  \([3, 3]\)  \(12.73_{p}\)  22.25  \(2^{6} 3^{8} 17^{2}\)  59  0.50  10  0  
18T141  6  \([3, 3]\)  9.34 \(_{p}\)  30.81  \(3^{8} 19^{4}\)  9  0.50  10  0  
12T133  4  \(\sqrt{3}\)  \([4, 2]\)  \(11.15_{g}\)  19.34  \(2^{6} 3^{7}\)  1  0.75  31.62  10 
12T132  \(4^{{{2}}}\)  \(\sqrt{3}\)  \([4, 2]\)  11.15 \(_{g}\)  33.50  \(2^{6} 3^{9}\)  1  0.75  31.62  0 
18T142  12  \([3, 3]\)  \(19.68_{p}\)  30.57  \(2^{18} 3^{26}\)  3  0.75  31.62  2  
\(C_3^2{\rtimes }\text {GL}_2(3)\)  432  10, 206  27.88  \(3^*11^*\)  76  453  
9T26  8  \([1, 2]\)  \(11.54_{g}\)  17.59  \(2^{6} 523^{3}\)  14  0.75  25.74  17  
18T157  8  \([2, 2]\)  12.14 \(_{p}\)  19.04  \(3^{7} 53^{4}\)  16  0.75  25.74  3  
24T1334  16  \([2, 1]\)  \(21.27_{g}\)  26.68  \(3^{26} 11^{10}\)  1  0.94  57.98  134  
\(S_3\wr C_3\)  648  14, 3706  33.56  \(2^*5^*13^*\)  150  1677  
18T197  \(6^{{{2}}}\)  \([3, 3]\)  9.70 \(_{v}\)  19.71  \(7^{4} 29^{3}\)  1  0.50  12.25  0  
18T202  6  \([4, 3]\)  9.70 \(_{v}\)  27.73  \(2^{6} 3^{9} 19^{2}\)  49  0.50  12.25  0  
9T28  6  \([3, 4]\)  \(9.70_{p}\)  12.20  \(3^{8} 503\)  335  0.33  5.31  0  
12T176  8  \([4, 2]\)  \(12.05_{g}\)  21.75  \(11^{4} 43^{4}\)  4  0.75  42.86  57  
36T1102  12  \([4, 3]\)  15.58 \(_{v}\)  29.90  \(2^{6} 5^{10} 13^{8}\)  1  0.75  42.86  2  
18T206  12  \([3, 4]\)  \(15.23_{v}\)  21.99  \(2^{6} 7^{10} 29^{4}\)  10  0.67  28.23  8  
24T1539  8  \(\sqrt{3}\)  \([8, 4]\)  17.07 \(_{v}\)  45.71  \(2^{14} 3^{19}\)  3  0.75  42.86  0 
648  13, 2206  40.81  \(2^*3^*17^*\)  200  838  
9T29  6  \([3, 3]\)  \(12.73_{p}\)  16.62  \(2^{8} 7^{2} 41^{2}\)  31  0.50  14.14  0  
18T223  6  \([3, 3]\)  9.70 \(_{v}\)  30.14  \(2^{4} 5^{2} 37^{4}\)  71  0.50  14.14  0  
24T1527  4  \(\sqrt{3}\)  \([4, 2]\)  \(12.05_{g}\)  32.34  \(2^{2} 3^{7} 5^{3}\)  18  0.75  53.18  43 
12T175  4  \(\sqrt{3}\)  \([4, 4]\)  \(7.60_{p}\)  9.23  \(11 \cdot 659\)  164  0.50  14.14  14 
36T1131  6  \(\sqrt{3}\)  \([6, 6]\)  11.95 \(_{v}\)  36.04  \(2^{2} 3^{8} 17^{4}\)  6  0.50  14.14  0 
36T1237  12  \([3, 3]\)  17.78 \(_{p}\)  44.72  \(2^{22} 3^{7} 17^{8}\)  1  0.75  53.18  6  
18T219  12  \([3, 3]\)  14.05 \(_{v}\)  33.23  \(3^{17} 107^{5}\)  5  0.75  53.18  65  
24T1540  8  \(\sqrt{3}\)  \([8, 4]\)  17.07 \(_{v}\)  49.37  \(2^{10} 3^{15} 7^{4}\)  2  0.75  53.18  1 
648  13, 1322  30.37  \(2^*269^*\)  200  4001  
9T30  6  \([3, 3]\)  \(9.70_{p}\)  10.67  \(11^{2} 23^{3}\)  3  0.50  14.14  1  
18T222  6  \([3, 3]\)  9.70 \(_{v}\)  23.27  \(31^{3} 73^{2}\)  57  0.50  14.14  0  
12T178  8  \([4, 2]\)  \(12.05_{g}\)  18.27  \(2^{10} 59^{4}\)  10  0.75  53.18  327  
12T177  \(8^{{{2}}}\)  \([4, 2]\)  12.05 \(_{g}\)  24.98  \(2^{10} 23^{6}\)  22  0.75  53.18  173  
36T1121  6  \(\sqrt{3}\)  \([6, 6]\)  11.95 \(_{v}\)  31.61  \(2^{8} 7^{2} 43^{3}\)  91  0.50  14.14  0 
36T1123  12  \([3, 3]\)  17.78 \(_{p}\)  28.87  \(2^{35} 5^{10}\)  2  0.75  53.18  71  
18T218  12  \([3, 3]\)  \(14.05_{v}\)  18.33  \(2^{10} 269^{5}\)  1  0.75  53.18  453  
\(S_3 \wr S_3\)  1296  22, –  36.26  \(2^*3^*\)  200  12,152  
18T320  6  \([3, 4]\)  8.80 \(_{p}\)  14.80  \(5 \cdot 23^{3} 173\)  8562  0.33  5.85  0  
18T312  6  \([4, 3]\)  8.79 \(_{p}\)  17.45  \(3^{5} 11^{2} 31^{2}\)  343  0.50  14.14  0  
9T31  6  \([3, 4]\)  \(9.70_{p}\)  10.38  \(31^{2} 1303\)  10,036  0.33  5.85  0  
18T303  6  \([4, 3]\)  8.79 \(_{p}\)  18.34  \(5^{5} 23^{3}\)  4  0.50  14.14  0  
12T213  8  \([4, 4]\)  \(11.29_{p}\)  13.38  \(5^{2} 7^{4} 131^{2}\)  397  0.50  14.14  3  
24T2895  8  \([4, 2]\)  12.79 \(_{g}\)  30.65  \(2^{8} 3^{4} 5^{6} 7^{4}\)  77  0.75  53.18  230  
18T315  12  \([3, 4]\)  \(13.59_{p}\)  22.81  \(2^{10} 23^{4} 37^{5}\)  72  0.67  34.20  105  
36T2216  12  \([3, 4]\)  13.79 \(_{p}\)  29.13  \(2^{10} 3^{17} 41^{4}\)  78  0.67  34.20  11  
36T2305  12  \([4, 3]\)  15.14 \(_{p}\)  31.73  \(2^{18} 331^{5}\)  58  0.75  53.18  151  
36T2211  12  \([6, 6]\)  14.64 \(_{p}\)  37.29  \(2^{16} 5^{8} 7^{10}\)  55  0.50  14.14  0  
36T2214  12  \([4, 3]\)  15.14 \(_{p}\)  32.07  \(2^{22} 3^{24}\)  11  0.75  53.18  136  
24T2912  16  \([8, 4]\)  18.82 \(_{p}\)  35.12  \(5^{12} 23^{12}\)  4  0.75  53.18  40 
Our second result unconditionally identifies initial segments:
8.2 Tables detailing results on lower bounds and initial segments
Our tables are organized by the standard doublyindexed lists of transitive permutation groups mTj, with degrees m running from 2 through 9. Within a degree, the blocks of rows are indexed by increasing j. There is no block to print if mTj has no faithful irreducible characters. For example, there is no block to print for groups having noncyclic center, such as \(4T2 = V = C_2 \times C_2\) or \(8T9 = D_4 \times C_2\). Also the block belonging to mTj is omitted if the abstract group G underlying mTj has appeared earlier. For example \(G=S_4\) has four transitive realization in degrees \(m \le 8\), namely 4T5, 6T7, 6T8, and 8T14; there is correspondingly a 4T5 line on our tables, but no 6T7, 6T8, or 8T14 lines.
8.2.1 Top row of the Gblock
The top row in the Gblock is different from the other rows, as it gives information corresponding to the abstract group G. Instead of referring to a faithful irreducible character, as the other lines do, many of its entries are the corresponding quantities for the regular character \(\phi _G\). The first four entries are a common name for the group G (if there is one), the order \(\phi _G(e) = G\), the symbol TW if G is known to have the universal tamewild property as defined in [11], and finally k, N. Here, k is the size of the rational character table, and N is number of vertices of the polytope \(P_G\) discussed in Sect. 6.2, or a dash if we did not compute N. The last four entries are the smallest root discriminant of a Galois G field, the factored form of the corresponding discriminant, a cutoff B for which the set \(\mathcal {K}(G;B)\) is known, and the size \(\mathcal {K}(G;B)\).
8.2.2 Remaining rows of the Gblock
Each remaining line of the Gblock corresponds to a type \((G,\chi )\). However the number of rows in the Gblock is typically substantially less than the number of faithful irreducible characters of G, as we list only one representative of each \({{\mathrm{Gal}}}(\overline{\mathbf {Q}}/\mathbf {Q}) \times \text{ Out }(G)\) orbit of such characters. As an example, \(S_6\) has eleven characters, all rational. Of the nine which are faithful, there are three which are fixed by the nontrivial element of \(\text{ Out }(S_6)\) and the others form three twoelement orbits. Thus the \(S_6\)block has six rows. In general, the information on a \((G,\chi )\) row comes in three parts, which we now describe in turn.
First four columns The first column gives the lexicographically first permutation group mTj for which the corresponding permutation character has \(\chi \) as a rational constituent. Then \(n_1=\chi _1(e)\) is the degree of an absolutely irreducible character \(\chi _1\) such that \(\chi \) is the sum of its conjugates. The number \(n_1\) is superscripted by the size of the \({{\mathrm{Out}}}(G)\) orbit of \(\chi \), in the unusual case when this orbit size is not 1. Next, the complex number z is a generator for the field generated by the values of the character \(\chi _1\), with no number printed in the common case that \(\chi _1\) is rationalvalued. The last entry gives the interval , where and \(\widehat{\chi }\) are the numbers introduced in the beginning of Sect. 5. In the range presented, the data of mTj, \(n_1\), z, and suffice to distinguish Galois types \((G,\chi )\) from each other.
Middle four columns The next four columns focus on minimal root conductors. In the first entry, \(\mathfrak {d}\) is the best conditional lower bound we obtained for root conductors, and the subscript \(i \in \{\ell ,s,q,g,p,v\}\) gives information on the corresponding auxiliary character \(\phi \). The first four possibilities refer to the methods of Sect. 5, namely linear, square, quadratic, and Galois. The last two, p and v, indicate a permutation character and a character coming from a vertex of the polytope \(P_G\). The best \(\phi \) of the ones we inspect is always at a vertex, except in the three cases on Table 4 where \(*\) is appended to the subscript. Capital letters S, Q, G, P, and V also appear as subscripts. These occur only for groups marked with TW, and indicate that the tamewild principle improved the lower bound. For most groups with fifteen or more classes, it was prohibitive to calculate all vertices, and the best of the other methods is indicated.
When the second entry is in roman type, it is the minimal root conductor and the third entry is the minimal conductor in factored form. When the second entry is in italic type, then it is the smallest currently known root conductor. The fourth entry gives the position of the source number field on the complete list ordered by Galois root discriminant. This information lets readers obtain further information from [10], such as a defining polynomial and details on ramification.
Last three columns The quantity \(\beta \) is the exponent we are using to pass from Galois number fields to Artin representations. Writing \(\widehat{\underline{\alpha }}= \widehat{\underline{\alpha }}(G,\chi ,\phi _G)\) and \(\underline{\alpha }= \underline{\alpha }(G,\chi ,\phi _G)\), one has the universal relation \(\widehat{\underline{\alpha }}\le \underline{\alpha }\). When equality holds then the common number is printed. To indicate that inequality holds, an extra symbol is printed. When we know that G satisfies TW then we can use the larger exponent and \(\underline{\alpha }_\bullet \) is printed. Otherwise we use the smaller exponent and \(\widehat{\underline{\alpha }}_\circ \) is printed. The column \(B^\beta \) gives the corresponding upper bound on our complete list of root conductors. Finally the column \(\#\) gives \(\mathcal {L}(G,\chi ; B^\beta )\), the length of the complete list of Artin Lfunctions we have identified. For the Lfunctions themselves, we refer to [14].
9 Discussion of tables
In this section, we discuss four topics, each of which makes specific reference to parts of the tables of the previous section. Each of the topics also serves the general purpose of making the tables more readily understandable.
9.1 Comparison of first Galois root discriminants and root conductors
Suppose first, for notational simplicity, that G is a group for which all irreducible complex characters take rational values only. When one fixes \(K^\mathrm{gal}\) with \({{\mathrm{Gal}}}(K^\mathrm{gal}/\mathbf {Q}) \cong G\) and lets \(\chi \) runs over all the irreducible characters of G, the root discriminant \(\delta _\mathrm{Gal}\) is just the weighted multiplicative average \(\prod _\chi = {\delta _\chi ^{\chi (e)^2/G}}\). Deviation of a root conductor \(\delta _\chi \) from \(\delta _\mathrm{Gal}\) is caused by nonzero values of \(\chi \). When \(\chi (e)\) is large and is small, \(\delta _\chi \) is necessarily close to \(\delta _\mathrm{Gal}\). One can therefore think generally of \(\delta _\mathrm{Gal}\) as a first approximation to \(\delta _\mathrm{\chi }\). The general principle of \(\delta _\mathrm{Gal}\) approximating \(\delta _\mathrm{\chi }\) applies to groups G with irrational characters as well.
Our first example of \(S_5\) illustrates both how the principle \(\delta _\mathrm{Gal} \approx \delta _{\chi }\) is reflected in the tables, and how it tends to be somewhat off in the direction that \(\delta _\mathrm{Gal} > \delta _{\chi }\). For a given \(K^\mathrm{gal}\), the variance of \(\delta _\chi \) about its \(\delta _\mathrm{Gal}\) is substantial and depends on the details of the ramification in \(K^\mathrm{gal}\). There are many \(K^\mathrm{gal}\) with root discriminant near the minimal root discriminant, all of which are possible sources of minimal root conductors. It is therefore expected that the minimal conductors \(\delta _1(S_5,\chi ) = \min \delta _\chi \) printed in the table, 6.33, 18.72, 16.27, 17.78, and 18.18, are substantially less than the printed minimal root discriminant \(\delta _1(S_5,\phi _{120}) \approx 24.18\). As groups G get larger, one can generally expect tighter clustering of the \(\delta _1(G,\chi )\) about \(\delta _1(G,\phi _G)\). One can see the beginning of this trend in our partial results for \(S_6\) and \(S_7\).
9.2 Known and unknown minimal root conductors
Our method of starting with a complete list of Galois fields is motivated by the principle from the previous subsection that the Galois root discriminant \(\delta _\mathrm{Gal}\) is a natural first approximation to \(\delta _\chi \). Indeed, as the tables show via nonzero entries in the \(\#\) column, this general method suffices to obtain a nonempty initial segment for most \((G,\chi )\). As our focus is primarily on the first root conductor \(\delta _1 = \delta _1(G,\chi )\), we do not pursue larger initial segments in these cases.
When the initial segment from our general method is empty, as reported by a 0 in the \(\#\) column, we aim to nonetheless present the minimal root conductor \(\delta _1\). Suppose there are subgroups \(H_m \subset H_k \subseteq G\), of the indicated indices, such that a multiple of the character \(\chi \) of interest is a difference of the corresponding permutation characters: \(c \chi = \phi _m  \phi _k\). Suppose one has the complete list of all degree m fields corresponding to the permutation representation of G on \(G/H_m\) and root discriminant \(\le B\). Then one can extract the complete list of \(\mathcal {L}(G,\chi ;B^{m/(mk)})\) of desired Artin Lfunctions.
For example, consider \(\chi _5\), the absolutely irreducible 5dimensional character of \(A_6\). The permutation character for a corresponding sextic field decomposes \(\phi _6 = 1+\chi _5\), and so the discriminant of the sextic field equals the conductor of \(\chi _5\). As an example with \(k>1\), consider the 6dimension character \(\chi _6\) for \(C_3\wr C_3 = 9T17\), which is the sum of a threedimensional character and its conjugate. The nonic field has a cubic subfield, and the characters are related by \(\phi _9 = \phi _3 + \chi _6\). In terms of conductors, \(D_9 = D_3 \cdot D_{\chi _6}\), where \(D_9\) and \(D_3\) are field discriminants. So, we can determine the minimal conductor of an Lfunction with type \((C_3\wr C_3,\chi _6)\) from a sufficiently long complete list of nonic number fields with Galois group \(C_3\wr C_3\).
This method, applied to both old and newer lists presented in [10], accounts for all but one of the \(\delta _1\) reported in Roman type on the same line as a 0 in the \(\#\) column. The remaining case of an established \(\delta _1\) is for the type \((\mathrm {GL}_3(2),\chi _7)\). The group \(\mathrm {GL}_3(2)\) appears on our tables as 7T5. The permutation representation 8T37 has character \(\chi _7+1\). Here the general method says that \(\mathcal {L}(\mathrm {GL}_3(2),\chi _7; 26.12)\) is empty. It is prohibitive to compute the first octic discriminant by searching among octic polynomials. In [9] we carried out a long search of septic polynomials, examining all local possibilities giving an octic discriminant at most 30. This computation shows that \(\mathcal {L}(\mathrm {GL}_3(2),\chi _7; 48.76) = 25\) and in particular identifies \(\delta _1 = 21^{8/7} \approx 32.44\).
The complete lists of Galois fields for a group first appearing in degree m were likewise computed by searching polynomials in degree m, targeting for small \(\delta _\mathrm{Gal}\). This single search can give many first root conductors at once. For example, the largest groups on our octic and nonic tables are \(S_4 \wr S_2 = 8T47\) and \(S_3 \wr S_3 = 9T31\). In these cases, minimal root conductors were obtained for 5 of the 10 and 7 of the 12 faithful \(\chi \) respectively. Searches adapted to a particular character \(\chi \) as in the previous paragraph can be viewed as a refinement of our method, with targeting being not for small \(\delta _\mathrm{Gal}\) but instead for small \(\delta _\chi \). Many of the italicized entries in the column \(\delta _1\) seem improvable to known minimal root conductors via this refinement.
9.3 The ratio \(\delta _1/{\mathfrak {d}}\)
When the translation from no Lfunctions to many Lfunctions is not abrupt, but there is an Lfunction with outlyingly small conductor, again \(\delta _1/{\mathfrak {d}}\) may be quite close to 1. As an example, for \((8T25,\chi _7)\), one has \({\mathfrak {d}} \approx 16.10\) and \(\delta _1 = 29^{6/7} \approx 17.93\) yielding \(\delta _1/{\mathfrak {d}} \approx 1.11\). However in this case the next root conductor is \(\delta _2 = 113^{6/7} \approx 57.52\), yielding \(\delta _2/{\mathfrak {d}} \approx 3.57\). Thus the close agreement is entirely dependent on the Lfunction with outlyingly small conductor. Even the second root conductor is somewhat of an outlier as the next three conductors are 71.70, 76.39, and 76.39, so that already \(\delta _3/{\mathfrak {d}} \approx 4.45\).
There are many \((G,\chi )\) in the table for which the ratio \(\delta _1/{\mathfrak {d}}\) is around 2 or 3. There is some room for improvement in our analytic lower bounds, for example changing the test function (3.1), varying \(\phi \) over all of \(P_G\), or replacing the exponent \(\widehat{\underline{\alpha }}\) with the best possible exponent b. However examples like the one in the previous paragraph suggest to us that in many cases the resulting increase in \(\mathfrak {d}\) towards \(\delta _1\) would be very small.
9.4 Multiply minimal fields
Invariants and defining polynomials for Galois number fields giving rise to minimal root discriminants for at least two rationally irreducible characters \(\chi \)

Further information on the characters \(\chi \) is given in Tables 3, 4, 5, 6 and 7. An interesting point, evident from repeated 1’s in the Gblock on these tables, is that five of the six fields \(K^\mathrm{gal}\) are also first on the list of G fields ordered by root discriminant. The exception is the \(S_6\) field on Table 8, which is only sixth on the list of Galois \(S_6\) fields ordered by root discriminant.
10 Lower bounds in large degrees
10.1 The constant \(\Omega \) as a limiting lower bound.
The next corollary makes use of the extreme character values and \(\widehat{\chi }\) introduced at the beginning of Sect. 5. It shows that if one restricts the type, then one can indeed increase \(\sqrt{\Omega }\) all the way to \(\Omega \). We formulate the corollary in the context of rationally irreducible characters, to stay in the main context we have set up. However via (2.4), it can be translated to a statement about absolutely irreducible characters.
Corollary 10.1
 A:
 B:
\(\widehat{\chi }_k/n_k \rightarrow 0\).
Proof
For the proof of Case B, the square auxiliary character would work equally well through (5.2). Also (10.1), (10.2), and Corollary 10.1 could all be strengthened by considering the placement of complex conjugation. For example, when restricting to the totally real case \(c=e\), the \(\Omega \)’s in (10.1), (10.2), and (10.3) are simply replaced by \(\Theta \approx 215.3325\).
10.2 Four contrasting examples
Four sequences of types, with Corollary 10.1 applicable to the first three
\(\varvec{G_k}\)  \(\varvec{\chi _k}\)  \(\varvec{\widehat{\chi }_k}\)  \(\varvec{n}\)  \(\varvec{G}\)  \(\varvec{A}\)  \(\varvec{B}\)  

\(\mathrm {PGL}_2(k)\)  \(\text{ Steinberg }\)  1  1  k  \(k^3k\)  \(\checkmark \)  \(\checkmark \) 
\(S_{k}\)  \(\text{ Reflection }\)  1  \(k3\)  \(k1\)  k!  \(\checkmark \)  
\(2_\epsilon ^{1+2k}\)  \(\text{ Spin }\)  \(2^k\)  0  \(2^k\)  \(2^{1+2k}\)  \(\checkmark \)  
\(2^k.S_k\)  \(\text{ Reflection }\)  k  \(k2\)  k  \(2^k\) k! 
10.2.1 The group \(\mathrm {PGL}_2(k)\) and its characters of degree \(k1\), k, and \(k+1\).
In the sequence \((\mathrm {PGL}_2(k),\chi _k)\) from the first line of Table 9, the index k is restricted to be a prime power. The permutation character \(\phi _{k+1}\) arising from the natural action of \(\mathrm {PGL}_2(k)\) on \(\mathbb {P}^1(\mathbb {F}_k)\) decomposes as \(1+\chi _k\) where \(\chi _k\) is the Steinberg character. Table 9 says that the ratios and \(\widehat{\chi }_k/n_k\) are both 1 / k, so Corollary 10.1 applies through both Hypotheses A and B.
The conductor of \(\chi _k\) is the absolute discriminant of the degree \(k+1\) number field with character \(\phi _{k+1}\). Thus, in this instance, (10.3) is already implied by the classical (10.1). However, the other nonabelian irreducible characters \(\chi \) of \(\mathrm {PGL}_2(k)\) behave very similarly to \(\chi _k\). Their dimensions are in \(\{k1,k,k+1\}\) and their values besides \(\chi (e)\) are all in \([2,2]\). So suppose for each k, an arbitrary nonabelian rationally irreducible character \(\chi _k\) of \(\mathrm {PGL}_2(k)\) were chosen, in such a way that the sequence \((\chi _k,\chi _k)\) is bounded. Then Corollary 10.1 would again apply through both Hypotheses A and B. But now the \(\chi _k\) are not particularly closely related to permutation characters.
10.2.2 The group \(S_{k}\) and its canonical characters
As with the last example, the permutation character \(\phi _{k}\) arising from the natural action of \(S_{k}\) on \(\{1,\ldots ,k\}\) decomposes as \(1 + \chi _k\) where \(\chi _k\) is the reflection character with degree \(k1\). The second line of Table 9 shows that Corollary 10.1 applies through Hypothesis A. In fact, using the linear auxiliary character underlying Hypothesis A here is essential; the limiting lower bound coming from the square or quadratic auxiliary characters is \(\sqrt{\Omega }\), and this lower bound is just 1 from the Galois auxiliary character.
Again in parallel to the previous example, the familiar sequence \((S_k,\chi _k)\) of types needs to be modified to make it a good illustration of the applicability of Corollary 10.1. Characters of \(S_k\) are most commonly indexed by partitions of k, with \(\chi _{(k)} =1\), \(\chi _{(k1,1)}\) being the reflection character, and \(\chi _{(1,1,\ldots ,1,1)}\) being the sign character. However an alternative convention is to include explicit reference to the degree k and then omit the largest part of the partition, so that the above three characters have the alternative names \(\chi _{k,()}\), \(\chi _{k,(1)}\), and \(\chi _{k,(1,\ldots ,1,1)}\). With this convention, one can prove that for any fixed partition \(\mu \) of a positive integer m, the sequence of types \((G_k,\chi _{k,\mu })\) satisfies Hypothesis A but not B.
10.2.3 The extraspecial group \(2_\epsilon ^{1+2k}\) and its degree \(2^k\) character
Fix \(\epsilon \in \{+,\}\). Let \(G_k\) be the extraspecial 2group of type \(\epsilon \) and order \(2^{1+2k}\), so that \(2^{1+2}_+\) and \(2^{1+2}_{}\) are the dihedral and quaternion groups respectively. These groups each have exactly one irreducible character of degree larger than 1, this degree being \(2^k\). There are just three character values, \(2^k\), 0, and \(2^k\). For these two sequences, Corollary 10.1 again applies, but now only through Hypothesis B.
10.2.4 The Weyl group \(2^k.S_k\) and its degree k reflection character
The Weyl group \(W(B_k) \cong 2^k.S_k\) of signed permutation matrices comes with its defining degree k character \(\chi _k\). Here, as indicated by the fourth line of Table 9, neither hypothesis of Corollary 10.1 applies.
However the conclusion (10.3) of Corollary 10.1 continues to hold as follows. Relate the character \(\chi _k\) in question to the two standard permutation characters of \(2^k.S_k\) via \(\phi _{2k} = \phi _k + \chi _k\). For a given \(2^k.S_k\) field, \(D_{\Phi _{2k}}=D_{\Phi _k}D_{\mathcal {X}_k}\). But, since \(\Phi _k\) corresponds to an index 2 subfield of the degree 2k number field for \(\Phi _{2k}\), we have \(D_{\Phi _k}^2\mid D_{\Phi _{2k}}\). Combining these we get \(D_{\Phi _k} \mid D_{\mathcal {X}_k}\) and hence \(\delta _{\Phi _k} < \delta _{\mathcal {X}_k}\). So (10.1) implies (10.3).
10.3 Concluding speculation
As we have illustrated in Sects. 10.2.1–10.2.3, both Hypothesis A and Hypothesis B are quite broad. This breadth, together with the fact that the conclusion (10.3) still holds for our last sequence, raises the question of whether (10.3) can be formulated more universally. While the evidence is far from definitive, we expect a positive answer. Thus we expect that the first accumulation point of the numbers \(\delta _1(G,\chi )\) is at least \(\Omega \), where \((G,\chi )\) runs over all types with \(\chi \) irreducible. Phrased differently, we expect that the first accumulation point of the root conductors of all irreducible Artin Lfunctions is at least \(\Omega \).
Declarations
Acknowledgements
DPR’s work on this paper was supported by Grant #209472 from the Simons Foundation and Grant DMS1601350 from the National Science Foundation.
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