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MIXED DEGREE NUMBER FIELD COMPUTATIONS
JOHN W. JONES AND DAVID P. ROBERTS
Abstract. We present a method for computing complete lists of number fields in cases where the Galois group, as an abstract group, appears as a Galois group in smaller degree. We apply this method to find the twenty-five octic fields with Galois group PSL 2 (7) and smallest absolute discriminant. We carry out a number of related computations, including determining the octic field with Galois group 2^3 : GL 3 (2) of smallest absolute discriminant.
- Introduction
1.1. Overview. Number theorists have computed number fields with minimal ab- solute discriminants for each of the thirty possible Galois groups in degrees at most
- In degrees 8 and 9, the minimal fields are known for the seventy-five solvable Galois groups. All these minimal fields are available, together with references to sources, on the Kl¨uners-Malle database [KM01]. The minimal fields are also avail- able, typically as first elements on long complete lists, at the online databases [JR14a, LMF18]. The Kl¨uners-Malle paper [KM01] also gives smallest known absolute discrimi- nants for the five nonsolvable octic groups and the four nonsolvable nonic groups. Despite the many years that have passed since its publication, rigorous minima have not been established for these nine groups. In this paper, we address two of the nine cases, proving that the absolute discriminants 3^878 and 5717^2 presented in [KM01] for the octic groups PSL 2 (7) and 2^3 : GL 3 (2) are indeed minimal. These cases are related through the exceptional isomorphism PSL 2 (7) ∼= GL 3 (2). One element of our approach for finding the PSL 2 (7) minimum was suggested already in [KM01]: any octic PSL 2 (7) field K 8 has the same Galois closure as two septic GL 3 (2) fields K 7 a and K 7 b. As the discriminants satisfy D 7 a = D 7 b | D 8 , one can in principle establish minimality of the octic discriminant 21^8 by conducting a search of all septic fields with absolute discriminant ≤ 218. We combine this with the method of targeted Hunter searches, which requires us to analyze, on a prime-by-prime basis, how discriminants either stay the same or increase when one passes from septic to octic fields. This targeting based on discriminants makes the computation feasible. We add several smaller refinements to make the computation run even faster. Our title refers to the general method of carrying out a carefully targeted search in one degree to obtain a complete list of fields in a larger degree. Section 2 gives background and then Section 3 describes the general method, using our case where the two degrees are 7 and 8 as an illustration. Section 4 presents our minimality
2010 Mathematics Subject Classification. Primary 11R21; Secondary 11Y40, 11R32. Key words and phrases. number field; discriminant. Roberts was supported by grant #209472 from the Simons foundation. 1
2 JOHN W. JONES AND DAVID P. ROBERTS
result for PSL 2 (7), improved in Theorem 1 to the complete list of twenty-five octic PSL 2 (7) fields with discriminant ≤ 308. This section also presents corollaries giving minimal absolute discriminants for certain related groups in degrees 16, 24, and 32. Section 5 gives a second illustration of the mixed degree method, now with degrees 5 and 6. Here we use the exceptional isomorphisms A 5 ∼= PSL 2 (5) and S 5 ∼= PGL 2 (5) and Theorem 2 considerably extends the known list of sextic PSL 2 (5) and PGL 2 (5) fields. We also explain in this section potential connections with asymptotic mass formulas and Artin representations. Our final section returns to groups related to the septic group GL 3 (2). Theorem 3 finds all alternating septics with discriminant ≤ 127. The long runtime of this search makes clear the importance of targeting for Theorem 1. However just the bound 127 is sufficient for our last corollary, which confirms minimality of the 2^3 : GL 3 (2) field with discriminant 5717^2.
1.2. Notation and conventions. We denote the cyclic group of order n by Cn. We use N :H to denote a semi-direct product with normal subgroup N and com- plement H. A number field is a finite extension of Q, which we consider up to isomorphism. If K/Q is such an extension with degree n, then its normal closure, Kg^ , is Galois over Q. Moreover, Gal(Kg^ /Q) comes with a natural embedding into Sn, which is well-defined up to conjugation. We denote the image of such an embedding, which is a transitive subgroup of Sn, by simply Gal(K). Transitive subgroups of Sn, considered up to conjugation, have been classi- fied and indexed for small n, and is available through Magma [BCP97] and Pari [PAR15], as well as though the Galois groups section of the LMFDB web site [LMF18], which provides information on each such conjugacy class of subgroups for n < 24. Here, we denote the jth subgroup by nT j. In §4.2 we use the classification of nearly 3 million transitive subgroups of S 32 which was completed more recently in [CH08]. When several non-isomorphic fields have the same splitting field, we refer to them as siblings. For example, fields K 7 a, K 7 b, and K 8 as in the overview are siblings. If K 1 and K 2 are siblings, then Gal(K 1 ) ∼= Gal(K 2 ) as abstract groups, but as described above, the two Galois groups typically come with different embeddings into Sn, possibly even for different n. We aim to exploit this difference where possible for the computations in this paper. While the nT j notation specifies both the degree n of the stem field and the conjugacy class of the subgroup in Sn, the numeric identifier conveys no information on the structure of the group. We use group names in the spirit of [CHM98], which assigns standard names for the groups nT j indicating their structures as permutation groups for n < 16. For example, 7T 5 = GL 3 (2) and 8T 37 = PSL 2 (7) have natural transitive actons on the projective spaces P^2 (F 2 ) and P^1 (F 7 ) of orders 7 and 8 respectively. The third group mentioned in the overview is the group of affine transformations of F^32. Our notation emphasizes its semidirect product structure: 8 T 48 = 2^3 : GL 3 (2). It is often enlightening to shift the focus from the absolute discriminant |D| of a degree n number field K to the corresponding root discriminant rd(K) = δ = |D|^1 /n. We generally try to indicate both, as in the numbers 21^8 and 30^8 of the overview.
4 JOHN W. JONES AND DAVID P. ROBERTS
G ↪→ Sm. Targets are triples ((e 1 ,... , et), cn, cm) where cn is the local discriminant exponent for the small degree fields searched, while cm is the local discriminant exponent of the larger degree fields actually sought. One uses the values pcm^ to decide which combinations of targets to search, and ((e 1 ,... , et), cn) to carry out the actual search in degree n. We describe how one deals with the two degrees here, often by using our first case with n = 7 and m = 8 as an example. Once we have the degree n polynomials in hand, we compute the corresponding degree m polynomials as resolvents using Magma [BCP97].
3.1. Tame ramification. The behavior of tame ramification under degree changes is straightforward. Let K be a degree n number field, G its Galois group, and p a tamely ramified prime. The inertia subgroup I for a prime above p is cyclic; let τ be a generator. Via the given inclusion G ⊆ Sn, we let e 1 , e 2 ,... , et be the cycle type of τ. These match the ej of the local target described above. The exponent of p in the discriminant of K is then given by
(2) cn =
∑^ t
j=
(ej − 1) = n − t.
When one is considering also a second degree m, one just runs through the above procedure a second time. In our first case, G ∼= GL 3 (2) ∼= PSL 2 (7), each row of Table 1 represents a candidate for τ. The row then gives the corresponding pair of partitions (λ 7 , λ 8 ) and pair of discriminant exponents (c 7 , c 8 ). These discriminant exponents are computed from the partitions via formula (2).
Table 1. Cycle types and discriminant exponents for GL 3 (2) ∼= PSL 2 (7) in degrees 7 and 8.
λ 7 λ 8 c 7 c 8 7 7 , 1 6 6 4 , 2 , 1 4 , 4 4 6 3 , 3 , 1 3 , 3 , 1 , 1 4 4 2 , 2 , 1 , 1 , 1 2 , 2 , 2 , 2 2 4
Note that if a prime p is tamely ramified in our pair of fields (K 7 , K 8 ), then its minimal contribution to the discriminant of the octic is p^4. Thus, when searching for octic fields with absolute discriminant ≤ B, we need only consider primes p ≤ 4
B.
Our largest search used B = 30^8 , so p ≤ 900. The relation D 7 | D 8 mentioned in the introduction is due to the fact that we always have c 7 ≤ c 8 , and that this inequality also holds for wildly ramified primes. In two of the tame cases, one has equality, but in the other two tame cases one has strict inequality. Our method using targeted searches makes use of the strictness of these latter inequalities.
3.2. Wild ramification. An explicit description of the behavior of wild p-adic ramification under degree changes becomes rapidly more complicated as ordp(G) increases. We describe just our case G ∼= GL 3 (2) ∼= PSL 2 (7) here, as this case represents the basic nature of the general case well.
MIXED DEGREE NUMBER FIELD COMPUTATIONS 5
Since |G| = 2^3 · 3 ·7, the only primes which can be wildly ramified in a G extension are 2, 3, and 7. For a subgroup of G to be an inertia group for a wildly ramified prime p, it must be an extension of a cyclic group of order prime to p by a non- trivial p-group. The candidates for a G = PSL 2 (7) extension are given in Table 2. They run over all of the non-trivial proper subgroups of PSL 2 (7) up to conjugation,
Table 2. Wild ramification data for PSL 2 (7).
p I D λ 7 λ 8 (c 7 , c 8 ) 7 C 7 :C 3 C 7 :C 3 7 7 , 1 (8, 8), (10, 10) C 7 C 7 , C 7 :C 3 7 7 , 1 (12, 12) 3 S 3 S 3 3 , 3 , 1 6 , 2 (6, 8), (10, 12) C 3 C 3 , S 3 3 , 3 , 1 3 , 3 , 1 , 1 (8, 8) 2 A 4 S 4 4 , 3 6 , 1 4 , 4 (6, 8), (10, 16) D 4 D 4 4 , 2 , 1 8 (12, 22), (14, 24) C 4 C 4 , D 4 4 , 2 , 1 4 , 4 (14, 22) V V, D 4 , A 4 2 , 2 , 2 , 1 4 , 1 , 1 , 1 4 , 4 (6, 12), (8, 16) C 2 C 2 , V, C 4 2 , 2 , 1 , 1 , 1 2 , 2 , 2 , 2 (4, 8), (6, 12)
with the exception of two conjugacy classes of subgroups isomorphic to S 4. Each subgroup in the table is a candidate for being the inertia group for a wild prime for only one prime. The horizontal lines separate the subgroups according to this prime. The second column gives the isomorphism type of the candidate for inertia, and the third column gives corresponding candidates for the decomposition group. Over other 2-adic ground fields, A 4 = I = D is possible, but not over Q 2 since there is no ramified C 3 extension of Q 2. The columns labeled λ 7 and λ 8 show the orbit sizes of the actions of I in the degree seven and eight representations respectively. There are two possibilities for inertia group A 4 and V in degree 7, so we give both. The orbit sizes are helpful in determining the data c 7 and c 8 in each case, and λ 7 is the partition of 7 needed for carrying out the targeted Hunter search. In most cases, it is clear from Galois theory how to interpret the orbit sizes. For example, inside a Galois A 4 field, there are unique subfields of degrees 3 and 4 up to isomorphism. So the 4 in the first A 4 entry is for the usual quartic representation, and the 3 is its resolvent cubic. More detailed computations with the groups allow us to resolve the two ambiguities, which are as follows.
- A Galois D 4 field has three quadratic subfields and three quartic subfields (up to isomorphism). In the degree 7 partition 4, 2 ,1, the 4 represents a quartic stem field, say defined by a polynomial f , and then the 2 represents the field obtained from a root of x^2 − Disc(f ).
- A V field has three quadratic subfields. In the line for V , the 2, 2 , 2 , 1 represents the product of these quadratic fields and Q 2. The last column gives a list of candidate pairs (c 7 , c 8 ), coming by analyzing the corresponding local extensions. Some cases can be done using just Galois theory and general properties of extensions of local fields. A simple approach, however, which applies to all cases is to make use of the complete lists of the relevant local fields [JR06, LMF18].
MIXED DEGREE NUMBER FIELD COMPUTATIONS 7
3.3.3. Savings from ord 3 (G) = 1 at p = 3. Cases when 3 is wildly ramified also offer an opportunity to reduce the search time by more refined targeting. As can be seen from the two relevant lines of Table 2, the decomposition subgroup is isomorphic to C 3 or S 3. In either case, the orbit partition for the decomposition group is (3, 3 , 1). Thus, a defining polynomial factors as the product of two cubics times a linear polynomial over Z 3. The savings comes from the fact that the two cubics have to define the same 3-adic field. So, the procedure here starts with computing possible polynomials for each ramified cubic extension of Q 3 modulo some 3r^. We take all products of the form (x+a)g 1 g 2 where the gi come from the list for a given field and a runs through all possibilities in Z/ 3 r^. In our actual search, we worked modulo 3^2. The resulting local targets are considerably smaller. For example, a target ((3, 3 , 1), 8) from our general method includes cases where the two cubic factors define non-isomorphic fields with discriminant ideal (3)^4 and also cases where the cubics have discriminant ideals (3)^3 and (3)^5. All these possibilities are not searched in our refinement.
3.3.4. Exploiting arithmetic equivalence at p = 2. The final refinement we use ex- ploits the fact that for each octic field sought, we need to find just one of its two siblings in degree 7. These pairs of septic fields are examples of arithmetically equivalent fields. The two fields K 7 a and K 7 b have the same Dedekind zeta func- tion, the same discriminant, and the same ramification partition at all odd primes. However, at p = 2 one can have λ 7 a 6 = λ 7 b. In Table 2, there are two orbit partitions for the inertia group A 4 , and again two orbit partitions for V. For each of these cases, if a septic GL 3 (2) field has inertia subgroup I and one orbit partition, its sibling has the other orbit partition for I. We save by targeting 4,3 and 4, 1 , 1 ,1, but not their transforms 6,1 and 2, 2 , 2 ,1.
3.3.5. Savings from global root numbers being 1. As we mentioned in §2.1, our code does not distinguish signatures. If it did, there would be an opportunity for yet further savings as follows. A separable algebra Kv over Qv has a local root number (Kv ) ∈ { 1 , i, − 1 , −i}. For v = ∞, one has (Rr^ Cs) = (−i)s. For v a prime p, one has (Kp) = 1 unless the inertia group Ip has even order. Further information about local root numbers is at [JR06, §3.3], with many root numbers calculated on the associated database. The savings comes from the reciprocity relation
v (Kv^ ) = 1, so that the signature is restricted by the behavior at ramifying primes. While we are not using local root numbers in our searches, we are using them in our interpretation of the output of our first case. Interesting facts here include the general formulas (K 7 a,v ) = (K 7 b,v ) and (K 8 ,v ) = 1. Also, from Tables 1 and 2, one has equality of discriminant exponents c 7 = c 8 at a prime p if and only if |Ip| is odd; so in this case the septic sign (K 7 a,p) = (K 7 b,p) is 1.
- Results for PSL 2 (7) and related groups
4.1. A complete list of PSL 2 (7) octics. Our search for PSL 2 (7) fields with rd(K) ≤ 21 took 41 CPU-hours and confirmed that the discriminant 21^8 given in [KM01] is indeed the smallest. The extended search through rd(K) ≤ 30 took approximately four CPU-months. In this extended search, we combined targets for a given prime in a subsearch whenever the contribution to the octic field discrim- inant is the same. In this sense, the computation consisted of 1471 subsearches of
8 JOHN W. JONES AND DAVID P. ROBERTS
varying difficulty. The fastest 380 cases took at most 10 seconds each, the median length case took 6.5 minutes, and the slowest ten cases took from 20 to 35 hours each. The slowest cases all involved searches where c 7 = c 8 for every ramifying prime. This larger search found twenty-five fields.
Theorem 1. There are exactly 25 octic fields with Galois group PSL 2 (7) and dis- criminant ≤ 308 , as given in Table 3. The smallest discriminant of such a field is
The full list of fields is also available in a computer-readable format by searching the websites [JR14a, LMF18]. The two septic siblings of the first octic PSL 2 (7) field are given by x^7 − 7 x^4 − 21 x^3 + 21x^2 + 42x − 9 and x^7 − 7 x + 3, the latter being the famous Trinks polynomial [Tri68].
K 8 δ 8 δ 7
1 x^8 − 4 x^7 + 7x^6 − 7 x^5 + 7x^4 − 7 x^3 + 7x^2 + 5x + 1 21. 00 23. 70 2 x^8 − x^7 + 7x^6 − x^5 + 33x^4 + x^3 + 61x^2 + 13x + 58 21. 21 23. 97 3 x^8 − 4 x^7 + 14x^6 − 24 x^5 + 29x^4 − 32 x^3 + 18x^2 − 16 x + 17 21. 54 18. 44 4 x^8 − 3 x^7 + 4x^6 + 2x^5 − 10 x^4 + 16x^3 − 20 x + 28 22. 37 25. 48 5 x^8 − 2 x^7 + 10x^6 − 17 x^5 + 28x^4 − 38 x^3 + 34x^2 − 17 x + 10 22. 45 25. 58 6 x^8 − 2 x^7 + 2x^6 − 8 x^5 + 16x^4 − 16 x^3 + 14x^2 − 10 x + 4 23. 16 26. 50 7 x^8 − 3 x^7 + 9x^6 − 21 x^5 + 44x^4 − 69 x^3 + 84x^2 − 84 x + 73 23. 39 26. 81 8 x^8 − 4 x^7 + 10x^6 − 12 x^5 − 7 x^4 + 44x^3 − 46 x^2 − 4 x + 95 24. 16 21. 02 9 x^8 − 3 x^7 + x^6 + 9x^5 − 3 x^4 − 57 x^3 + 133x^2 − 132 x + 76 24. 23 27. 91 10 x^8 − 4 x^7 + 14x^6 − 28 x^5 + 49x^4 − 56 x^3 + 56x^2 − 14 x + 7 24. 25 22. 92 11 x^8 − x^7 + 5x^6 − 19 x^5 + 31x^4 − 47 x^3 + 47x^2 − 17 x + 4 25. 14 29. 11 12 x^8 − x^7 + 14x^4 − 28 x^3 + 28x^2 − 14 x + 14 26. 32 26. 52 13 x^8 − 4 x^7 + 11x^6 − 17 x^5 + 37x^4 − 78 x^3 + 132x^2 − 153 x + 72 26. 78 31. 29 14 x^8 − x^7 − 7 x^6 − 7 x^5 + 7x^4 + 49x^3 + 77x^2 + 31x + 4 26. 84 31. 37 15 x^8 − x^7 + x^6 − 11 x^5 + 11x^4 + 35x^3 + 45x^2 + 35x + 10 26. 97 27. 26 16 x^8 − 2 x^7 + 4x^6 + 2x^5 + 27x^4 − 46 x^3 + 84x^2 − 10 x + 59 27. 01 22. 41 17 x^8 − 3 x^7 + 6x^6 + 2x^5 + 6x^3 + 4x^2 + 6x + 6 27. 17 31. 82 18 x^8 − 2 x^7 + 2x^6 − 14 x^5 + 46x^4 − 86 x^3 + 126x^2 − 118 x + 49 27. 35 18. 14 19 x^8 − 4 x^7 + 14x^6 − 28 x^5 + 49x^4 − 56 x^3 + 56x^2 − 24 x + 5 28. 00 20. 41 20 x^8 + 28x^4 + 112x^2 − 32 x + 84 28. 00 24. 88 21 x^8 − 4 x^7 + 14x^6 − 28 x^5 + 63x^4 − 84 x^3 + 98x^2 − 52 x + 19 28. 00 24. 88 22 x^8 − 4 x^7 + 49x^4 − 42 x^3 + 77x^2 − 31 x + 19 28. 86 20. 77 23 x^8 − 4 x^7 + 9x^6 − 12 x^5 + 4x^4 − 4 x^3 + 13x^2 + 4x + 1 29. 05 31. 64 24 x^8 + 10x^6 + 12x^4 − 28 x^3 − 58 x^2 + 28x + 51 29. 22 27. 14 25 x^8 − x^7 − 4 x^6 − 12 x^5 + 8x^4 + 64x^3 + 16x^2 − 32 x + 24 29. 94 9. 39 Table 3. The 25 octic fields with Galois group PSL 2 (7) and root discriminant ≤ 30, along with their root discriminants, and the root discriminants of the corresponding septic siblings.
Figure 1 gives a visualization of how the ordering by discriminant of septic fields and the ordering by discriminant of octic fields, all having the simple group of order 168 as their Galois group, seem to have little to do with one another. The 25 PSL 2 (7) octics of Theorem 1 give the 24 points beneath the δ 8 = 30 line, the
10 JOHN W. JONES AND DAVID P. ROBERTS
[PAR15]. They are all cyclic and Table 4 gives their orders. The fact that all but three of these class groups are non-trivial is already remarkable. By way of contrast, the first 620 S 5 quintic fields ordered by absolute discriminant all have trivial class group. Using Theorem 1, one can get complete lists of fields with root discriminant ≤ 30 for many other Galois groups via class field theory. We restrict ourselves to four Galois groups, chosen because they interact interestingly with the class groups in Table 4. Each Galois group is even so absolute discriminants coincide with discriminants. Fields in Corollaries 1, 2, 3, and 4 correspond to entries in columns t, u, v, and w respectively in Table 4, these entries indicating ramification over the octic base. Most entries are 1, indicating that these class fields correspond to quotient groups of the class groups. However each column has entries larger than 1, so that each complete list of fields also includes ramified extensions not seen from class groups. To get the first two Galois groups, consider the unramified tower K 32 /K 16 /K 8 coming from the first field in Table 4, K 8 , and its cyclic class group of order four. Defining equations can be computed using Magma [BCP97] or Pari’s [PAR15] class field theory commands. Following the conventions of §1.2, let G 16 = Gal(K 16 ) and G 32 = Gal(K 32 ) be the corresponding Galois groups. Then K 16 and K 32 are the unique fields of degree 16 and 32 respectively with smallest discriminant for these Galois groups. In fact, G 16 is just the Cartesian product PSL 2 (7) × C 2 = 16T 714. More interestingly, G 32 = 32T 34620 is a non-split double cover of G 16 , having SL 2 (7) = 16T 715 as a subgroup with quotient group C 2. One can carry out a similar analysis for all twenty-five base fields, allowing ramified towers K 32 /K 16 /K 8 as well, with discriminants denoted D 32 , D 16 , and D 8. Using Pari [PAR15], we first compute those extensions with root discriminant ≤ 30. There are 56 possible K 16 in all and 163 such K 32. As the corollaries indicate, these numbers are reduced when we extract the fields with the Galois groups sought. To use small numbers only to describe ramification, write D 16 = D^28 t^2 and D 32 = D^216 u^4. This analysis gives the following two consequences of Theorem 1.
Corollary 1. There are exactly 25 number fields with Galois group PSL 2 (7)×C 2 = 16 T 714 and discriminant ≤ 3016. Base octics and ramification invariants t are given in Table 4. The smallest discriminant of such a field is 2116 and the field has defining polynomial
x^16 − 4 x^15 + 9x^14 − 14 x^13 + 14x^12 − 14 x^10 + 8x^9 + 45x^8 − 82 x^7
- 49x^6 + 63x^5 − 112 x^4 + 49x^3 + 99x^2 − 130 x + 100.
Corollary 2. There are exactly 14 number fields of degree 32 with Galois group 32 T 34620 and discriminant ≤ 3032. Base octics and ramification invariants u are given in Table 4. The smallest discriminant of such a field is 2132 and the field has defining polynomial
x^32 − x^31 + 2x^30 + x^29 + 8x^28 − 7 x^27 + 21x^26 − 9 x^25 − 12 x^24 + 248x^23 − 548 x^22 − 65 x^21 + 2653x^20 − 4879 x^19 + 2564x^18 + 4198x^17 − 7780 x^16
- 3593x^15 + 4020x^14 − 7014 x^13 + 4935x^12 − 2042 x^11 + 929x^10 − 787 x^9
- 695x^8 − 215 x^7 + 70x^6 − 42 x^5 + 15x^4 − 15 x^3 + 2x^2 + x + 1.
MIXED DEGREE NUMBER FIELD COMPUTATIONS 11
D 8 D 7 h = a ` c e t u v w
1 3878 3678 4 = 2 2 1 , 7 , 7 1, 7 2 2634534 2632534 4 = 2 2 1 1 2 , 2 , 3 , 3 3 216294 210294 1 4 4 2634594 2632594 4 = 2 2 1 1 5 36974 34974 2 = 2 1 6 2636116 2634116 6 = 2 3 1 3 1 7 3454116 3254116 4 = 2 2 1 1 8 216116 210116 1 4 3 2 9 361134 341134 2 = 2 1 3 10 283478 263278 2 = 2 1 4 11 2636434 2634434 2 = 2 1 3 12 265478 265278 2 = 2 1 13 342394 322394 4 = 2 2 1 1 14 263678 263478 2 = 2 1 15 2656234 2654234 8 = 4 2 1 16 2854116 2652116 2 = 2 1 17 2638294 2636294 4 = 2 2 1 1 18 2^8114174 26112174 2 = 2 1 19 21678 2878 2 = 2 1 20 21678 21078 1 21 21678 21078 2 = 2 1 22 78174 78172 2 = 2 1 23 282114 242114 4 = 2 2 1 1 24 2474614 2472614 4 = 2 2 1 1 25 263174 263172 6 = 2 3 1 1 Table 4. Discriminants of the octic PSL 2 (7) fields of Theorem 1 and their septic siblings, the class number h of each octic, and ramification invariants t, u, v, and w of abelian extensions. Field numbers are the same as in Table 3. The boldface conventions and the factorization h = a`ce are explained in the paragraph containing equation (3).
The first line of Table 4 indicates two degree 32 fields. The one highlighted in Corol- lary 2 has ramification invariants (t, u) = (1, 1) and the other one has ramification invariants (t, u) = (1, 7). To get a third Galois group, note that the sixth field K 8 has class number divisible by 3, yielding an extension K 24 /K 8. Let G 24 = Gal(K 24 ). The Galois group G 24 is in fact 24T 284, which is PSL 2 (7) itself, but now in its action on cosets of C 7. So PSL 2 (7) octics are in bijection with 24T 284 fields via an elementary resolvent construction. In this case, we relate discriminants via D 24 = D^38 v^4. Inspecting the twenty-five 24T 284 fields coming from the twenty-five octics says in particular that the above class field K 24 in fact has the minimal discriminant:
Corollary 3. There are exactly three number fields with Galois group 24 T 284 and discriminant ≤ 3024. Base octics and ramification invariants are given in Table 4. The smallest discriminant of such a field is (66^3 /^4 )^22 ≈ 23. 1622 and the field has
MIXED DEGREE NUMBER FIELD COMPUTATIONS 13
Table 5. Tame ramification data for A 5 ∼= PSL 2 (5) and S 5 ∼= PGL 2 (5) in degrees 5 and 6.
λ 5 λ 6 c 5 c 6 5 5 , 1 4 4 3 , 1 , 1 3 , 3 2 4 2 , 2 , 1 2 , 2 , 1 , 1 2 2 4 , 1 4 , 1 , 1 3 3 3 , 2 6 3 5 2 , 1 , 1 , 1 2 , 2 , 2 1 3
λn determines the corresponding discriminant exponent cn via formula (2). The behavior of wild ramification in this 5-to-6 context is analogous to the case of GL 3 (2) and PSL 2 (7) discussed earlier and so we omit the detailed analysis. From [JR14b], one knows that the bounds suggested by the tame table hold in general: D 5 ≤ D 6 ≤ D^25 for A 5 ∼= PSL 2 (5) and |D 5 | ≤ |D 6 | ≤ |D 5 |^3 for S 5 ∼= PGL 2 (5). As pointed out in [JR14b], the first bound |D 5 | ≤ |D 6 | implies that a complete table of quintic fields up through discriminant bound B determines the correspond- ing complete table of sextic fields up through B. In contrast to the situation for GL 3 (2) ∼= PSL 2 (7), this observation and existing tables of fields give non-empty complete lists of fields in the larger degree. In fact, taking B = 12, 000 ,000 from our extension [JR14a] of [SPDyD94], one gets 78 PSL 2 (5) sextics and 34 PGL 2 (5) sextics with root discriminant at most B^1 /^6 ≈ 15 .13. In Table 5 there are three instances when c 5 < c 6. Accordingly, we can use targeting to substantially reduce the quintic search space for the sextic fields sought. The result for root discriminant δ 6 ≤ 35 is as follows.
Theorem 2. Among sextic fields with absolute discriminant ≤ 356 , exactly 2361 have Galois group PSL 2 (5) and 3454 have Galois group PGL 2 (5).
Lists of fields can be retrieved by searching the websites [JR14a, LMF18]. In parallel with the figure for our first case, Figure 2 illustrates our second case.
The regularity near the bottom boundary δ 6 = δ 55 /^6 is easily explained, as follows. For any pair (K 5 , K 6 ), the ratio D 6 /D 5 is always a perfect square r^2. The pair
gives rise to a point on the curve δ 6 = r^1 /^3 δ 5 / 6 5.^ In the^ S^5 ∼= PGL^2 (5) case, the curves corresponding to r = 1, 2 ,... , 14 , 15 are all clearly visible. The first “missing curve,” clearly visible as a gap, corresponds to r = 16 = 2^4. This curve is missing because none of the 2-adic possibilities for (c 5 , c 6 ) satisfy c 6 − c 5 = 8. In the A 5 ∼= PSL 2 (5) case, there are fewer 2-adic possibilities and the first four visible gaps correspond to r = 4, 8, 12, and 16.
5.2. Connections with expected mass formulas. Let N Fn(G, x) denote the set of isomorphism classes of degree n number fields K with Gal(K) = G ⊆ Sn and root discriminant at most x. Here G is well-defined, as a subgroup of Sn, up to conjugation. From the quintic search in [JR14a], one knows
(5) |N F 5 (A 5 , 26)| = 539, |N F 5 (S 5 , 26)| = 726862.
14 JOHN W. JONES AND DAVID P. ROBERTS
0 26 70
0 δ (^5)
35
70
δ (^6)
0 26 70
0 δ (^5)
35
70
δ (^6)
Figure 2. Root discriminant pairs (δ 5 , δ 6 ) associated to A 5 ∼= PSL 2 (5) (left) and S 5 ∼= PGL 2 (5) (right), including all pairs in the window with δ 5 ≤ 26 from [JR14a, LMF18], and all pairs with δ 6 ≤ 35 from Theorem 2.
The ratio 539/ 726862 ≈ 0 .00074 is an instance of the familiar informal principle “Sn fields are common but An fields are rare.” In this light, the much larger ratio 2361 / 3454 ≈ 0 .68356 from Theorem 2 is surprising. However, the fact that N F 6 (PSL 2 (5), 35) and N F 6 (PGL 2 (5), 35) have such sim- ilar sizes can be explained as follows. For g ∈ Sn with cycle type n 1 , n 2 ,... , nk, let g =
∑k j=1(nj^ −^ 1). For a transitive permutation group^ G^ ⊆^ Sn^ and a conjugacy class C ⊆ G, define C to be g for any g ∈ C. Define aG to be the reciprocal of the minimum of the C over non-identity conjugacy classes C, and define bG to be the number of classes obtaining this minimum. Then Malle conjectured an asymptotic growth rate
|N Fn(G, x)| ∼ cGxnaG^ log(x)bG−^1 ,
for some constant cG [Mal04]. Note that we are presenting Malle’s conjecture in a renormalized form, since here x is a bound on root discriminant while in [Mal04] it is a bound on absolute discriminant. While Kl¨uners [Kl¨u05] has found a counterexample to the general statement of the conjecture, the problem is related to the presence of roots of unity in subfields, and Malle’s conjecture is still expected to hold in the cases considered here. For An, the two minimizing classes have cycle types 2^2 , 1 n−^4 and 3, 1 n−^3 , while for Sn the unique minimizing class is 2, 1 n−^2. Thus, consistent with numerical data like (5), one expects very different growth rates:
(6) |N Fn(An, x)| ∼ cAn xn/^2 log x, |N Fn(Sn, x)| ∼ cSn xn.
For n ≤ 5, this growth rate is proved for Sn with identified constants, and it is known that the growth for An is indeed slower; see [Bha10] for S 5 and [BCT15] for A 5.
16 JOHN W. JONES AND DAVID P. ROBERTS
for the runtime for confirming the first PSL 2 (7) octic field by computing septic fields without targeting is approximately (21^8 / 127 )^9 /^46. 5 / 12 ≈ 3 million CPU- years. To get Theorem 1’s complete list through discriminant 30^8 would then take (30^8 / 127 )^9 /^46. 5 / 12 ≈ 2 billion CPU-years, as opposed to the four CPU-months with mixed degree targeted searching.
6.2. The first 23 : GL 3 (2) field. As observed in [KM01], a sufficiently long com- plete list of septic GL 3 (2) fields can be used to determine the first octic 2^3 : GL 3 (2) field. The splitting field of a 2^3 : GL 3 (2) polynomial contains (up to isomorphism), two subfields K 7 a and K 7 b of degree 7, two subfields K 14 a and K 14 b of degree 14 and Galois group 14T 34, and two subfields K 8 a and K 8 b of degree 8 and Galois group 8T 48. One of the septic fields is contained in both K 14 a and K 14 b, and the other is contained in neither. Since the septic fields are arithmetically equivalent, they have the same discrim- inants, i.e., D 7 a = D 7 b. The other indices can be adjusted so that
(7) D 7 xD 8 x = D 14 x for x ∈ {a, b}.
This comes from a character relation on the relevant permutation characters. Be- cause of the asymmetry in the field inclusions described above, the fact that GL 3 (2) fields come in arithmetically equivalent pairs does not play a role in our computa- tions, and so we have forty-six septic ground fields to consider separately. Accord- ingly, we drop x from the notation, always taking the correct octic resolvent so that (7) holds. Of the forty-six septic GL 3 (2) fields with discriminant ≤ 127 , only one has a non-trivial narrow class group. This field has narrow class number two, and is K 7 = Q[x]/f (x) with
(8) f (x) = x^7 − x^6 − x^5 − 2 x^4 − 7 x^3 − x^2 + 3x + 1.
The unramified quadratic extension turns out to be simply K 14 = Q[x]/f (−x^2 ), which has Galois group 14T 34. So K 14 and K 7 both have root discriminant 57172 /^7 ≈ 11 .84. By (7), the sibling K 8 of K 14 has the even smaller root dis- criminant 5717^1 /^4 ≈ 8 .70. In general, given all septic GL 3 (2) fields up to some discriminant bound B, one can get all 14T 34 fields up to the discriminant bound B^2 via quadratic extensions. If d is the relative discriminant of K 14 /K 7 , then D 14 = D 72 NK 7 /Q(d), and so NK 7 /Q(d) must be at most B^2 /D^27. In the same way, the stronger bound NK 7 /Q(d) ≤ B/D 7 is necessary and sufficient for the resolvent 8T 48 field to have discriminant ≤ B. Taking B = 12^7 now, the quotient B/D 7 decreases from 12^7 /(13^21092 ) ≈ 17. 85 for the first two ground fields K 7 to 12^7 /(2^67432 ) ≈ 1 .01 for the last two ground fields. For the first twenty-six ground fields, computation shows that there are no 14T 34 overfields satisfying NK 7 /Q(d) ≤ B^2 /D^27. For the last twenty fields, already B/D 7 <
2 and so the lack of overfields, except for the unramified one above, follows from the narrow class numbers being 1. Hence we have the following corollary of Theorem 3.
Corollary 5. The field Q[x]/f (−x^2 ) from (8) is the only degree fourteen field with Galois group 14 T 34 and discriminant ≤ 1214. Its sibling, with defining polynomial
x^8 − 4 x^7 + 8x^6 − 9 x^5 + 7x^4 − 4 x^3 + 2x^2 + 1,
MIXED DEGREE NUMBER FIELD COMPUTATIONS 17
is the only the octic field with Galois group 23 : GL 3 (2) = 8T 48 and discriminant ≤ 127. These two fields have discriminants 57174 ≈ 11. 8414 and 57172 ≈ 8. 708 respectively.
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