SEPTIC FIELDS WITH DISCRIMINANT ±2^{a}3^{b}

JOHN W. JONES AND DAVID P. ROBERTS

Abstract. We classify septic number fields which are unramified outside of {∞, 2, 3} by a targeted Hunter search; there are exactly 10 such fields, all with Galois group S_{7}. We also describe separate computations which strongly suggest that none of these fields come from specializing septic genus zero three point covers.

1. Introduction

Let S ⊂ {∞, 2, 3, 5, … } be a finite set of primes and let n be a positive integer. The set of isomorphism classes of degree n extensions of Q which are unramified outside of S has long been known to be finite by a classical theorem of Hermite, yet little else is known about these sets in general. We are particularly interested in the case where S is a small set of small primes. Number fields in these sets are extremely rare in lists of number fields with absolute discriminant less than a given bound. On the other hand, they appear in a variety of places in mathematics. For discussions of why fields ramified at only a few small primes are interesting, see [JR1], [Har].

Here we consider the case S = {∞, 2, 3}, i.e. fields with discriminant ±2^{a}3^{b}. In [JR1], we presented the previously known cases of degrees n ≤ 5 and worked out the computationally much harder case n = 6. In this paper, we settle the computationally yet harder case n = 7, thereby completing the following table, giving the number of fields unramified away from ∞, 2 and 3 for small n:

n 1 2 3 4 5  6 7

Total number of fields 1 7 9 62 6 398 10 Number of nonsolvable fields 5 62 10.

It is fair to say that nonsolvable fields just barely exist for S = {∞, 2, 3} in these degrees. Certainly for S = {∞, 2}, {∞, 3}, or {2, 3} in degrees ≤ 7, all fields are solvable; indeed the nonsolvable fields in our case S = {∞, 2, 3} are all wildly ramified at 2 and 3. On the other hand, for S = {∞, 2, 3, 5} there are 1, 415 nonsolvable quintic fields [Jon], and surely many nonsolvable sextic and septic fields too.

Section 2 discusses how our search technique is an improvement on [JR1]. Section 3 presents a defining polynomial for each of the ten fields. Also it discusses quantitatively how the ten fields were found early in the search and analyzes the wild ramification at 2 and 3. It is natural to ask if our septic fields have conceptual sources, like many of the fields in [JR1] do. Section 4 reports on our attempts to produce some of the ten fields by specializing 3-point covers of P^{1}. We are able to produce one field by specializing a nonic cover and present evidence that none of the fields can be produced by specializing septic covers.

2. The computer search

The computer search was based on Hunter’s theorem, a method used extensively for determining all number fields of a given degree with absolute discriminant less than a prescribed bound. See [Coh, §9.3] for a full discussion of this approach.

For septic fields, a Hunter search with bound B enumerates a long but finite list of polynomials guaranteed to contain the characteristic polynomial of at least one primitive element for each septic field with absolute discriminant ≤ B. The number of polynomials examined grows roughly in proportion to B^{9/4} for septic fields.

For this paper, we need to consider absolute discriminants up to 2^{14}3^{11}. A Hunter search with this value for B would be prohibitively long. On the other hand, we are only interested in a few isolated discriminant values, and for each candidate discriminant value, the defining polynomials will satisfy p-adic conditions for p = 2 and 3. We refer to a Hunter search which incorporates these p-adic conditions as targeted. The primary difference in methodology for this search as compared with [JR1] is how we implemented the targeting.

Here, as in [JR1], we divide the search into smaller searches, each indexed by a 2-adic ramification structure and a 3-adic ramification structure. A p-adic ramification structure (see [JR1, §2]) contains the information of ramification exponents for the factorization of the ideal (p) in the ring of integers O_{K} of a number field K, along with corresponding contributions to the discriminant.

In each smaller search of [JR1], we searched for polynomials f(x) = x^{n}+a_{1}x^{n−1}+ ⋯+a_{n} ∈ Z[x] which could arise as characteristic polynomials f_{η} of elements η ∈ I, where I is a particular ideal of O_{K}. The field K is unknown, but satisfies the prescribed ramification properties which translate to congruences on the coefficients a_{i} of f. The attraction of working in I rather than O_{K} is that one has strong congruences having the very simple form p^{e} | a_{i}. The method of [JR1] uses only linear congruences on individual coefficients of this sort. It therefore oversearches substantially, as is explained via an example in [JR1, §2.5].

The improvement suggested in [JR1, §2.5] is to use all congruences on the vector (a_{1}, …, a_{n}) of coefficients, not just linear congruences on individual coefficients a_{i}. Here we indeed use all such congruences. Having done this, there is no longer any point in restricting η to lie in the search ideal I. Rather we let η be anywhere in O_{K}. Note that the timing analysis in [JR2] of each of these smaller searches applies to our new method.

To illustrate how ramification properties of the fields sought translate to congruences on coefficients, consider looking for fields which 2-adically are of the form K_{t} × K_{w}, where K_{t} is a cubic field of discriminant 2^{2} while K_{w} is a quartic field of discriminant 2^{10}. These invariants force the two fields to be totally ramified, which simplifies our discussion. The cubic field K_{t} is necessarily tamely ramified while the quartic field K_{w} must be wildly ramified, which explains our notation. Let O_{t} ⊃ I_{t} and O_{w} ⊃ I_{w} be the corresponding local rings of integers and maximal ideals.

Consider first K_{t}. An element of I_{t} has characteristic polynomial congruent to x^{3} modulo 2. Every element of O_{t} is either in I_{t}, or is a translate by 1 from an element of the I_{t}, hence every element of O_{t} has characteristic polynomial congruent modulo 2 to (x + a)^{3} where a ∈ {0, 1}.

Consider next K_{w}. Elements of I_{w} have characteristic polynomials of the form

x^{4} + 4Ax^{3} + 4Bx^{2} + 8Cx + 2D,

where A, B, C, D ∈ Z_{2}. The numerical coefficients 4, 4, and 8 correspond to what we called Ore congruences in [JR1]; they come from the fact that totally ramified quartics can have discriminant as low as 2^{4} while here our quartic discriminant is 2^{10}. To get all elements of O_{w}, we again have to allow for a shift by 1. We get polynomials of the form

(x + E)^{4} + 4A(x + E)^{3} + 4B(x + E)^{2} + 8C(x + E) + 2D, (2.1)

where A, B, C, D ∈ Z_{2} and E ∈ {0, 1}. Looking modulo 8, we have

(x + E)^{4} + 4A(x + E)^{3} + 4B(x + E)^{2} + 2D,

where A, B, E ∈ {0, 1} and D ∈ {0, 1, 2, 3}.

So, septic polynomials for fields with the given ramification structure at 2 are congruent modulo 8 to a polynomial of the form

((x + a)^{3} + 2bx^{2} + 2cx + 2d) ((x + E)^{4} + 4A(x + E)^{3} + 4B(x + E)^{2} + 2D) ,

where a, A, B, E ∈ {0, 1} and b, c, d, D ∈ {0, 1, 2, 3}. Enumerating all such polynomials modulo 8 we find 2560 possibilities for the reduction of f modulo 8. However, the coefficient a_{1} of x^{6} is constrained to be simply 0, 1, 2, or 3 for a Hunter search. This cuts the list down to 1280 possibilities.

This number can be reduced a bit further by handling the wildly ramified extension more carefully. If D is odd and A is even in equation (2.1), then the polynomial necessarily defines a quartic extension of Q_{2} with discriminant 2^{11}, not 2^{10}. Eliminating those cases and carrying through as above brings the list to 1024 possibilities, 256 for each allowed value of a_{1}. This data is stored in one of our many 2-adic input files. Similarly, we created a 3-adic input file for each possible 3-adic ramification structure.

When conducting the actual search for a given combination of ramification structures at 2 and 3, the computer program starts by reading the appropriate file of 2-adic bounds and the appropriate file of 3-adic bounds. We then conduct Hunter searches using the method of [Poh] to deduce the archimedean bounds on coefficients. Note, there is still a remaining issue of how to integrate the p-adic bounds with the archimedean ones.

Looking over all ramification structures, we have a total of 22, 782 congruence vectors modulo powers of 2 and 101, 699 congruence vectors modulo powers of 3. If we were to look directly at each possible combination and conduct a Hunter search for f congruent to the resultant f̄ modulo 2^{j}3^{k}, we would have 2, 316, 906, 618 such searches. The run time would then be dominated by setting up many searches only to find that strong congruences on the coefficients of f were incompatible with the archimedean inequalities. In other words, we would have spent most of the computer time initializing searches which would end up looking at no polynomials. To avoid this, we adopted the following strategy.

A Hunter search program is built around nested loops, one for each a_{i}. For the outermost 3 loops, we simply loop through all values for (a_{1}, a_{2}, a_{7}) of a Hunter search for the targeted discriminant, skipping only those triples which are incompatible with all of our congruence vectors for 2 or for 3. Then, we find the congruence vectors modulo 2 and 3 which are consistent with the current values of a_{1}, a_{2}, and a_{7}. This takes very little time since the input files of p-adic bounds have been appropriately presorted. Then, we step through the relatively few consistent pairs of congruence vectors, running for each pair a search for the remaining coefficients a_{3}, a_{4}, a_{5}, a_{6}.

The main program was written in C making use of the Pari library. The searches were conducted on a Sun workstation, inspected approximately 700 million polynomials, and had a run time of approximately 13 hours. For comparison, we had our new programs rerun the sextic searches of [JR1]. While the original sextic searches took approximately several months, the rerun searches took approximately 2 hours.

3. The ten fields

The ten fields are Q[x]/f_{i}(x) with defining polynomials f_{i}(x) as in Table 3.1. We have chosen the defining polynomials to minimize the quantity r appearing in

Table 3.1. Defining polynomials

f_{1}(x) = x^{7} + x^{6} + 3x^{5} + 5x^{4} + 2x^{3} + 12x^{2} + x + 7
f_{2}(x) = x^{7} − x^{6} − x^{4} − 2x^{3} + 4x + 2
f_{3}(x) = x^{7} − 2x^{6} + 6x^{5} − 10x^{4} + 14x^{3} − 12x^{2} + 4x + 4
f_{4}(x) = x^{7} + 2x^{6} + 27x^{3} + 54x^{2} + 42x + 12
f_{5}(x) = x^{7} − 3x^{6} + 9x^{5} − 3x^{4} + 12x − 36
f_{6}(x) = x^{7} − 3x^{5} + x^{4} + 3x^{3} − 6x^{2} − x + 9
f_{7}(x) = x^{7} − 2x^{6} + 3x^{5} + x^{4} − 5x^{3} + 12x^{2} − 7x + 5
f_{8}(x) = x^{7} − 3x^{6} + 3x^{5} + 3x^{4} + 3x^{3} − 9x^{2} − 11x − 3
f_{9}(x) = x^{7} − 3x^{6} + 3x^{5} + x^{4} − 3x^{3} − 3x^{2} − 5x − 3
f_{10}(x) = x^{7} + 2x^{6} + 3x^{5} + 4x^{4} + 5x^{3} + 6x^{2} + 7x + 8

equation (3.1) below.

The Galois group corresponding to each of the ten fields is the full symmetric group S_{7}. Another common feature shared by the ten fields is that they all have the maximal possible number of complex places, namely three. No complex places is the meaning of unramified at ∞, so in this sense all ten fields are maximally ramified at ∞. Table 3.2 presents more numeric invariants of the fields.

The column D of Table 3.2 gives the discriminant of each field. Because the fields all have an odd number of complex places, all the discriminants are negative. Somewhat surprisingly, the maximum locally allowed exponent at 2, namely 14, does not arise.

All 10 fields were found near the very beginning of the search. Also the coefficients of our defining polynomials are quite small. The next block of three columns gives numerical quantities related to these two observations as follows. Given a degree n polynomial, f(x) = x^{n} + a_{1}x^{n−1} + ⋯ ∈ Z[x], associate to it the traceless polynomial g(x) = f(x − a_{1}/n) = x^{n} + 0x^{n−1} + ⋯ ∈ Q[x]. Let α_{1}, …, α_{n} ∈ C be the roots of f and let β_{1}, …, β_{n} ∈ C be the roots of g. Define the root-length \sqrt{T_{2}} and the reduced root-length r of f by

\sqrt{T_{2}} = \sqrt{∑_{j=1}^{n} |α_{j}|^{2}} and r = \sqrt{∑_{j=1}^{n} |β_{j}|^{2}}. (3.1)

Table 3.2. Some invariants of the 10 fields

root-length  p-adic  p-adic

information factors

slope content

#  D  r  R Earliness 2  3  2  3  GRD
1 −2^{9} 3^{11} 3.70 5.05  4, 580 3_{9}^{2}  1_{11}^{6} 3220  \frac{5}{2}210 51.41
2 −2^{9} 3^{11} 3.16 5.05 321, 000 1_{9}^{4} 3_{0}^{1} 1_{11}^{6} \frac{7}{2}320 \frac{5}{2}210 72.71
3 −2^{13}3^{9} 3.71 5.30 15, 100 1_{11}^{4}1_{2}^{3} 1_{9}^{6} 431 0 2\frac{3}{2}1  50.09
4 −2^{13}3^{11} 4.58 6.37  7, 420 1_{10}^{4}1_{3}^{2} 1_{11}^{6} \frac{7}{2}320 \frac{5}{2}210 72.71
5 −2^{12}3^{11} 4.89 6.01  318 1_{10}^{4}1_{2}^{2} 1_{11}^{6} \frac{7}{2}320 \frac{5}{2}21  72.71
6 −2^{11}3^{8} 3.63 4.31  105 1_{11}^{4}3_{0}^{1} 2_{8}^{3} 4300 2210  50.41
7 −2^{10}3^{10} 3.49 4.88  9, 010 1_{10}^{4}3_{0}^{1} 2_{10}^{3} \frac{7}{2}320 \frac{5}{2} \frac{3}{2}10 64.35
8 −2^{10}3^{10} 3.99 4.88 251 1_{10}^{6} 2_{10}^{3} \frac{8}{3} \frac{8}{3}21 0 \frac{5}{2} \frac{3}{2}10 46.84
9 −2^{12}3^{10} 3.48 5.48 206, 000 1_{10}^{4}1_{2}^{3} 1_{10}^{6} \frac{7}{2}321 0 \frac{9}{4} \frac{9}{4}110 68.29
10 −2^{12}3^{10} 3.56 5.48 115, 000 1_{9}^{4} 1_{3}^{2} 1_{10}^{6} \frac{7}{2}32  \frac{9}{4} \frac{9}{4}110 66.35

These two quantities are related by

T_{2} = \frac{a_{1}^{2}}{n} + r^{2}.

In Hunter searches one normally restricts attention to polynomials with 0 ≤ a_{1} ≤ n/2. Thus, the difference between the quantities \sqrt{T_{2}} and r is rather small. It is standard in the literature to work with T_{2}. We will work instead with r, as it figures directly into (3.2) and (3.3) below.

Hunter’s theorem, which is the theoretical foundation for archimedean bounds we use, asserts that any septic field with discriminant D has a defining polynomial f(x) ∈ Z[x] satisfying

r ≤ R := (\frac{64|D|}{7})^{1/12} . (3.2)

Let V_{r} be the volume in traceless coefficient space of the image of the ball in traceless root space of radius r, notions being as in [JR2]. Then V_{r} grows as a constant times r^{27}. The exponent 27 comes from multiplying the 2n − 2 appearing in the first displayed equation of [JR2, Section 4] by the (n+2)/4 in [JR2, Prop 5.1] and specializing to the case n = 7; see also [Coh, p. 448] for a similar statement in terms of T_{2}. The table prints r and R to two decimal places and

(3.3) Earliness := (R/r)^{27}

to three significant figures. Note that if we had we decreased our cutoffs so as to decrease search volumes by a factor of say 100, we would have still found all 10 fields, as 100 < 105. The discussion of this paragraph emphasizes that the main mathematical result of this paper is not that we have found 10 septic fields with discriminant ±2^{a}3^{b}, but rather that we have proved there are no more. Also our discussion suggests that in other situations where complete searches would be prohibitively long, very much shorter searches have a good chance of finding most of the fields sought.

For exactly two of the ten fields, the polynomial in Table 3.1 which minimizes r^{2} does not minimize T_{2}. In these cases, polynomials minimizing T_{2} are

f_{5}^{∗}(x) = x^{7} − x^{6} + 18x^{3} − 18x^{2} − 24x − 48
f_{8}^{∗}(x) = x^{7} − 3x^{5} + 2x^{4} + 3x^{3} − 12x^{2} + 19x − 6.

Numerical quantities compare as follows:

T_{2} |a_{1}| r^{2} Earliness
f_{5}^{∗} 24.10 1 23.96  253
f_{5} 24.84 3 23.56  318
f_{8}^{∗} 15.89 0 15.89  241
f_{8} 17.13 3 15.84  251.

Thus the true earliness of Fields 5 and 8 is not captured by what would be traditional defining polynomials, f_{5}^{∗} and f_{8}^{∗}.

Returning to our description of Table 3.2, we consider next the two columns under “p-adic factors,” which give some basic 2-adic and 3-adic information about each field K. The p-adic algebra K ⊗ Q_{p} factors as a product of p-adic fields. Each of the factor fields has an inertial degree f, a ramification degree e, and a discriminantal exponent c. We represent such a factor field by f_{c}^{e}. For example, a 1_{0}^{1} would represent a factor isomorphic to Q_{p} itself, but we omit printing these degree one factors. Note that, rather remarkably, all ten fields factor 3-adically as a sextic field times Q_{3}.

The next two columns give the slope content of the local algebras K_{p}, following the conventions of [JR1]. For example, the decomposition group of Field 8 at 2 has the form D_{2} = S_{4} ×S_{2} while the decomposition group D_{3} has the form C_{3}^{2}.C_{4}. The slope-filtrations of these groups are

\begin{matrix}\{e\} = I_{∞} & ⊂^{2^{2}} & I_{8/3} & ⊂^{2} & I_{2} & ⊂^{3} & I_{1} & ⊂^{2} & I_{0} = D_{2} \\ \{e\} = I_{∞} & ⊂^{3^{2}} & I_{9/4} & & & ⊂^{2} & I_{1} & ⊂^{2} & I_{0} = D_{3}. \end{matrix} (3.4)

A minimal subquotient I_{s}/I_{t} is associated with the slope s, and its order is given above the corresponding inclusion in (3.4). For all ten fields, the orders |D_{2}| and |D_{3}| are a power of 2 times a power of 3. If the slope s contributes 2^{A}3^{B} to |D_{2}| we print it A times in roman and B times in italics. If s contributes 2^{A}3^{B} to |D_{3}| then we print it A times in italics and B times in roman. So s > 1, which corresponds to wild ramification, can be printed only in roman; s = 1, which corresponds to tame ramification, can be printed only in italics; s = 0, corresponding to unramified extensions of Q_{p}, can be printed in either.

Finally, the entry under GRD is the Galois root discriminant, i.e. Δ^{1/7!} where Δ is the discriminant of a Galois closure of the given septic field, the 7! arising as the order of the Galois group S_{7}. The GRD is a numerical invariant of ramification which lets one meaningfully compare fields with different Galois groups. It is of the form 2^{α}3^{β} with α and β weighted averages of the slopes just examined. Slope s appears with weight (|I_{s}| − |I_{t}|)/(|I_{s}||I_{t}|), with t being the next greater appearing slope. In the case of Field 8,

α = \frac{3}{4} · \frac{8}{3} + \frac{1}{8} · 2 + \frac{1}{24} · 1 = 2\frac{7}{24}
β = \frac{8}{9} · \frac{9}{4} + \frac{1}{18} · 1 = 2\frac{1}{18}.

So for Field 8, the GRD is 2^{α}3^{β} ≈ 46.84. For a more detailed discussion of slope content and Galois root discriminants, see [JR1, Section 1.2]. Some more p-adic information on our ten fields is available in the format of [JR1] at [Jon].

Our main result can be compared with the main result of [Bru], which says that there are no nonsolvable septic fields with discriminant of the form ±7^{a}. The maximal locally-allowed GRD in our situation is 2^{37/12}3^{13/6} ≈ 91.61 while the maximal locally-allowed GRD in the situation of [Bru] is 7^{83/42} ≈ 46.78. In the light of the fields found by our search and their GRD’s, the fact that [Bru] found no nonsolvable fields is not too surprising. In fact, at present the lowest GRD we know of for an S_{7} septic field is the 46.84 from Field 8.

4. Comparison with three-point covers

We investigated the question of whether any of the 10 septic fields with discriminant ±2^{a}3^{b} can be obtained from a family of fields arising from a three-point cover of P^{1}. Interestingly, the only field which we able to produce in this way came from a degeneration of a degree 9 cover. After discussing this example, we consider degree 7 covers and explain in some detail why the most promising cover did not produce any of our 10 fields.

As a technical note, all the covers considered here have genus zero, and in fact correspond to maps from a projective line P_{x}^{1} with coordinate x to a projective line P_{t}^{1} with coordinate t, with ramification only above t = 0, 1, ∞, everything being defined over Q. We restricted to genus zero because one cannot yet systematically compute covers in higher genus. However moduli algebras as in [Mal2] tend to have large degrees in higher genus cases, and no higher genus cover looks particularly promising as a source for septic fields with discriminant ±2^{a}3^{b}.

Consider the two-variable polynomial

f_{t}(x) = x^{9} − 9tx + 8t. (4.1)

Its discriminant with respect to x is

D_{t} = −2^{24}3^{18}t^{8}(t − 1). (4.2)

Since D_{1} = 0, the polynomial f_{1}(x) has a repeated root; in fact, its factorization into irreducibles is

f_{1}(x) = (x − 1)^{2}(x^{7} + 2x^{6} + 3x^{5} + 4x^{4} + 5x^{3} + 6x^{2} + 7x + 8).

The septic factor has polynomial discriminant −2^{16}3^{12} and the septic field that it defines is Field 10 from the previous section. One couldn’t ask for more simplybehaved coefficients, and this septic polynomial is an example of an abc-polynomial, in the sense of [Bor].

The polynomial f_{t}(x) in (4.1) can be regarded as a family of separable nonic polynomials in Q[x], indexed by t ∈ Q − {0, 1}. One sees from (4.2) that for some of these t, the discriminant of f_{t}(x) has the form ±2^{a}3^{b}. For other t, the polynomial discriminant will have an extra square factor but the algebra discriminant of Q[x]/f_{t}(x) will be of the form ±2^{a}3^{b}. We report on this example in [Rob]: from 21 specialization points of the first type and 14 specialization points of the second type, one gets 25 S_{9} fields and 10 A_{9} fields with discriminant ±2^{a}3^{b}, all distinct.

Similarly, many of the fields of degree ≤ 6 in [JR1] come from specializing families degenerating only at t = 0, 1, and ∞. For example, x^{4} − 4tx + 3t has discriminant −2^{8}3^{3}t^{3}(t−1). The 22 S_{4} fields listed in [JR1], except the totally real one, come from specializing this family. For degree six examples with more complicated defining equations, see [Rob].

We expected to similarly find some of the 10 fields by septic covers. Using the computations in [Mal2] to eliminate many cases, we found that there are ten septic genus zero covers such that any septic genus zero cover which could possibly lead to a field of the sort we seek is a base change [Mal1] of one of these ten covers. We will describe the details of our investigation for only one of these covers, the one that came the closest to producing a septic field of discriminant ±2^{a}3^{b}.

This cover is defined by the partition data

λ_{0} = 331
(4.3)  λ_{1} = 2221
λ_{∞} = 43.

To find a corresponding polynomial, one looks for relatively prime polynomials g_{0}(x), g_{1}(x), g_{∞}(x) satisfying

g_{0}(x) + g_{1}(x) + g_{∞}(x) = 0,

with the multiplicities of the roots of g_{c} being given by λ_{c}. The unique solution, up to a fractional linear change in the x-variable and a constant scale factor, is

g_{0}(x) = (256x^{2} − 448x + 189)^{3}x
g_{1}(x) = −(4096x^{3} − 8704x^{2} + 5400x − 729)^{2}(x − 1)
g_{∞}(x) = 27(28x − 27)^{3}.

The cover F : P_{x}^{1} → P_{t}^{1} is given by F (x) = −g_{0}(x)/g_{∞}(x). Our x-variable is normalized by the requirement that F (c) = c for c = 0, 1, ∞. The corresponding family of septic polynomials is

f_{t}(x) = g_{0}(x) + tg_{∞}(x) (4.4)

which has discriminant

D_{t} = −2^{118}3^{60}7^{7}t^{4}(t − 1)^{3}. (4.5)

The problem is that one has to choose t ∈ Q − {0, 1} such that 7 is not a factor of the discriminant of Q[x]/f_{t}(x) and no extraneous primes are introduced. This problem with 7 is present in all nine other covers as well. For seven of the other nine covers, one has to get rid of 5 too.

Table 4.1 summarizes our p-adic analysis of the family (4.4). Consider first t ∈ Q_{2} − {0, 1}, defining a 2-adic algebra Q_{2}[x]/f_{t}(x). The specialization point t is 2-adically closest to one of three cusps c = 0, 1, ∞. Suppose it is m-close to 0, meaning it has the form 2^{m}u with u a unit and m ≥ 1. Then the first data line of Table 4.1 gives d with disc(Q_{2}[x]/f_{t}(x)) = 2^{d}. The parentheses indicate repetition, e.g. m = 9, 10, 11 gives d = 8, 8, 6, etc. Suppose next that t is m-close to ∞, meaning t = u2^{−m}, with m ≥ 1. If m = 1, the table says disc(Q_{2}[x]/f_{t}(x)) = 2^{13}. If m > 1, then Q_{2}[x]/f_{t}(x) splits naturally into a cubic and a quartic. This splitting is reflected in the table by the use of two lines for ∞. The presence of several Table 4.1. p-adic variation of the discriminantal exponents of Q_{p}[x]/f_{t}(x) with f_{t}(x) = (256x^{2} − 448x + 189)^{3}x + 27t(28x − 27)^{3}

gen  c  1  2  3  4 5 6 7 8 9 10
p = 2  0 12  6  10  8 4 (8 8 6)
1 13  (8 9, 11)
∞ 13 (0 2 2)
(8, 4, 2 11 6, 4 11)
p = 3  0 (10  10  6, 4)
1 7  6  (3 4)
6  ∞ (3, 1  5  5)
(0  3  2  3)
p = 7  0 11  9  7  (5 3 5)
1 10  7  (4 5)
5 , 3  ∞ 12  11  10  9 8 7 (0 2 2)
(0 3 2 3)
p ≠ 2, 3, 7 0 (4 4 0)
1 (3  0)
0  ∞ (2  2  0)
(3  2  3  0)

numbers in a slot on Table 4.1 indicates an ambiguity. For example, if m ≡ 2 (4) then the quartic algebra has discriminant 2^{d} with d = 8, 4 or 2. Finally, t may be a 2-adic unit, in which case t = 1 + 2^{m}u for some m ≥ 1. For the remaining primes, conventions are similar; however here t ∈ Z_{p} may reduce to an element of F_{p}−{0, 1}, in which case we say that t is generic. Ramification at primes p ≠ 2, 3, 7 is always tame and the last block on Table 4.1 follows from general considerations of tame ramification. The blocks for primes p = 2, 3, and 7 reflect wild ramification and needed to be computed directly from (4.4). We should add that we have not written out a proof of the correctness of these blocks. However we are convinced that the information given is correct because of a number of theoretical consistencies and because the data agrees with numerical computation of discriminants for a great many t ∈ Q − {0, 1}.

Table 4.1 says that 7 does not divide the discriminant of Q[x]/f_{t}(x) iff t is mclose to ∞ for m ≡ 7 (12). We actually do not need the 2- and 3-blocks of Table 4.1 at the moment, because we are not trying to get rid of 2 or 3. Note, however, that they say that 2 and 3 cannot be eliminated. Similar analysis of the other nine covers says that one cannot eliminate 7 in one of them and cannot eliminate 5 in two others. So, besides the cover we are treating in detail, there are now only six more to consider.

Next, there is a global component to our analysis. Let t ≠ 0, 1 be a rational number and write it in lowest terms as −au^{3}/cw^{12} with a, u, c, and w being integers with a cube-free and c twelfth-power-free. Define integers b and v by

au^{3} + bv^{2} + cw^{12} = 0 (4.6) and the requirement that b be square-free. Impose the normalization condition that u, v, and w are all positive and exactly one of a, b, and c is negative. Then the integers a, b, c, u, v, and w are all uniquely determined by t. The previous local analysis using the last block of Table 4.1 says that Q[x]/f_{t}(x) is ramified within {2, 3, 7} iff

(4.7) all primes dividing abc are in {2, 3, 7}.

The local analysis using the 7-block says that 7 does not divide disc(Q[x]/f_{t}(x)) iff

(4.8) 7^{7} exactly divides c in (4.6).

We carried out a computer search which found four t-values corresponding to the following solutions of (4.6) satisfying (4.7) and (4.8):

\begin{aligned}3 & 139^{3} & − & 2^{4} & 745^{2} & + & 7^{7} & = & 0 \\ & 197^{3} & − & 2 & 2249^{2} & + & 3 7^{7} & = & 0 \\ 2 & 113^{3} & − & & 3209^{2} & + & 3^{2}7^{7} & = & 0 \\ & 2351^{3} & + & & 46745^{2} & − & 2^{11}3^{2}7^{7} & = & 0. \end{aligned} (4.9)

The six other covers have their analogs of (4.6)–(4.8). However the form of these analogs make global solutions seem much less likely than solutions to (4.6)–(4.8). For while 7 behaves similarly, in four cases one has to kill 5 also. Also the smaller two exponents in the analogs of (4.6) are always larger than the 2 and 3 in (4.6). The importance of having small exponents is clear from the solutions (4.9) found for our cover. At any rate, computer searches did not find any global solutions for these other six systems.

We know that for the t-values corresponding to (4.9), the algebra Q[x]/f_{t}(x) is ramified exactly at 2 and 3. However, unfortunately from the point of view of constructing septic fields, all four f_{t}(x) are reducible over Q, factoring into a linear factor and a sextic factor! Our experience with specializing three point covers tells us these factorizations constitute extremely unusual behavior. Much more typical is the nonic case (4.1), where the 35 specialization points behave generically, as we discussed at the beginning of this section.

Note that the analysis just completed did not make use of our list of ten fields. Nor did it make use of of the 2- and 3-blocks on Table 4.1. But now let’s compare Table 3.2 with Table 4.1. We will consider only the 3-adic information, because the situation is remarkably simple here. Note that Fields 1–5 on Table 3.2 have discriminant of the form −2^{a}3^{b} with b either 9 or 11. But, by Table 4.1, specializations of our cover can have 3-adic discriminantal exponent only 1, 3, 4, 5, 6, 7, 8, or 10. More subtlety, Field 6 has b = 8, but 3-adically factors into a sextic field and Q_{3}. But by Table 4.1, specializations of our cover with 3-adic discriminantal exponent 8 factor into a wild cubic field and a tame quartic field. Finally, Fields 7–10 all have 3-adic discriminantal exponent b = 10 and factor into a sextic field and Q_{3}. But one can check by a 3-adic Newton polygon calculation that any specialization of our cover with b = 10 factors 3-adically as a cubic field times a cubic field times Q_{3}. In short, no specialization of our cover matches one of our 10 fields even 3-adically. So, while we were unable to produce any of the 10 fields by septic 3-point covers, the local analysis of the cover examined above combined with our classification of septic fields unramified outside {∞, 2, 3} explains why this cover unexpectedly split at our four specialization points.

References

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http://math.la.asu.edu/~jj/numberfields

[JR1] Jones, J. and Roberts, D., Sextic number fields with discriminant (−1)^{j}2^{a}3^{b}, in Number Theory: Fifth Conference of the Canadian Number Theory Association, CRM Proceedings and Lecture Notes, 19, American Math. Soc., 141–172 (1999).

[JR2] , Timing analysis of targeted Hunter searches, in ANTS III, Springer Lecture Notes in Computer Science, 1423, 412–423 (1998)

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[Mal2] , Fields of definition of some three point ramified field extensions, The Grothendieck theory of dessins d’enfants, L. Schneps, ed., London Math. Soc. Lecture Note Ser. 200, Cambridge Univ. Press, 1994.

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[Rob] Roberts, D., Number fields with discriminant ±2^{a}3^{b}: examples from three point covers,

preprint, http://cda.morris.umn.edu/~roberts

Department of Mathematics, Arizona State University, Box 871804, Tempe, AZ 85287 E-mail address: jj@asu.edu

Division of Science and Mathematics, University of Minnesota-Morris, Morris, MN 56267

E-mail address: roberts@morris.umn.edu