Research Frontier 2013

The Ultimate Revival of Ayadi's Thesis

on 3-Class Field Towers of Type (3,3)

over Cyclic Cubic Number Fields



Karl-Franzens University Graz, left side

Statistics and Minima:

  • Inspired by Amandine Leriche and Henri Cohen,
    we use a very convenient parametrization of
    generating polynomials for cyclic cubic fields ,
    P(X) = X3 - 3*e*X - e*u,
    and we apply our New Algorithm for determining the
    Second p-Class Group Gp2(K)
    of a number field K
    by computing the Kernels and Targets
    of its Transfers to maximal subgroups.

    We set p = 3 and obtain the complete
    statistical evaluation of the smallest
    cyclic cubic number fields K
    whose conductors f are divisible by
    two or three primes (2 ≤ t ≤ 3) and
    whose 3-class groups are of type (3,3).
    The distribution visualizes the population
    of vertices on the coclass graphs G(3,1) and G(3,2)
    of 3-groups of coclass either 1 or 2
    by the groups G32(K).


  • For t = 3, we use graphs for visualizing
    cubic residue characters between the
    three prime divisors of the conductor f,
    which were introduced by George Gras on p. 21 of
    l-Classes of Ideals (Chapter VI, Section 3) ,
    and the categories defined on p. 45 of Ayadi's Thesis
    Cyclic Cubic Fields (Chapter 4, Section 4.2) .




Karl-Franzens University Graz, centre with 8 figures

  • The Construction Process:

    *************************************************************************

    First Step: with the aid of PARI/GP we compute a list of
    the first 250 conductors of cyclic cubic number fields K,
    having a 3-class group Cl3(K) of type (3,3).
    These conductors occur in the range 657 ≤ f ≤ 26467.
    --------------------------------------------------------------

    Second Step: by means of MAGMA we iterate through the list
    of generating polynomials computed in the first step and determine
    1. the 3-principalization kernel of K in its four
      unramified cyclic cubic extensions N1,N2,N3,N4
      (the transfer kernel type TKT),
    2. the structure of the 3-class groups of N1,N2,N3,N4
      (the transfer target type TTT).
    --------------------------------------------------------------

    Third Step: we use our unambiguous criteria
    to identify the second 3-class group G32(K) of K
    by the combination of TKT and TTT in the following table.
    --------------------------------------------------------------

    Remark for non-specialists: For 3-groups,
    we use identifiers of the SmallGroups Library
    provided by GAP and MAGMA .
    They are of the shape < order, counter >.
    <81,7> is 3-Sylow subgroup of A9
    and has an abelian maximal subgroup of type (3,3,3).
    <9,2> ≅ (3,3) is the abelian root of the
    unique coclass tree of coclass graph G(3,1)
    consisting of 3-groups of maximal class.
    The other groups are also vertices of G(3,1)
    with three exceptions: <243,8>
    in the stem of isoclinism family Φ6
    which is root of one of the three coclass trees
    with metabelian mainlines on coclass graph G(3,2)
    consisting of 3-groups of second maximal class,
    and the vertices <729,41> and <729,34…39>
    at depth 1 of branch B(35) of
    coclass tree T(<243,3>) on G(3,2).

    G32(K) TKT Cl3(N1) Cl3(N2) Cl3(N3) Cl3(N4) Cl3(F31(K)) Coclass
    <9,2> (0,0,0,0) (3) (3) (3) (3) 1 1
    <27,4> (1,1,1,1) (3,3) (9) (9) (9) (3) 1
    <81,7> (2,0,0,0) (3,3,3) (3,3) (3,3) (3,3) (3,3) 1
    <81,8> (2,0,0,0) (3,9) (3,3) (3,3) (3,3) (3,3) 1
    <81,10> (1,0,0,0) (3,9) (3,3) (3,3) (3,3) (3,3) 1
    <81,9> (0,0,0,0) (3,9) (3,3) (3,3) (3,3) (3,3) 1
    <243,28…30> (0,0,0,0) (3,9) (3,3) (3,3) (3,3) (3,9) 1
    <243,25> (2,0,0,0) (9,9) (3,3) (3,3) (3,3) (3,9) 1
    <243,27> (1,0,0,0) (9,9) (3,3) (3,3) (3,3) (3,9) 1
    <243,8> (0,2,3,1) (3,9) (3,9) (3,9) (3,9) (3,3,3) 2
    <243,3> (0,0,4,3) (3,9) (3,9) (3,3,3) (3,3,3) (3,3,3) 2
    <729,34…36> (0,0,4,3) (3,9) (3,9) (3,3,3) (3,3,3) (3,3,3,3) 2
    <729,37…39> (0,0,4,3) (3,9) (3,9) (3,3,3) (3,3,3) (3,3,9) 2
    <729,41> (4,0,4,3) (9,9) (3,9) (3,3,3) (3,3,3) (3,3,9) 2


    Remarks:
    1. Our new criteria use the TTT to locate a member
    of an infinite sequence sharing a common TKT.
    Of course, they would have been useless
    in times before the availability of Pari/GP and Magma.

    2. There are no selection rules in terms of branches of G(3,1)
    for the groups G32(K) of cyclic cubic fields K.
    (For real quadratic fields K of type (3,3)
    we have proved that only every other branch is admissible.)
    Therefore, cyclic cubic fields of type (3,3) reveal a
    similar behavior as quadratic fields of type (2,2).

    3. However, the Weak Leaf Conjecture seems to be satisfied
    for cyclic cubic and quadratic fields of type (3,3).
    The two instances which occurred up to now are
    the distinction between <81,9> and its immediate descendants <243,28…30>
    and between <243,3> and its immediate descendants <729,34…39>.
    Descendants are of the same TKT but of higher defect of commutativity.

    4. The decision between <81,9> and the indistinguishable batch
    <243,28…30> closely approaches the limits
    of current hardware and software technology.
    This is one of the two instances where the 3-class group of
    the Hilbert 3-class field F31(K) of K must be determined.
    Since F31(K) is of absolute degree 27, the demand of memory is gigantic.

    5. There seem to be differences in the behaviour of operating systems.
    Whereas Windows 7 Pro strictly protects its own memory resources
    and Magma frequently is forced to terminate complaining
    "Cannot complete task, since OS doesn't grant access to required memory",
    Mac OS X is obviously more generous and lets Magma use system memory.
    Magma completes its task but the OS remains in an instable state
    or even in a dead lock where Mac OS X doesn't show any reactions.






*************************************************************************

  • Statistics:

    For the calculation of relative frequencies (percentages),
    we need some absolute frequencies as references.
    Due to the explanations given in the section "Details" below,
    the 250 conductors in the range 657 ≤ f ≤ 26467 consist of
    19 with t = 2 and 9 | f, corresponding to 38 fields,
    69 with t = 2 and (f,3) = 1, corresponding to 138 fields,
    since fields with t = 2 arise in doublets, and
    84 with t = 3 and 9 | f, corresponding to 10*3 + 7*2 + 67*4 = 312 fields,
    78 with t = 3 and (f,3) = 1, corresponding to 14*3 + 11*2 + 53*4 = 276 fields,
    since fields with t = 3 may arise in triplets (Category I)
    or in doublets (Category II) or in quartets (Category III).

    G32(K) <9,2> <27,4> <81,7> <243,28…30> <243,25> <243,8> <81,8> <81,10> <729,37…39> <243,27> <729,41> <729,34…36>
    t = 2, 9 | f 18/38 (47%) 20/38 (53%) 0 0 0 0 0 0 0 0 0 0
    t = 2, (f,3) = 1 96/138 (70%) 42/138 (30%) 0 0 0 0 0 0 0 0 0 0
    t = 3, 9 | f 220/312 (71%) 0 46/312 (15%) 11/312 4/312 6/312 6/312 12/312 0 3/312 2/312 2/312
    t = 3, (f,3) = 1 184/276 (67%) 0 46/276 (17%) 2/276 1/276 15/276 8/276 16/276 4/276 0 0 0


    The 3-tower consists of a single stage
    if and only if G32(K) is abelian.

    For t = 2, 9 | f,
    the extraspecial group <27,4> is populated most densely with 53%.

    For t = 2, (f,3) = 1,
    the single stage 3-towers are dominating with 70%.

    For t = 3, 9 | f,
    the single stage 3-towers are dominating with 71%.

    For t = 3, (f,3) = 1,
    the single stage 3-towers are dominating with 67%.


  • Minimal Conductors:

    G32(K) <9,2> <27,4> <81,7> <243,28…30> <243,25> <243,8> <81,8> <81,10> <729,37…39> <243,27> <729,41> <729,34…36>
    t = 2, 9 | f 657 2439
    t = 2, (f,3) = 1 1267 5971
    t = 3, 9 | f 819 4599 4977 4977 21177 8001 8001 20367 22581 25929
    t = 3, (f,3) = 1 1729 3913 10621 10621 7657 9709 9709 20293


    Remarks:
    1. The minimal conductors 657, 1267, 2439, 5971 were given
    in the joint paper by Ayadi, Azizi, Ismaïli, p. 474 .

    2. In his Thesis, p. 63 f. , Ayadi only indicated that the conductors
    4977, 10621 with graph 1, and 7657, 8001 with graph 2
    of category I give rise to triplets of 3-class rank 2
    but he didn't determine the isomorphism type of G32(K).






  • *************************************************************************

  • Details for 250 conductors:

    Although we split the conductors into 4 basic types,
    the counting number No. refers to the
    entire sequence, ordered by increasing values.

    Cases with non-abelian group G32(K) are printed
    in red boldface font for coclass 1,
    in green boldface font for coclass 2.

    1. The 88 conductors f with two prime divisors, t = 2

    According to the multiplicity formula m(f) = (3-1)t-1,
    there are 2 cyclic cubic fields K sharing the common conductor f.
    If one of them has 3-class group (3,3), then the same is true for the other.
    Necessarily, the prime divisors of the conductor f
    are mutual cubic residues with respect to each other.
    Thus, we do not need graphs and categories in the case t = 2.
    It turns out that both members of the doublet have the same TKT κ(K)
    and the same 3-class field tower group G32(K),
    which can be determined with the aid of the following theorem.

    Theorem 1 ( Ayadi , 1995).
    Let f be a conductor divisible by exactly two primes, t = 2, such that
    Cl3(K) ≅ (3,3) for both cyclic cubic fields K with conductor f.
    Denote by n the number of prime divisors of the norm NK|Q(α) of
    any non-trivial primitive ambiguous principal ideal (α) of any of the two fields K.
    Then the second 3-class group G32(K) of both fields K
    is given by <9,2> with TKT a.1, κ(K) = (0,0,0,0), if n = 2,
    and by <27,4> with TKT A.1, κ(K) = (1,1,1,1), if n = 1.

    Examples. We have
    NK|Q(α) = 32*73, n = 2 for f = 657 = 32*73,
    NK|Q(α) = 7*181, n = 2 for f = 1267 = 7*181,
    NK|Q(α) = 271, n = 1 for f = 2439 = 32*271,
    NK|Q(α) = 853, n = 1 for f = 5971 = 7*853,
    according to Ayadi, Azizi, Ismaïli, p. 474 .


    1.1. The 19 conductors f divisible by nine, 9 | f

    No. f factors TKT κ(K) G32(K)
    1 657 32*73 a.1 (0000) <9,2>
    12 2439 32*271 A.1 (1111) <27,4>
    14 2763 32*307 a.1 (0000) <9,2>
    29 4707 32*523 A.1 (1111) <27,4>
    34 5193 32*577 a.1 (0000) <9,2>
    38 5517 32*613 a.1 (0000) <9,2>
    50 6813 32*757 a.1 (0000) <9,2>
    65 8271 32*919 A.1 (1111) <27,4>
    72 8919 32*991 A.1 (1111) <27,4>
    73 9081 32*1009 a.1 (0000) <9,2>
    84 10053 32*1117 A.1 (1111) <27,4>
    116 13779 32*1531 A.1 (1111) <27,4>
    118 13941 32*1549 A.1 (1111) <27,4>
    127 14589 32*1621 A.1 (1111) <27,4>
    141 16047 32*1783 a.1 (0000) <9,2>
    177 19611 32*2179 a.1 (0000) <9,2>
    182 20259 32*2251 A.1 (1111) <27,4>
    187 20583 32*2287 a.1 (0000) <9,2>
    194 21069 32*2341 A.1 (1111) <27,4>



    1.2. The 69 conductors f coprime to three, (f,3) = 1

    No. f factors TKT κ(K) G32(K)
    4 1267 7*181 a.1 (0000) <9,2>
    5 1339 13*103 a.1 (0000) <9,2>
    6 1561 7*223 a.1 (0000) <9,2>
    11 2359 7*337 a.1 (0000) <9,2>
    16 2869 19*151 a.1 (0000) <9,2>
    17 2947 7*421 a.1 (0000) <9,2>
    18 2977 13*229 a.1 (0000) <9,2>
    19 3241 7*463 a.1 (0000) <9,2>
    22 3811 37*103 a.1 (0000) <9,2>
    33 5053 31*163 a.1 (0000) <9,2>
    35 5263 19*277 a.1 (0000) <9,2>
    37 5473 13*421 a.1 (0000) <9,2>
    40 5677 7*811 a.1 (0000) <9,2>
    42 5971 7*853 A.1 (1111) <27,4>
    45 6181 7*883 a.1 (0000) <9,2>
    47 6487 13*499 a.1 (0000) <9,2>
    52 7087 19*373 A.1 (1111) <27,4>
    54 7147 7*1021 a.1 (0000) <9,2>
    57 7519 73*103 A.1 (1111) <27,4>
    59 7663 79*97 a.1 (0000) <9,2>
    63 8047 13*619 A.1 (1111) <27,4>
    66 8299 43*193 A.1 (1111) <27,4>
    67 8401 31*271 a.1 (0000) <9,2>
    76 9253 19*487 A.1 (1111) <27,4>
    83 9943 61*163 A.1 (1111) <27,4>
    90 10819 31*349 a.1 (0000) <9,2>
    91 10963 19*577 A.1 (1111) <27,4>
    92 11089 13*853 A.1 (1111) <27,4>
    98 11563 31*373 a.1 (0000) <9,2>
    101 12061 7*1723 a.1 (0000) <9,2>
    108 12931 67*193 a.1 (0000) <9,2>
    109 13117 13*1009 a.1 (0000) <9,2>
    110 13129 19*691 a.1 (0000) <9,2>
    117 13927 19*733 a.1 (0000) <9,2>
    120 14203 7*2029 A.1 (1111) <27,4>
    125 14521 13*1117 a.1 (0000) <9,2>
    128 14701 61*241 A.1 (1111) <27,4>
    134 15223 13*1171 a.1 (0000) <9,2>
    135 15577 37*421 a.1 (0000) <9,2>
    136 15613 13*1201 A.1 (1111) <27,4>
    137 15673 7*2239 A.1 (1111) <27,4>
    140 16021 37*433 a.1 (0000) <9,2>
    147 16897 61*277 A.1 (1111) <27,4>
    155 17587 43*409 a.1 (0000) <9,2>
    159 18019 37*487 a.1 (0000) <9,2>
    162 18319 7*2617 A.1 (1111) <27,4>
    169 18961 67*283 a.1 (0000) <9,2>
    171 19093 61*313 a.1 (0000) <9,2>
    172 19189 31*619 a.1 (0000) <9,2>
    174 19279 13*1483 a.1 (0000) <9,2>
    178 19651 43*457 a.1 (0000) <9,2>
    188 20671 7*2953 a.1 (0000) <9,2>
    197 21259 7*3037 a.1 (0000) <9,2>
    198 21451 19*1129 A.1 (1111) <27,4>
    199 21469 7*3067 a.1 (0000) <9,2>
    201 21691 109*199 a.1 (0000) <9,2>
    208 22141 7*3163 A.1 (1111) <27,4>
    211 22357 79*283 a.1 (0000) <9,2>
    213 22633 13*1741 a.1 (0000) <9,2>
    214 22849 73*313 a.1 (0000) <9,2>
    216 23233 7*3319 a.1 (0000) <9,2>
    217 23257 13*1789 A.1 (1111) <27,4>
    219 23383 67*349 A.1 (1111) <27,4>
    222 23611 7*3373 a.1 (0000) <9,2>
    230 24349 13*1873 A.1 (1111) <27,4>
    234 24703 7*3529 A.1 (1111) <27,4>
    236 25081 7*3583 a.1 (0000) <9,2>
    242 25699 31*829 a.1 (0000) <9,2>
    249 26359 43*613 a.1 (0000) <9,2>


    The conductors f ≤ 26359 in our Sections 1.1 and 1.2
    overlap with those conductors f = p*q which are given in
    Table 1 on p. 40 (p < 1000, q < 1000) and Table 4 on p. 42 (p ≤ 151, q < 1000)
    of the Section "Tables Numériques" in the paper by George Gras .
    We point out that Table 1 also contains cases
    with class numbers divisible by 27
    whereas Table 4 is restricted to 3-class groups of type (3,3).






  • 2. The 162 conductors f with three prime divisors, t = 3

    According to the multiplicity formula m(f) = (3-1)t-1,
    there are 4 cyclic cubic fields K sharing the common conductor f.
    The members of a multiplet do not necessarily have equal 3-class ranks.
    This forces us to introduce three categories of quadruplets.

    Category I: three members have 3-class rank 2, the other has rank 3.
    Category II: only two members have 3-class rank 2, the other two have rank 3.
    Category III: all four members have 3-class rank 2.

    However, there arises an additional complication:
    In general, the members of rank 2 neither have the same TKT κ(K)
    nor the same 3-class field tower group G32(K).
    Therefore we use exponents denoting iteration.
    A criterion for G32(K) to be abelian
    is given in the following theorem [& conjecture].

    Theorem 2 ( Ayadi , 1995) [& Conjecture 2 (2013)].
    Let f be a conductor divisible by exactly three primes, t = 3, such that
    Cl3(K) ≅ (3,3) for all four cyclic cubic fields K with conductor f.
    Then the second 3-class group G32(K) of all four fields K
    is abelian of type <9,2> ≅ (3,3) with TKT a.1, κ(K) = (0,0,0,0), if [and only if]
    there are no mutual cubic residues among the prime divisors of f,
    that is, f belongs to graphs 1,…,4 of category III.
    [G32(K) is never abelian for categories I and II.]

    For the group <81,7> with second largest density of population we suggest the following:

    Conjecture 3 (2013).
    Let f be a conductor divisible by exactly three primes, t = 3, such that
    Cl3(K) ≅ (3,3) for only two of the four cyclic cubic fields K with conductor f,
    that is, f belongs to graph 1 or 2 of category II.
    Then the second 3-class group G32(K) of both fields K
    is given by <81,7> ≅ Syl3(A9) with TKT a.3*, κ(K) = (2,0,0,0),
    provided that f belongs to graph 2 when 9 | f.


    2.1. The 84 conductors f divisible by nine, 9 | f

    No. f factors category graph TKTs κ(K) G32(K)
    2 819 32*7*13 III 2: 13 → 7 a.14 (0000)4 <9,2>4
    3 1197 32*7*19 III 3: 7 → 19 → 9 a.14 (0000)4 <9,2>4
    8 1953 32*7*31 III 1: δ ≠ 0 a.14 (0000)4 <9,2>4
    9 2223 32*13*19 III 2: 19 → 9 a.14 (0000)4 <9,2>4
    10 2331 32*7*37 III 2: 37 → 9 a.14 (0000)4 <9,2>4
    13 2709 32*7*43 III 2: 43 → 7 a.14 (0000)4 <9,2>4
    21 3627 32*13*31 III 2: 31 → 13 a.14 (0000)4 <9,2>4
    23 3843 32*7*61 III 2: 9 → 61 a.14 (0000)4 <9,2>4
    26 4221 32*7*67 III 2: 9 → 67 a.14 (0000)4 <9,2>4
    27 4329 32*13*37 III 2: 37 → 9 a.14 (0000)4 <9,2>4
    28 4599 32*7*73 III 7: 9 ↔ 73 ← 7 a.3*4 (2000)4 <81,7>4
    31 4977 32*7*79 I 1: δ = 0 a.3,a.12 (2000),(0000)2 <243,25>,<243,28…30>2
    32 5031 32*13*43 III 1: δ ≠ 0 a.14 (0000)4 <9,2>4
    36 5301 32*19*31 III 3: 31 → 19 → 9 a.14 (0000)4 <9,2>4
    44 6111 32*7*97 III 2: 97 → 7 a.14 (0000)4 <9,2>4
    46 6327 32*19*37 II 2: 19 → 9 ← 37 → 19 a.3*2 (2000)2 <81,7>2
    48 6489 32*7*103 III 2: 9 → 103 a.14 (0000)4 <9,2>4
    51 6867 32*7*109 III 2: 109 → 9 a.14 (0000)4 <9,2>4
    53 7137 32*13*61 III 2: 9 → 61 a.14 (0000)4 <9,2>4
    56 7353 32*19*43 III 2: 19 → 9 a.14 (0000)4 <9,2>4
    60 7839 32*13*67 III 2: 9 → 67 a.14 (0000)4 <9,2>4
    61 8001 32*7*127 I 2: 9 ← 127 → 7 a.3,a.22 (2000),(1000)2 <81,8>,<81,10>2
    68 8541 32*13*73 III 6: 9 ↔ 73 → 13 a.3*4 (2000)4 <81,7>4
    69 8757 32*7*139 III 2: 139 → 7 a.14 (0000)4 <9,2>4
    75 9243 32*13*79 III 2: 79 → 13 a.14 (0000)4 <9,2>4
    79 9513 32*7*151 III 2: 9 → 151 a.14 (0000)4 <9,2>4
    81 9891 32*7*157 III 2: 7 → 157 a.14 (0000)4 <9,2>4
    85 10269 32*7*163 III 2: 163 → 9 a.14 (0000)4 <9,2>4
    86 10323 32*31*37 III 3: 31 → 37 → 9 a.14 (0000)4 <9,2>4
    87 10431 32*19*61 III 3: 19 → 9 → 61 a.14 (0000)4 <9,2>4
    94 11349 32*13*97 I 1: δ = 0 a.3,a.22 (2000),(1000)2 <81,8>,<81,10>2
    95 11403 32*7*181 III 6: 7 ↔ 181 → 9 a.3*4 (2000)4 <81,7>4
    96 11457 32*19*67 III 3: 19 → 9 → 67 a.14 (0000)4 <9,2>4
    99 11997 32*31*43 III 1: δ ≠ 0 a.14 (0000)4 <9,2>4
    100 12051 32*13*103 III 7: 13 ↔ 103 ← 9 a.3*4 (2000)4 <81,7>4
    102 12159 32*7*193 III 2: 9 → 193 a.14 (0000)4 <9,2>4
    103 12483 32*19*73 III 7: 19 → 9 ↔ 73 a.3*4 (2000)4 <81,7>4
    104 12537 32*7*199 III 2: 199 → 9 a.14 (0000)4 <9,2>4
    106 12753 32*13*109 I 2: 9 ← 109 → 13 a.3,a.22 (2000),(1000)2 <81,8>,<81,10>2
    112 13293 32*7*211 III 2: 211 → 7 a.14 (0000)4 <9,2>4
    113 13509 32*19*79 III 2: 19 → 9 a.14 (0000)4 <9,2>4
    119 14049 32*7*223 III 5: 7 ↔ 223 a.14 (0000)4 <243,28…30>4
    122 14319 32*37*43 III 3: 43 → 37 → 9 a.14 (0000)4 <9,2>4
    123 14427 32*7*229 III 1: δ ≠ 0 a.14 (0000)4 <9,2>4
    130 14859 32*13*127 III 2: 127 → 9 a.14 (0000)4 <9,2>4
    133 15183 32*7*241 III 1: δ ≠ 0 a.14 (0000)4 <9,2>4
    142 16263 32*13*139 I 1: δ = 0 a.3,a.12 (2000),(0000)2 <243,25>,<243,28…30>2
    145 16587 32*19*97 I 2: 9 ← 19 → 97 a.3,a.22 (2000),(1000)2 <81,8>,<81,10>2
    148 17019 32*31*61 III 3: 9 → 61 → 31 a.14 (0000)4 <9,2>4
    149 17073 32*7*271 III 5: 9 ↔ 271 a.3*4 (2000)4 <81,7>4
    153 17451 32*7*277 III 1: δ ≠ 0 a.14 (0000)4 <9,2>4
    156 17613 32*19*103 III 4: 19 → 9 → 103 → 19 a.14 (0000)4 <9,2>4
    157 17667 32*13*151 III 3: 9 → 151 → 13 a.14 (0000)4 <9,2>4
    158 17829 32*7*283 I 1: δ = 0 a.3,a.22 (2000),(1000)2 <81,8>,<81,10>2
    164 18369 32*13*157 III 2: 157 → 13 a.14 (0000)4 <9,2>4
    166 18639 32*19*109 II 2: 109 → 9 ← 19 → 109 a.3*2 (2000)2 <81,7>2
    167 18693 32*31*67 III 2: 9 → 67 a.14 (0000)4 <9,2>4
    170 19071 32*13*163 III 3: 13 → 163 → 9 a.14 (0000)4 <9,2>4
    175 19341 32*7*307 III 6: 9 ↔ 307 → 7 a.3*4 (2000)4 <81,7>4
    179 19719 32*7*313 III 2: 7 → 313 a.14 (0000)4 <9,2>4
    184 20313 32*37*61 II 2: 61 → 9 ← 37 → 61 a.3*2 (2000)2 <81,7>2
    185 20367 32*31*73 III 5: 9 ↔ 73 a.2,a.13 (1000),(0000)3 <243,27>,<243,28…30>3
    191 20853 32*7*331 III 2: 7 → 331 a.14 (0000)4 <9,2>4
    195 21177 32*13*181 I 2: 9 ← 181 → 13 c.213 (0231)3 <243,8>3
    196 21231 32*7*337 III 5: 7 ↔ 337 a.22,a.32 (1000)2,(2000)2 <243,27>2,<243,25>2
    202 21717 32*19*127 II 2: 9 → 127 ← 19 → 9 a.3*2 (2000)2 <81,7>2
    206 21987 32*7*349 III 2: 349 → 7 a.14 (0000)4 <9,2>4
    207 22041 32*19*97 I 1: δ = 0 a.3,a.22 (2000),(1000)2 <81,8>,<81,10>2
    209 22311 32*37*67 III 3: 37 → 9 → 67 a.14 (0000)4 <9,2>4
    212 22581 32*13*193 II 1: 9 → 193 ← 13 d.192 (4043)2 <729,41>2
    215 23121 32*7*367 II 1: 7 → 367 ← 9 a.3*2 (2000)2 <81,7>2
    218 23283 32*13*199 III 2: 199 → 9 a.14 (0000)4 <9,2>4
    220 23499 32*7*373 III 2: 7 → 373 a.14 (0000)4 <9,2>4
    221 23607 32*43*61 III 2: 9 → 61 a.14 (0000)4 <9,2>4
    224 23769 32*19*139 III 2: 19 → 9 a.14 (0000)4 <9,2>4
    226 23877 32*7*379 I 2: 7 ← 379 → 9 c.213 (0231)3 <243,8>3
    229 24309 32*37*73 III 9: 37 → 9 ↔ 73 → 37 a.3*4 (2000)4 <81,7>4
    233 24687 32*13*211 III 2: 13 → 211 a.14 (0000)4 <9,2>4
    235 25011 32*7*397 III 2: 397 → 9 a.14 (0000)4 <9,2>4
    244 25767 32*7*409 III 1: δ ≠ 0 a.14 (0000)4 <9,2>4
    245 25821 32*19*151 III 9: 9 → 151 ↔ 19 → 9 a.3*4 (2000)4 <81,7>4
    246 25929 32*43*67 II 1: 9 → 67 ← 43 b.102 (0043)2 <729,37…39>2
    247 26091 32*13*223 III 2: 13 → 223 a.14 (0000)4 <9,2>4
    248 26307 32*37*79 III 2: 37 → 9 a.14 (0000)4 <9,2>4
    251 26523 32*7*421 III 5: 7 ↔ 421 a.2,a.13 (1000),(0000)3 <243,27>,<243,28…30>3



    2.2. The 78 conductors f coprime to three, (f,3) = 1

    No. f factors category graph TKTs κ(K) G32(K)
    7 1729 7*13*19 III 3: 13 → 7 → 19 a.14 (0000)4 <9,2>4
    15 2821 7*13*31 III 3: 31 → 13 → 7 a.14 (0000)4 <9,2>4
    20 3367 7*13*37 III 2: 13 → 7 a.14 (0000)4 <9,2>4
    24 3913 7*13*43 II 1: 13 → 7 ← 43 a.3*2 (2000)2 <81,7>2
    25 4123 7*19*31 II 1: 7 → 19 ← 31 a.3*2 (2000)2 <81,7>2
    30 4921 7*19*37 II 1: 7 → 19 ← 37 a.3*2 (2000)2 <81,7>2
    39 5551 7*13*61 III 2: 13 → 7 a.14 (0000)4 <9,2>4
    41 5719 7*19*43 III 3: 43 → 7 → 19 a.14 (0000)4 <9,2>4
    43 6097 7*13*67 III 2: 13 → 7 a.14 (0000)4 <9,2>4
    49 6643 7*13*73 III 4: 13 → 7 → 73 → 13 a.14 (0000)4 <9,2>4
    55 7189 7*13*79 III 3: 79 → 13 → 7 a.14 (0000)4 <9,2>4
    58 7657 13*19*31 I 2: 13 ← 31 → 19 c.213 (0231)3 <243,8>3
    62 8029 7*31*37 III 2: 31 → 37 a.14 (0000)4 <9,2>4
    64 8113 7*19*61 III 2: 7 → 19 a.14 (0000)4 <9,2>4
    70 8827 7*13*97 II 1: 13 → 7 ← 97 a.3*2 (2000)2 <81,7>2
    71 8911 7*19*67 III 2: 7 → 19 a.14 (0000)4 <9,2>4
    74 9139 13*19*37 III 2: 37 → 19 a.14 (0000)4 <9,2>4
    77 9331 7*31*43 III 2: 43 → 7 a.14 (0000)4 <9,2>4
    78 9373 7*13*103 III 6: 7 ← 13 ↔ 103 a.3*4 (2000)4 <81,7>4
    80 9709 7*19*73 I 2: 19 ← 7 → 73 a.3,a.22 (2000),(1000)2 <81,8>,<81,10>2
    82 9919 7*13*109 III 3: 109 → 13 → 7 a.14 (0000)4 <9,2>4
    88 10507 7*19*79 III 2: 7 → 19 a.14 (0000)4 <9,2>4
    89 10621 13*19*43 I 1: δ = 0 a.3,a.12 (2000),(0000)2 <243,25>,<243,28…30>2
    93 11137 7*37*43 I 2: 7 ← 43 → 37 a.3,a.22 (2000),(1000)2 <81,8>,<81,10>2
    97 11557 7*13*127 II 1: 13 → 7 ← 127 a.3*2 (2000)2 <81,7>2
    105 12649 7*13*139 II 1: 13 → 7 ← 139 a.3*2 (2000)2 <81,7>2
    107 12901 7*19*97 III 4: 7 → 19 → 97 → 7 a.14 (0000)4 <9,2>4
    111 13237 7*31*61 III 2: 61 → 31 a.14 (0000)4 <9,2>4
    114 13699 7*19*103 II 1: 7 → 19 ← 103 a.3*2 (2000)2 <81,7>2
    115 13741 7*13*151 III 3: 151 → 13 → 7 a.14 (0000)4 <9,2>4
    121 14287 7*13*157 III 4: 7 → 157 → 13 → 7 a.14 (0000)4 <9,2>4
    124 14497 7*19*109 III 3: 7 → 19 → 109 a.14 (0000)4 <9,2>4
    126 14539 7*31*67 III 1: δ ≠ 0 a.14 (0000)4 <9,2>4
    129 14833 7*13*163 I 2: 7 ← 13 → 163 a.3,a.22 (2000),(1000)2 <81,8>,<81,10>2
    131 14911 13*31*37 I 2: 13 ← 31 → 37 a.3,a.22 (2000),(1000)2 <81,8>,<81,10>2
    132 15067 13*19*61 III 1: δ ≠ 0 a.14 (0000)4 <9,2>4
    138 15799 7*37*61 III 2: 37 → 61 a.14 (0000)4 <9,2>4
    139 15841 7*31*73 III 2: 7 → 73 a.14 (0000)4 <9,2>4
    143 16471 7*13*181 III 9: 13 → 7 ↔ 181 → 13 a.3*4 (2000)4 <81,7>4
    144 16549 13*19*67 III 1: δ ≠ 0 a.14 (0000)4 <9,2>4
    146 16891 7*19*127 III 4: 127 → 7 → 19 → 127 a.14 (0000)4 <9,2>4
    150 17143 7*31*79 III 1: δ ≠ 0 a.14 (0000)4 <9,2>4
    151 17329 13*31*43 III 2: 31 → 13 a.14 (0000)4 <9,2>4
    152 17353 7*37*67 I 1: δ = 0 a.3,a.22 (2000),(1000)2 <81,8>,<81,10>2
    154 17563 7*13*193 I 2: 7 ← 13 → 193 c.213 (0231)3 <243,8>3
    160 18031 13*19*73 III 2: 73 → 13 a.14 (0000)4 <9,2>4
    161 18109 7*13*199 III 2: 13 → 7 a.14 (0000)4 <9,2>4
    163 18361 7*43*61 III 2: 43 → 7 a.14 (0000)4 <9,2>4
    165 18487 7*19*139 III 3: 139 → 7 → 19 a.14 (0000)4 <9,2>4
    168 18907 7*37*73 III 3: 7 → 73 → 37 a.14 (0000)4 <9,2>4
    173 19201 7*13*211 II 2: 211 → 7 ← 13 → 211 a.3*2 (2000)2 <81,7>2
    176 19513 13*19*79 III 2: 79 → 13 a.14 (0000)4 <9,2>4
    180 20083 7*19*151 III 7: 7 → 19 ↔ 151 a.3*4 (2000)4 <81,7>4
    181 20167 7*43*67 I 2: 7 ← 43 → 67 c.213 (0231)3 <243,8>3
    183 20293 7*13*223 III 8: 13 → 7 ↔ 223 ← 13 b.104 (0043)4 <729,34…36>4
    186 20461 7*37*79 III 1: δ ≠ 0 a.14 (0000)4 <9,2>4
    189 20683 13*37*43 III 2: 43 → 37 a.14 (0000)4 <9,2>4
    190 20839 7*13*229 III 6: 7 ← 13 ↔ 229 a.3*4 (2000)4 <81,7>4
    192 20881 7*19*157 I 2: 19 ← 7 → 157 c.213 (0231)3 <243,8>3
    193 21049 7*31*97 I 2: 7 ← 97 → 31 c.213 (0231)3 <243,8>3
    200 21679 7*19*163 II 1: 7 → 19 ← 163 a.3*2 (2000)2 <81,7>2
    203 21793 19*31*37 II 2: 37 → 19 ← 31 → 37 a.3*2 (2000)2 <81,7>2
    204 21931 7*13*241 III 2: 13 → 7 a.14 (0000)4 <9,2>4
    205 21973 7*43*73 II 2: 7 → 73 ← 43 → 7 a.3*2 (2000)2 <81,7>2
    210 22351 7*31*103 III 2: 31 → 103 a.14 (0000)4 <9,2>4
    223 23653 7*31*109 III 2: 109 → 31 a.14 (0000)4 <9,2>4
    225 23779 7*43*79 III 2: 43 → 7 a.14 (0000)4 <9,2>4
    227 23959 13*19*97 III 2: 19 → 97 a.14 (0000)4 <9,2>4
    228 24073 7*19*181 III 9: 19 → 181 ↔ 7 → 19 a.3*4 (2000)4 <81,7>4
    231 24583 13*31*61 III 3: 61 → 31 → 13 a.14 (0000)4 <9,2>4
    232 24661 7*13*271 I 2: 7 ← 13 → 271 a.3,a.22 (2000),(1000)2 <81,8>,<81,10>2
    237 25123 7*37*97 I 2: 7 ← 97 → 37 a.3,a.22 (2000),(1000)2 <81,8>,<81,10>2
    238 25207 7*13*277 I 2: 7 ← 13 → 277 a.3,a.22 (2000),(1000)2 <81,8>,<81,10>2
    239 25327 19*31*43 III 2: 31 → 19 a.14 (0000)4 <9,2>4
    240 25441 13*19*103 III 6: 19 ← 103 ↔ 13 a.3*4 (2000)4 <81,7>4
    241 25669 7*19*193 III 2: 7 → 19 a.14 (0000)4 <9,2>4
    243 25753 7*13*283 III 2: 13 → 7 a.14 (0000)4 <9,2>4
    250 26467 7*19*199 III 2: 7 → 19 a.14 (0000)4 <9,2>4


    Remarks and Corrections:
    1. The most exotic phenomena are the occurrences of conductors
    f = 7657, 17563, 20167, 20881, 21049, resp. 20293, with (f,3) = 1,
    and f = 21177, 23877, resp. 22581, resp. 25929, with 9 | f,
    which give rise to second 3-class groups of coclass 2.

    2. It is clear that the investigations have to be extended further
    to see whether groups of coclass bigger than 2 are possible.

    3. Due to theorems by Blackburn , resp. by Heider and Schmithals ,
    the length of the 3-class field tower is exactly two,
    for all cases having a non-abelian group G32(K),
    except for <729,34…39> and <729,41>,
    where the length is unknown and probably equal to three.

    4. In the joint paper by Ayadi, Azizi, Ismaïli, p. 474 ,
    there are two misprints and two errors
    in the Numerical Examples after Theorem 4.4.
    2 misprints:
    Upper bound for the conductors is 6000 instead of 16000
    and conductor 2843 should read 3843.
    2 errors:
    Conductor 3367 of category III, graph 2 is missing
    from the listing of conductors in category III
    having one of the graphs 1,2,3,4 and abelian G32(K).
    Instead, the conductor 3913 of category II, graph 1
    with non-abelian G32(K) is given erroneously.
    This seems to be a systematic error, since 3913 doesn't
    show up among the examples on p. 73 of Ayadi's Thesis either.




    Karl-Franzens University Graz, right side
    Daniel C. Mayer
    Explanation:

    • Since 2002, initiated by Aïssa Derhem,
      we are planning this long desired extension
      of Mohammed Ayadi's Ph. D. Thesis
      on 3-principalization in unramified cyclic extensions
      of degree 3 over cyclic cubic fields K of type (3,3).

      Except for the complete lack of statistical information,
      the conductors divisible by two primes (t = 2) were settled
      by a nice but rather simple theory which gives criteria
      for only two possible second 3-class groups,
      either G32(K) = <9,2> or <27,4>,
      in terms of the ambiguous principal ideals of K.
      See our presentation in Joint Research 2002 .

      However, here we show that the theory of
      conductors divisible by three primes (t = 3)
      is of considerable complexity, as expected,
      and reveals a wealth of new configurations.
      In spite of the complexity of possible cubic residue
      conditions between the prime divisors of the conductor f,
      nobody knew whether exotic variants of the
      second 3-class groups G32(K) are to be expected.
      Of course, we hoped that they will appear.
      And, indeed, they did: all the groups G32(K)
      different from <9,2> or <27,4>
      were completely unknown up to now.

    *
    Web master's e-mail address:
    contact@algebra.at
    *

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