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A269322 Discriminants of real quadratic fields with second 3-class group <729,49> 1
534824, 1030117, 2661365, 2733965, 3194013, 3259597, 3268781, 3928632, 4006033, 4593673, 5180081, 5250941, 5327080, 5489661, 5909813, 6115852, 6290549, 7102277, 7712184, 7738629, 7758589, 7857048, 7943761, 8243113, 8747997, 8899661, 9583736, 9907837 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The Artin transfer homomorphisms of the assigned second 3-class group M with SmallGroups identifier <729,49> [Besche, Eick, O'Brien] determine the capitulation type (0,1,2,2) (TKT without fixed points) of the real quadratic field K in its four unramified cyclic cubic extensions N_i|K (i=1,...,4) and the abelian type invariants of the 3-class groups Cl(3,K)=(3,3) (whence A269322 is a subsequence of A269319) and [Cl(3,N_i)]=[(9,9),(3,3,3),(9,3),(9,3)] (TTT or IPAD). Conversely, the group M=<729,49> is determined uniquely not only by its Artin pattern AP(G)=(TTT,TKT) but even by the TTT component alone [Mayer, 2015], where TKT, TTT, IPAD are abbreviations for transfer kernel type, transfer target type, index-p abelianization data, respectively [Mayer, 2016]. Consequently, the MAGMA program has to determine only the TTT component of the Artin pattern. This is an instance of the "Principalization algorithm via class group structure" [Mayer, 2014] and saves a considerable amount of CPU time, since the determination of the TKT component is very delicate.

An important Theorem by I.R. Shafarevich [Mayer, 2015, Thm.5.1] disables the metabelian group M=<729,49> as a candidate for the 3-class tower group G, since the relation rank of M is too big. In [Mayer, 2015] it is proved that exactly the two non-metabelian groups <2187,284> and <2187,291> [Besche, Eick, O'Brien] are permitted for G, and the decision is possible with the aid of iterated IPADs of second order (which require computing 3-class groups of number fields with absolute degree 18). Since the derived length of both groups is equal to 3, the Hilbert 3-class field tower of all these real quadratic fields has exact length 3.

The MAGMA program requires A269319 as its input data.

LINKS

Table of n, a(n) for n=1..28.

H. U. Besche, B. Eick, and E. A. O'Brien, The SmallGroups Library - a library of groups of small order, 2005, an accepted and refereed GAP package, available also in MAGMA.

M. R. Bush, private communication, 11 July 2015.

D. C. Mayer, The second p-class group of a number field, Int. J. Number Theory 8 (2012), no. 2, 471-505.

D. C. Mayer, Principalization algorithm via class group structure, J. Thèor. Nombres Bordeaux 26 (2014), no. 2, 415-464.

D. C. Mayer, New number fields with known p-class tower, Tatra Mt. Math. Publ. 64 (2015), 21-57.

D. C. Mayer, Artin transfer patterns on descendant trees of finite p-groups, Adv. Pure Math. 6 (2016), no. 2, 66-104.

EXAMPLE

The leading term, 534824, and thus the first real quadratic field K with capitulation type c.18, (0,1,2,2), has been identified on 20 August 2009. However, it required six further years to determine the pro-3 Galois group G=<2187,291>, with metabelianization M=G/G''=<729,49>, of the Hilbert 3-class field tower of K in August 2015. The first 28 terms of A269322 up to 10^7 have been determined in [Mayer, 2012] and [Mayer, 2014]. The 347, resp. 4318, terms up to 10^8, resp. 10^9, have been computed by [Bush].

Concerning the two possibilities for the 3-class tower group, 534824 is the smallest term with associated group G=<2187,291> and 1030117 is the smallest term with associated group G=<2187,284>. (See [Mayer, 2015] for more details.)

PROG

(MAGMA) SetClassGroupBounds("GRH"); p:=3; dList:=A269319; for d in dList do

ZX<X>:=PolynomialRing(Integers()); K:=NumberField(X^2-d); O:=MaximalOrder(K); C, mC:=ClassGroup(O); sS:=Subgroups(C: Quot:=[p]); sI:=[]; for j in [1..p+1] do Append(~sI, 0); end for; n:=Ngens(C); g:=(Order(C.(n-1)) div p)*C.(n-1); h:=(Order(C.n) div p)*C.n; ct:=0;

for x in sS do ct:=ct+1; if g in x`subgroup then sI[1]:=ct; end if; if h in x`subgroup then sI[2]:=ct; end if; for e in [1..p-1] do if g+e*h in x`subgroup then sI[e+2]:=ct; end if; end for; end for;

sA:=[AbelianExtension(Inverse(mQ)*mC) where Q, mQ:=quo<C|x`subgroup>: x in sS]; sN:=[NumberField(x): x in sA]; sR:=[MaximalOrder(x): x in sA];

sF:=[AbsoluteField(x): x in sN]; sM:=[MaximalOrder(x): x in sF];

sM:=[OptimizedRepresentation(x): x in sF];

sA:=[NumberField(DefiningPolynomial(x)): x in sM]; sO:=[Simplify(LLL(MaximalOrder(x))): x in sA];

TTT:=[]; epsilon:=0; polarization1:=3; polarization2:=3; for j in [1..#sO] do CO:=ClassGroup(sO[j]); Append(~TTT, pPrimaryInvariants(CO, p));

if (3 eq #pPrimaryInvariants(CO, p)) then epsilon:=epsilon+1; end if;

val:=Valuation(Order(CO), p); if (2 eq val) then polarization2:=val; elif (4 le val) then if (3 eq polarization1) then polarization1:=val; else polarization2:=val; end if; end if; end for; if (3 eq polarization2) and (4 eq polarization1) and (1 eq epsilon) then printf "%o, ", d; end if; end for;

CROSSREFS

Subsequence of A269319

Sequence in context: A010037 A061730 A235076 * A227497 A183650 A273812

Adjacent sequences:  A269319 A269320 A269321 * A269323 A269324 A269325

KEYWORD

nonn,hard

AUTHOR

Daniel Constantin Mayer, Mar 09 2016

STATUS

approved

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Last modified November 19 08:48 EST 2018. Contains 317347 sequences. (Running on oeis4.)