A269323 Discriminants of real quadratic fields with second 3-class group <729,54>.

540365, 945813, 1202680, 1695260, 1958629, 3018569, 3236657, 3687441, 4441560, 5512252, 5571377, 5701693, 6027557, 6049356, 6054060, 6274609, 6366029, 6501608, 6773557, 7573868, 8243464, 8251521, 9054177, 9162577, 9967837

Offset: 1

Author

Daniel Constantin Mayer, Mar 10 2016

Keywords

• nonn
• hard

Comments

The Artin transfer homomorphisms of the assigned second 3-class group M with SmallGroups identifier <729,54> [Besche, Eick, O'Brien] determine the capitulation type (2,0,3,4) (TKT with two fixed points 3 and 4) 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 A269323 is a subsequence of A269319) and [Cl(3,N_i)]=[(9,9),(9,3),(9,3),(9,3)] (TTT or IPAD). Conversely, the group M=<729,54> 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, it suffices that the MAGMA program only determines 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=<729,54> 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 <2187,307> and <2187,308> [Besche, Eick, O'Brien] are permitted for G, and the decision is possible with the aid of multi-layered iterated IPADs of second order (which require computing 3-class groups of number fields with absolute degree 54). Since the derived length of both groups is equal to 3, the Hilbert 3-class field tower of all these real quadratic fields has certainly exact length 3.

The MAGMA program requires A269319 as its input list.

Examples

			The leading term, 540365, and thus the first real quadratic field K with capitulation type c.21, (2,0,3,4), has been identified on 01 January 2008 [Mayer, 2007/2008]. However, it required seven further years to determine the pro-3 Galois group G=<2187,307|308>, with metabelianization M=G/G''=<729,54>, of the Hilbert 3-class field tower of K in August 2015. (See [Mayer, 2015] for more details.) The first 25 terms of A269323 up to 10^7 have been determined in [Mayer, 2012] and [Mayer, 2014]. The 358, resp. 4377, terms up to 10^8, resp. 10^9, have been computed by [Bush].
		

Links

• 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 real quadratic base field K with discriminant d=540365, Targets 2007/2008.
• 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.

Crossrefs

Subsequence of A269319

Programs

Magma

SetClassGroupBounds("GRH"); p:=3; dList:=A269319; for d in dList do
ZX:=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: 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 (0 eq epsilon) then printf "%o, ",d; end if; end for;