

A269320


Discriminants of real quadratic fields with 3class tower group <81,10>.


2



72329, 94636, 153949, 189237, 206776, 209765, 214028, 219461, 275881, 390876, 400369, 431761, 460817, 486581, 548549, 551692, 552392, 602521, 698556, 775480, 775661, 781177, 782876, 804648, 831484, 836493, 893689, 907629, 907709, 957484, 959629, 980108, 993349, 994008
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OFFSET

1,1


COMMENTS

The Artin transfer homomorphisms of the assigned 3class tower group G with SmallGroups identifier <81,10> [Besche, Eick, O'Brien] determine the capitulation type (1,0,0,0) (TKT with fixed point 1) of the real quadratic field K in its four unramified cyclic cubic extensions N_iK (i=1,...,4) and the abelian type invariants of the 3class groups Cl(3,K)=(3,3) (whence A269320 is a subsequence of A269319) and [Cl(3,N_i)]=[(9,3),(3,3),(3,3),(3,3)] (TTT or IPAD). Conversely, the group G=<81,10> is determined uniquely by its Artin pattern AP(G)=(TTT,TKT) [Mayer, 2014, Fig.3.1, Tbl.4.1], where TKT, TTT, IPAD are abbreviations for transfer kernel type, transfer target type, indexp abelianization data, respectively [Mayer, 2016]. The MAGMA program has to determine both components of the Artin pattern, since there are infinitely many 3groups with TKT a.2, (1,0,0,0), and there are three groups with IPAD [(3,3);(9,3),(3,3),(3,3),(3,3)]. (This is one of the few cases where the "Principalization algorithm via class group structure" [Mayer, 2014] is unable to distinguish between TKT a.2, (1,0,0,0), and a.3, (2,0,0,0). A zero always denotes a total capitulation.) Since the group G=<81,10> has derived length dl(G)=2, the Hilbert 3class field tower of these real quadratic fields consists of exactly two stages.
It must be pointed out that the MAGMA program must be executed on a machine with Linux operating system, since the MAGMA versions starting with V2.218 are not available for Mac OS and MS Windows. MAGMA version V2.217 will fail at discriminant 751657. (Bug corrected 13 November 2015 by the MAGMA group, Univ. of Sydney, on our request. See the Change Log of V2.218.)
The MAGMA program requires A269319 as its input data.


LINKS

Table of n, a(n) for n=1..34.
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.
F.P. Heider and B. Schmithals, Zur Kapitulation der Idealklassen in unverzweigten primzyklischen Erweiterungen, J. Reine Angew. Math. 336 (1982), 125.
D. C. Mayer, All known examples for principalization types, Memorial 2009.
D. C. Mayer, Principalization algorithm via class group structure, J. ThÃ¨or. Nombres Bordeaux 26 (2014), no. 2, 415464.
D. C. Mayer, Artin transfer patterns on descendant trees of finite pgroups, Adv. Pure Math. 6 (2016), no. 2, 66104.


EXAMPLE

The leading two terms, 72329, 94636, have been identified by [Heider, Schmithals] (up to 10^5). The first 34 terms up to 10^6 have been determined in the time between 2006 and 2009 [Mayer, 2009]. The 535 terms up to 10^7, computed 13 January 2016, are not published officially yet. They constitute a refinement of the numerical results in [Mayer, 2014].


PROG

(MAGMA) SetClassGroupBounds("GRH"); p:=3; dList:=A269319; for d in dList do
ZX<X>:=PolynomialRing(Integers()); K:=NumberField(X^2d); 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.(n1)) div p)*C.(n1); 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..p1] 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<Cx`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; TKT:=[]; for j in [1..#sR] do Collector:=[]; I:=sR[j]!!mC(g); if IsPrincipal(I) then Append(~Collector, sI[1]); end if;
I:=sR[j]!!mC(h); if IsPrincipal(I) then Append(~Collector, sI[2]); end if;
for e in [1..p1] do I := sR[j]!!mC(g+e*h); if IsPrincipal(I) then Append(~Collector, sI[e+2]); end if; end for;
if (2 le #Collector) then Append(~TKT, 0); else Append(~TKT, Collector[1]); end if; end for; TAB:=[]; image:=[]; fixedpoints:=0; capitulations:=0;
for j in [1..#TKT] do if (j eq TKT[j]) then Append(~TAB, "A"); fixedpoints:=fixedpoints+1;
elif (0 eq TKT[j]) then Append(~TAB, "A"); capitulations:=capitulations+1;
else Append(~TAB, "B"); end if;
if not (TKT[j] in image) then Append(~image, TKT[j]); end if; end for;
if (2 eq polarization2) and (3 eq polarization1) and (0 eq epsilon) and (1 eq fixedpoints) then printf "%o, ", d; end if; end for;
On 04 April 2016, MAGMA version V2.2111 was released for Mac OS, and is able to execute the PROG.  Daniel Constantin Mayer, Apr 16 2016


CROSSREFS

Subsequence of A269319.
Sequence in context: A254904 A257206 A229322 * A126658 A251340 A183788
Adjacent sequences: A269317 A269318 A269319 * A269321 A269322 A269323


KEYWORD

nonn,hard


AUTHOR

Daniel Constantin Mayer, Mar 09 2016


STATUS

approved



