A250237 Fundamental discriminants d uniquely characterizing all complex biquadratic fields Q(sqrt(-3),sqrt(d)) which have 3-class group of type (3,3) and abelian 3-class field tower of length 1.

229, 257, 316, 321, 473, 568, 697, 761, 785, 892, 940, 985, 993, 1016, 1229, 1304, 1345, 1384, 1436, 1509, 1765, 1929, 2024, 2089, 2101, 2233, 2296, 2505, 2920, 2993

Offset: 1

Author

Daniel Constantin Mayer, Nov 15 2014

Keywords

• hard
• nonn

Comments

This is the beginning of an investigation of the maximal unramified pro-3 extension of complex bicyclic biquadratic fields containing the third roots of unity which have an elementary 3-class group of rank two.

For the discriminants d in A250237, the 3-class field tower of K=Q(sqrt(-3),sqrt(d)) is abelian, terminating with the first stage at the Hilbert 3-class field already. An equivalent condition is that the second 3-class group G of K is given by G=SmallGroup(9,2). Another equivalent condition in terms of a fundamental system of units has been given by Yoshida.

Examples

			A250237 covers the dominant part of A250236. The smallest discriminant d in A250236 with non-abelian 3-class field tower of length bigger than 1 is given by d=A250238(1)=469, the initial term of the disjoint sequence A250238.
		

References

• 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 4 package, available also in MAGMA.

Links

• D. C. Mayer, The second p-class group of a number field, Int. J. Number Theory 8 (2) (2012), 471-505.
• E. Yoshida, On the 3-class field tower of some biquadratic fields, Acta Arith. 107 (2003), no. 4, 327-336.

Crossrefs

A006832, A250235, A250236 are supersequences, A250238, A250239, A250240, A250241, A250242 are disjoint sequences.

Programs

Magma

SetClassGroupBounds("GRH"); for n := 229 to 3000 do cnd := false; if (1 eq n mod 4) and IsSquarefree(n) then cnd := true; end if; if (0 eq n mod 4) then r := n div 4; if IsSquarefree(r) and ((2 eq r mod 4) or (3 eq r mod 4)) then cnd := true; end if; end if; if (true eq cnd) then R := QuadraticField(n); E := QuadraticField(-3); K := Compositum(R,E); C, mC := ClassGroup(K); if ([3,3] eq pPrimaryInvariants(C, 3)) then s := Subgroups(C: Quot := [3]); a := [AbelianExtension(Inverse(mq)*mC) where _, mq := quo : x in s]; b := [NumberField(x) : x in a]; d := [MaximalOrder(x) : x in a]; b := [AbsoluteField(x) : x in b]; c := [MaximalOrder(x) : x in b]; c := [OptimizedRepresentation(x) : x in b]; b := [NumberField(DefiningPolynomial(x)) : x in c]; a := [Simplify(LLL(MaximalOrder(x))) : x in b]; if IsNormal(b[2]) then H := Compositum(NumberField(a[1]),NumberField(a[2])); else H := Compositum(NumberField(a[1]),NumberField(a[3])); end if; O := MaximalOrder(H); CH := ClassGroup(LLL(O)); if (0 eq #pPrimaryInvariants(CH, 3)) then n, ", "; end if; end if; end if; end for;