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A247696
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Minimal absolute discriminants a(n) of complex quadratic fields with 3-class group of type (3,3), 3-principalization type E.9 (2334), and second 3-class group G of odd nilpotency class cl(G)=2(n+2)+1.
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6
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OFFSET
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0,1
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COMMENTS
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The 3-principalization type (transfer kernel type, TKT) E.9 (2334) is not a permutation and has two fixed points.
The nilpotency condition cl(G)=2n+5 for the second 3-class group is equivalent to a transfer target type, TTT (called IPAD by Boston, Bush and Hajir) of the shape [(3,9),(3^{n+2},3^{n+3}),(3,9)^2].
The second 3-class group G is one of two vertices of depth 1 on the coclass tree with root SmallGroup(243,8) contained in the coclass graph G(3,2).
All these fields possess a Hilbert 3-class field tower of exact length 3.
A247696 is an extremely sparse subsequence of A242878 and it is exceedingly hard to compute a(n) for n>0.
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LINKS
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EXAMPLE
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For a(0)=9748, we have the ground state of TKT E.9 with TTT [(3,9),(9,27),(3,9)^2] and cl(G)=5.
For a(1)=297079, we have the first excited state of TKT E.9 with TTT [(3,9),(27,81),(3,9)^2] and cl(G)=7.
For a(2)=1088808, we have the second excited state of TKT E.9 with TTT [(3,9),(81,243),(3,9)^2] and cl(G)=9.
For a(3)=11091140, we have the third excited state of TKT E.9 with TTT [(3,9),(243,729),(3,9)^2] and cl(G)=11.
For a(4)=94880548, we have the fourth excited state of TKT E.9 with TTT [(3,9),(729,2187),(3,9)^2] and cl(G)=13.
a(0) and a(1) are due to D. C. Mayer (2012).
a(2), a(3) and a(4) are due to N. Boston, M. R. Bush and F. Hajir (2013).
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CROSSREFS
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KEYWORD
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hard,more,nonn
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AUTHOR
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STATUS
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approved
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