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A244575
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Absolute discriminants of complex quadratic fields with 3-class group of type (3,3,3), thus having an infinite class tower.
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1
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4447704, 4472360, 4818916, 4897363, 5067967, 5769988, 7060148, 8180671, 8721735, 8819519, 8992363, 9379703, 9487991, 9778603
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OFFSET
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1,1
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COMMENTS
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I do not know who actually discovered a(1)=4447704. It is neither mentioned in Diaz y Diaz (1973) nor in Buell (1976). Maybe it can be found in Shanks (1976). MAGMA required 18 hours CPU time for the first 14 terms.
Meanwhile, it came to my knowledge that a(1)=4447704 and all the other terms below 10^7 are given in Appendice 1, pp. 66-77, of the Thesis of Diaz y Diaz (1978). a(1) is not contained in Shanks (1976). - Daniel Constantin Mayer, Sep 28 2014.
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REFERENCES
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F. Diaz y Diaz, Sur le 3-rang des corps quadratiques, Publ. math. d'Orsay, No. 78-11, Univ. Paris-Sud (1978).
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LINKS
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EXAMPLE
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a(1)=4447704 is the minimal absolute discriminant with elementary abelian 3-class group of type (3,3,3), whereas the smaller A244574(1)=3321607 has non-elementary (9,3,3).
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PROG
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(Magma) for d := 1 to 10^7 do a := false; if (3 eq d mod 4) and IsSquarefree(d) then a := true; end if; if (0 eq d mod 4) then r := d div 4; if IsSquarefree(r) and ((2 eq r mod 4) or (1 eq r mod 4)) then a := true; end if; end if; if (true eq a) then K := QuadraticField(-d); C := ClassGroup(K); if ([3, 3, 3] eq pPrimaryInvariants(C, 3)) then d, ", "; end if; end if; end for;
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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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