

A359871


Absolute discriminants of imaginary quadratic number fields with elementary bicyclic 5class group (5,5)


2



11199, 12451, 17944, 30263, 33531, 37363, 38047, 39947, 42871, 53079, 54211, 58424, 61556, 62632, 63411, 64103, 65784, 66328, 67031, 67063, 67128, 69811, 72084, 74051, 75688, 83767, 84271, 85099, 85279, 87971, 89751, 90795, 90868, 92263, 98591, 99031, 99743
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OFFSET

1,1


COMMENTS

The maximal unramified pro5extension, that is, the Hilbert 5class field tower, of these imaginary quadratic fields must have a Schur sigmagroup as its Galois group. The tower has an unbounded number of stages at least equal to two, and may even be infinite.


REFERENCES

F.P. Heider, B. Schmithals, Zur Kapitulation der Idealklassen in unverzweigten primzyklischen Erweiterungen, J. reine angew. Math. 336 (1982), 1  25.
D. C. Mayer, The distribution of second pclass groups on coclass graphs, J. Théor. Nombres Bordeaux 25 (2013), no. 2, 401456. (Sec. 3.5.2, p. 448)


LINKS



EXAMPLE

On page 22 of their 1982 paper, FranzPeter Heider and Bodo Schmithals gave the smallest prime discriminant 12451 and determined two of the six capitulation kernels in unramified cyclic quintic extensions. On 03 November 2011, Daniel C. Mayer determined the abelian type invariants, and thus indirectly the coarse capitulation type, of these six extensions for all 37 discriminants in the range between 11199 and 99743, with computational aid by Claus Fieker. In particular, 89751 was the minimal occurrence of the identity capitulation (see A359291). In his 2012 Ph.D. thesis, Tobias Bembom independently recomputed the capitulation in this range, without being able to detect the identity capitulation for 89751. It must be pointed out that in his table on pages 129 and 130, the minimal discriminant 11199=3*3733 is missing, whereas the discriminant 81287 is superfluous and must be cancelled, since its 5class group is nonelementary bicyclic of type (25,5).


PROG

(Magma)
for d := 2 to 10^5 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 ([5, 5] eq pPrimaryInvariants(C, 5)) then d, ", "; end if; end if; end for;


CROSSREFS



KEYWORD

easy,nonn


AUTHOR



STATUS

approved



