A359871 Absolute discriminants of imaginary quadratic number fields with elementary bicyclic 5-class group (5,5).
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
Offset: 1
Examples
On page 22 of their 1982 paper, Franz-Peter 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 5-class group is non-elementary bicyclic of type (25,5).
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 p-class groups on coclass graphs, J. Théor. Nombres Bordeaux 25 (2013), no. 2, 401-456. (Sec. 3.5.2, p. 448)
Links
- T. Bembom, The capitulation problem in class field theory, Dissertation, Univ. Göttingen, 2012. (Sec. 6.3, p. 128)
- D. C. Mayer, Pentadic quantum class groups
- D. C. Mayer, The distribution of second p-class groups on coclass graphs, arXiv:1403.3833 [math.NT], 2014. (Sec. 3.5.2-3.5.3, pp. 448-450)
Programs
-
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;
Comments