A359872 Absolute discriminants of imaginary quadratic number fields with elementary bicyclic 7-class group (7,7).

63499, 118843, 124043, 149519, 159592, 170679, 183619, 185723, 220503, 226691, 227387, 227860, 236931, 240347, 240655, 247252, 260111, 268739, 272179, 275636, 294935, 299627, 301211, 308531, 318547, 346883, 361595, 366295, 373655, 465719, 489576, 491767, 501576, 506551, 511988, 518879, 528243, 546792, 553791

Offset: 1

Author

Daniel Constantin Mayer, Jan 16 2023

Keywords

• easy
• nonn

Comments

The maximal unramified pro-7-extension, that is, the Hilbert 7-class field tower, of these imaginary quadratic fields must have a Schur sigma-group as its Galois group. The tower has an unbounded number of stages at least equal to two, and may even be infinite.

Examples

			On 06 January 2012, Daniel C. Mayer determined the abelian type invariants (ATI), and thus indirectly the coarse capitulation type, of the eight unramified cyclic septic extensions for all 70 discriminants in the range between -63499 and -751288. On page 133 of his 2012 Ph.D. thesis, Tobias Bembom independently recomputed the capitulation for the two discriminants -63499 and -159592. In the time between 09 and 16 August 2014, Daniel C. Mayer computed the fine capitulation type of all 94 discriminants in the range -63499 and -991720 without any hit of the identity capitulation. Since the fine capitulation requires much more CPU time than the ATI, Mayer conducted an extensive search for the identity capitulation, identified by eight ATI of the shape (7,7,7), in the range from 10^6 to 6578723, with an eventual successful hit of the identity capitulation for -5073691 (the 555th term of A359872) on 26 October 2019 (see A359296).
		

References

• Daniel 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

• Tobias Bembom, The capitulation problem in class field theory, Dissertation, Univ. Göttingen, 2012. (Sec. 6.3, p. 128)
• Daniel C. Mayer, Heptadic quantum class groups
• Daniel C. Mayer, The distribution of second p-class groups on coclass graphs, arXiv:1403.3833 [math.NT], 2014. (Sec. 3.5.4, pp. 450-451)

Crossrefs

Cf. A359296 (subsequence), A242863 (3,3), A359871 (5,5).

Programs

Magma

for d := 2 to 10^6 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 ([7,7] eq pPrimaryInvariants(C,7)) then d, ", "; end if; end if; end for;