A359291
Absolute discriminants of imaginary quadratic fields with elementary bicyclic 5-class group and capitulation type the identity permutation.
Original entry on oeis.org
89751, 235796, 1006931, 1996091, 2187064
Offset: 1
The first imaginary quadratic field with 5-class group (5,5) and identity capitulation (123456) has discriminant -89751 and was discovered by Daniel C. Mayer on 03 November 2011. It has ordinal number 31 in the sequence A359871 of all imaginary quadratic fields with 5-class group (5,5). The discriminant -89751 appears in the table on page 130 in the Ph.D. thesis of Tobias Bembom, 2012. However, contrary to his assertion in Remark 2 on page 129, his method was not able to detect the identity capitulation. Consequently, Bembom only found a (non-identity) permutation (135246) but did not solve Taussky's problem.
- A. Azizi et al., 5-Class towers of cyclic quartic fields arising from quintic reflection, Ann. math. Québec 44 (2020), 299-328. (p. 314)
- 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)
- A. Azizi et al., 5-Class towers of cyclic quartic fields arising from quintic reflection, arXiv:1909.03407 [math.NT], 2019.
- T. Bembom, The capitulation problem in class field theory, Dissertation, Univ. Göttingen, 2012. (Sec. 6.3, p. 128)
- D. C. Mayer, The distribution of second p-class groups on coclass graphs, arXiv:1403.3833 [math.NT], 2014.
- O. Taussky-Todd, A remark concerning Hilbert's Theorem 94, J. reine angew. Math. 239/240 (1970), 435-438.
A359872
Absolute discriminants of imaginary quadratic number fields with elementary bicyclic 7-class group (7,7).
Original entry on oeis.org
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
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).
- 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)
- 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)
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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;
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