A359296 Absolute discriminants of imaginary quadratic fields with elementary bicyclic 7-class group and capitulation type the identity permutation.

4973316, 5073691

Offset

1,1

Comments

An algebraic number field with this capitulation type has a 7-class field tower of precise length 2 with Galois group isomorphic to the Schur sigma-group SmallGroup(16807,7). It is a solution to the problem posed by Olga Taussky-Todd in 1970.

Links

Table of n, a(n) for n=1..2.

D. C. Mayer, The distribution of second p-class groups on coclass graphs, J. Théor. Nombres Bordeaux 25 (2013), no. 2, 401-456. Sec. 3.5.4, p. 450.

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.

Example

The second, respectively first, imaginary quadratic field with 7-class group (7,7) and identity capitulation (12345678) has discriminant -5073691, respectively -4973316, and was discovered by Daniel C. Mayer on 26 October 2019, respectively 09 November 2019. It has ordinal number 555, respectively 545, in the sequence of all imaginary quadratic fields with 7-class group (7,7).

Crossrefs

Cf. A359291.

Sequence in context: A157804 A151646 A210318 * A227155 A106785 A034607

Adjacent sequences: A359293 A359294 A359295 * A359297 A359298 A359299

Keyword

nonn,more,hard,bref

Author

Daniel Constantin Mayer, Dec 24 2022

Status

approved