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

4973316, 5073691

Offset: 1

Author

Daniel Constantin Mayer, Dec 24 2022

Keywords

• nonn
• more
• hard
• bref

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.

Examples

			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).
		

Links

• 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.

Crossrefs

Cf. A359291.