A380102 Minimal absolute discriminants |d| of imaginary quadratic number fields K = Q(sqrt(d)), d < 0, with elementary bicyclic 3-class group Cl_3(K)=(3,3) and second 3-class group M=Gal(F_3^2(K)/K) of assigned even coclass cc(M)=2,4,6,8,...

3896, 27156, 423640, 99888340

Offset

1,1

Comments

The coclass cc(M) for the field K with discriminant d = -a(n) is 2*n, and for each field K with absolute discriminant |d| < a(n), the coclass cc(M) is less than 2*n.

Links

Table of n, a(n) for n=1..4.

Siham Aouissi and Daniel C. Mayer, Coclass of the second 3-class group, arXiv:2508.17510 [math.NT], 2025. See p. 10.

Daniel Constantin Mayer, The distribution of second p-class groups on coclass graphs, arXiv:1403.3833 [math.NT], 2014; J. Théor. Nombres Bordeaux 25 (2013), 401-456.

Daniel Constantin Mayer, Magma program "ImagCoClass.m" with endless loop

Formula

According to Theorem 3.12 on page 435 of "The distribution of second p-class groups on coclass graphs", the coclass of the group M is given by cc(M)+1=log_3(h_3(L_2)), where h_3(L_2) is the second largest 3-class number among the four unramified cyclic cubic extensions L_1,..,L_4 of the quadratic field K, and log_3 denotes the logarithm with respect to the basis 3.

Example

The coclass cannot be odd for imaginary quadratic fields. We have cc(M)=2 for d=-3896, cc(M)=4 for d=-27156, cc(M)=6 for d=-423640, cc(M)=8 for d=-99888340.

Prog

(Magma) // See Links section.

Crossrefs

Cf. A242862, A242863 (supersequences). Analog of A379524 for real quadratic fields.

Sequence in context: A242863 A247691 A242873 * A135202 A204147 A252138

Adjacent sequences: A380099 A380100 A380101 * A380103 A380104 A380105

Keyword

nonn,hard,more

Author

Daniel Constantin Mayer, Jan 12 2025

Status

approved