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

Author

Daniel Constantin Mayer, Jan 12 2025

Keywords

• nonn
• hard
• more

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.

Examples

			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.
		

Links

• D. C. 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

Crossrefs

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

Programs

Magma

// See Links section.

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.