This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A380104 #9 Jan 25 2025 23:01:19 %S A380104 30,90,418,1626 %N A380104 Minimal conductors c of complex dihedral normal closures K = L(zeta_3) of pure cubic number fields L = Q(d^1/3), d > 1 cubefree, 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 coclass cc(M)=0,1,2,3,... %C A380104 The coclass cc(M) for one of the fields K with conductor c = a(n) is n-1, and for each field K with conductor c < a(n), the coclass cc(M) is less than n-1. Among the 3-groups M of coclass cc(M)=1, we distinguish the abelian 3-group A=(3,3) by formally putting cc(A)=0, in accordance with the FORMULA. This is a significant difference to quadratic fields, which are firstly uniquely determined by their discriminant, and secondly cannot have an abelian second 3-class group. %H A380104 D. C. Mayer, <a href="http://arxiv.org/abs/1403.3833">The distribution of second p-class groups on coclass graphs</a>, arXiv:1403.3833 [math.NT], 2014; J. Théor. Nombres Bordeaux 25 (2013), 401-456. %H A380104 Daniel Constantin Mayer, <a href="/A380104/a380104.m.txt">Magma program "PureCoClass.m" with endless loop</a> %F A380104 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(E_2)), where h_3(E_2) is the second largest 3-class number among the four unramified cyclic cubic extensions E_1,..,E_4 of the complex dihedral field K, and log_3 denotes the logarithm with respect to the basis 3. An exception is the abelian 3-group A=(3,3) with correct cc(A)=1, where the FORMULA yields cc(A)=0. %e A380104 We have M abelian for c=30=2*3*5 (a singlet), cc(M)=1 for c=90=2*3^2*5 (two fields in a quartet), cc(M)=2 for c=418=2*11*19, cc(M)=3 for c=1626=2*3*271. %o A380104 (Magma) // See Links section. %Y A380104 Analog of A379524 for real quadratic fields. %K A380104 nonn,hard,more %O A380104 1,1 %A A380104 _Daniel Constantin Mayer_, Jan 15 2025