cp's OEIS Frontend

Generally, the 3-class ranks s of the real quadratic field R=Q(sqrt(d)) and r of the complex quadratic field C=Q(sqrt(-3d)) are related by the inequalities s <= r <= s+1. This reflection theorem was proved by Scholz and independently by Reichardt using a combination of class field theory and Kummer theory over the bicyclic biquadratic compositum K=R*E of R with Eisenstein's cyclotomic field E=Q(sqrt(-3)) of third roots of unity.

In particular, the biquadratic field K=Q(sqrt(-3),sqrt(d)) has a 3-class group of type (3,3) if and only if s=r and R and C both have 3-class groups of type (3).

Therefore, the discriminants in the sequence A250236 uniquely characterize all complex biquadratic fields containing the third roots of unity which have an elementary 3-class group of rank two.

The discriminant of K=R*E is given by d(K)=3^2*d^2 if gcd(3,d)=1 and simply by d(K)=d^2 if 3 divides d.

This is the beginning of an investigation of the maximal unramified pro-3 extension of complex bicyclic biquadratic fields containing the third roots of unity which have an elementary 3-class group of rank two.

For the discriminants d in A250237, the 3-class field tower of K=Q(sqrt(-3),sqrt(d)) is abelian, terminating with the first stage at the Hilbert 3-class field already. An equivalent condition is that the second 3-class group G of K is given by G=SmallGroup(9,2). Another equivalent condition in terms of a fundamental system of units has been given by Yoshida.

For the discriminants d in A250239, the 3-class field tower of K=Q(sqrt(-3),sqrt(d)) has exactly two stages and the second 3-class group G of K is given by the metabelian 3-group G=SmallGroup(729,95) with transfer kernel type a.1, (0,0,0,0), transfer target type [(9,27),(3,3)^3] and coclass 1. This is the first excited state on the coclass-1 graph.

The reason the 3-class field tower of K must stop at the second Hilbert 3-class field is Blackburn's Theorem on two-generated 3-groups G whose commutator subgroup G' also has two generators. In fact, the group G=SmallGroup(729,95) has commutator subgroup G'=(9,9), two-generated.

For the discriminants d in A250240, the 3-class field tower of K=Q(sqrt(-3),sqrt(d)) has at least three stages and the second 3-class group G of K is given by G=SmallGroup(729,37), which is called the non-CF group A by Ascione, Havas and Leedham-Green. It has many properties (transfer kernel type b.10, (0,0,4,3), and transfer target type [(3,9)^2,(3,3,3)^2]) coinciding with those of SmallGroup(729,34), called the non-CF group H. Both are immediate descendants of SmallGroup(243,3) and can only be distinguished by their commutator subgroup G', which is of type (3,3,9) for A, and (3,3,3,3) for H.

Since the verification of the structure of G' requires computation of the 3-class group of the Hilbert 3-class field of K, which is of absolute degree 36 over Q, the construction of A250240 is extremely tough.

Whereas the metabelian 3-group A is rather well behaved, possessing six terminal immediate descendants only, the notorious group H is famous for giving rise to three infinite coclass trees with non-metabelian mainlines and horrible complexity.

In 66.2 hours of CPU time, Magma computed all 34 discriminants d up to the bound 50000. Starting with d=38057, Magma begins to struggle considerably, since an increasing amount of time (NOT included above) is used for swapping to the hard disk. - Daniel Constantin Mayer, Dec 02 2014

The given Magma PROG works correctly up to 10000. However, for ranges beyond 10000, a complication arises, since the non-CF group B = SmallGroup(729,40) also has a commutator subgroup of type (3,3,9) and must be sifted with the aid of its different transfer target type [(9,9),(3,9),(3,3,3)^2]. Up to 50000, this occurs three times for d in {17609,30941,31516}. - Daniel Constantin Mayer, Dec 05 2014

The group G=SmallGroup(729,37) has p-multiplicator rank m(G)=5. By Theorem 6 of I. R. Shafarevich (with misprint corrected) the relation rank of the 3-class tower group H is bounded by r(H) <= d(H) + r + 1 = 2 + 1 + 1 = 4, where d(H) denotes the generator rank of H and r is the torsionfree unit rank of K. Thus, G with r(G) >= m(G) = 5 cannot be the 3-class tower group of K and the tower must have at least three stages. - Daniel Constantin Mayer, Sep 24 2015

For the discriminants d in A250241, the 3-class field tower of K=Q(sqrt(-3),sqrt(d)) has at least three stages and the second 3-class group G of K is given by G=SmallGroup(729,34), which is called the non-CF group H by Ascione, Havas and Leedham-Green. It has properties very similar to those of SmallGroup(729,37), called the non-CF group A. Both are immediate descendants of SmallGroup(243,3) and can only be distinguished by their commutator subgroup G', which is of type (3,3,3,3) for H, and (3,3,9) for A.

Since the verification of the structure of G' requires computation of the 3-class group of the Hilbert 3-class field of K, which is of absolute degree 36 over Q, the construction of A250241 is extremely tough.

In 40.5 hours of CPU time, Magma computed all 25 discriminants d up to the bound 50000. Starting with d=37916, Magma begins to struggle considerably, since an increasing amount of time (NOT included above) is used for swapping to the hard disk. A very powerful machine would be required for continuing beyond 50000. - Daniel Constantin Mayer, Dec 02 2014

The group G=SmallGroup(729,34) has p-multiplicator rank m(G)=5. By Theorem 6 of I. R. Shafarevich (with misprint corrected) the relation rank of the 3-class tower group H is bounded by r(H) <= d(H) + r + 1 = 2 + 1 + 1 = 4, where d(H) denotes the generator rank of H and r is the torsionfree unit rank of K. Thus, G with r(G) >= m(G) = 5 cannot be the 3-class tower group of K and the tower must have at least three stages. - Daniel Constantin Mayer, Sep 24 2015

For the discriminants d in A250242, the 3-class field tower of K=Q(sqrt(-3),sqrt(d)) has at least three stages and the second 3-class group G of K is of coclass 2, given by either SmallGroup(2187,247)-#1;5 or SmallGroup(2187,247)-#1;9. (Note that these groups G are of order 6561 and lie outside of the SmallGroups Library, whence we must use the terminology for descendants defined in the ANUPQ package of Magma and GAP.) Both have transfer kernel type b.10, (0,0,4,3), and full transfer target type [(3,3);(9,27),(3,9),(3,3,3)^2;(3,9,27)]). Both are immediate descendants of the mainline group SmallGroup(2187,247) on the coclass tree with root SmallGroup(729,40) and cannot be distinguished by any known arithmetical criteria. Their commutator subgroup G' is of type (3,9,27).

Since the verification of the structure of G' (used by the given Magma PROG) requires computation of the 3-class group of the Hilbert 3-class field of K, which is of absolute degree 36 over Q, the construction of A250242 is rather expensive.

Both groups G=SmallGroup(2187,247)-#1;5 and G=SmallGroup(2187,247)-#1;9 have p-multiplicator rank m(G)=5. By Theorem 6 of I. R. Shafarevich (with misprint corrected) the relation rank of the 3-class tower group H is bounded by r(H) <= d(H) + r + 1 = 2 + 1 + 1 = 4, where d(H) denotes the generator rank of H and r is the torsionfree unit rank of K. Thus, G with r(G) >= m(G) = 5 cannot be the 3-class tower group of K and the tower must have at least three stages. - Daniel Constantin Mayer, Sep 24 2015