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These real quadratic fields have class number divisible by 3 but not divisible by 9. Therefore, this sequence does not contain the discriminant 1129, since the corresponding quadratic field has cyclic 3-class group (9). However, this sequence contains the discriminant 697 whose corresponding quadratic field has class number 6=2*3. Note that 697 is not a member of the sequence A094612, where an exact class number 3 is required.

According to the Artin reciprocity law of class field theory, these real quadratic fields possess a cyclic cubic Hilbert 3-class field as their maximal unramified abelian 3-extension.

According to the Hasse formula d(K)=f^2*d for the discriminant d(K) of a non-Galois totally real cubic field in terms of the conductor f and the associated discriminant d of the real quadratic subfield of the normal closure of K, the sequence A006832 contains all discriminants d of real quadratic fields with class number divisible by 3, since they give rise to a totally real cubic field with conductor f=1 and discriminant d(K)=f^2*d=d. In particular, A006832 contains A250235.

Square terms in A006832.

Numbers of the form k^2 where A160498(k) >= 2.

Each term k^2 is associated with A343003(k) cyclic cubic fields.

Let D be a discriminant of a cubic field F, then F is a cyclic cubic field if and only if D is a square. For D = k^2, k must be of the form (p_1)*(p_2)*...*(p_t) or 9*(p_1)*(p_2)*...*(p_{t-1}) with distinct primes p_i == 1 (mod 3), in which case there are exactly 2^(t-1) = 2^(omega(k)-1) (cyclic) cubic fields with discriminant D. See Page 17, Theorem 2.7 of the Ka Lun Wong link.

Numbers k such that k^2 is in A006832.

Numbers k such that A160498(k) >= 2.

Each term k is associated with A343003(k) cyclic cubic fields.

Let D be a discriminant of a cubic field F, then F is a cyclic cubic field if and only if D is a square. For D = k^2, k must be of the form (p_1)*(p_2)*...*(p_t) or 9*(p_1)*(p_2)*...*(p_{t-1}) with distinct primes p_i == 1 (mod 3), in which case there are exactly 2^(t-1) = 2^(omega(k)-1) (cyclic) cubic fields with discriminant D. See Page 17, Theorem 2.7 of the Ka Lun Wong link.

A cubic field is cyclic if and only if its discriminant is a square. Hence all terms are squares.

Numbers of the form k^2 where A160498(k) = 2.

Numbers of the form k^2 where k is in A002476 U {9}. That is to say, numbers of the form k^2 where k = 9 or is a prime congruent to 1 modulo 3.

In general, there are exactly 2^(t-1) (cyclic) cubic fields with discriminant k^2 if and only if k is of the form (p_1)*(p_2)*...*(p_t) or 9*(p_1)*(p_2)*...*(p_{t-1}) with distinct primes p_i == 1 (mod 3), See A343000 for more detailed information.

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 A250238, 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(81,9) with transfer kernel type a.1, (0,0,0,0), transfer target type [(3,9),(3,3)^3] and coclass 1. Actually, this is the ground 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(81,9) has two-generated commutator subgroup G' of type (3,3).

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