| Displaying 1-6 of 6 results found.
page
1
Fundamental discriminants d such that the real quadratic field Q(sqrt(d)) and the complex quadratic field Q(sqrt(-3d)) both have cyclic 3-class groups of order 3.
+10
7
229, 257, 316, 321, 469, 473, 568, 697, 761, 785, 892, 940, 985, 993, 1016, 1229, 1304, 1345, 1384, 1436, 1489, 1509, 1708, 1765, 1929, 1937, 2024, 2089, 2101, 2177, 2233, 2296, 2505, 2557, 2589, 2677, 2920, 2941, 2981, 2993
COMMENTS
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.
REFERENCES
G. Eisenstein, Beweis des Reciprocitätssatzes für die cubischen Reste in der Theorie der aus den dritten Wurzeln der Einheit zusammengesetzten Zahlen, J. Reine Angew. Math. 27 (1844), 289-310.
EXAMPLE
A250236 is a proper subsequence of A250235. For instance, it does not contain the discriminant d=733, resp. 1373, although the corresponding real quadratic field R=Q(sqrt(d)) has 3-class group (3). The reason is that the 3-dual complex quadratic field C=Q(sqrt(-3d)) of R has 3-class group (9), resp. (27).
PROG
(Magma) for d := 2 to 3000 do a := false; if (1 eq d mod 4) and IsSquarefree(d) then a := true; end if; if (0 eq d mod 4) then r := d div 4; if IsSquarefree(r) and ((2 eq r mod 4) or (3 eq r mod 4)) then a := true; end if; end if; if (true eq a) then R := QuadraticField(d); E := QuadraticField(-3); K := Compositum(R, E); C := ClassGroup(K); if ([3, 3] eq pPrimaryInvariants(C, 3)) then d, ", "; end if; end if; end for;
Fundamental discriminants d uniquely characterizing all complex biquadratic fields Q(sqrt(-3),sqrt(d)) which have 3-class group of type (3,3) and abelian 3-class field tower of length 1.
+10
7
229, 257, 316, 321, 473, 568, 697, 761, 785, 892, 940, 985, 993, 1016, 1229, 1304, 1345, 1384, 1436, 1509, 1765, 1929, 2024, 2089, 2101, 2233, 2296, 2505, 2920, 2993
COMMENTS
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.
REFERENCES
H. U. Besche, B. Eick, and E. A. O'Brien, The SmallGroups Library - a Library of Groups of Small Order, 2005, an accepted and refereed GAP 4 package, available also in MAGMA.
EXAMPLE
A250237 covers the dominant part of A250236. The smallest discriminant d in A250236 with non-abelian 3-class field tower of length bigger than 1 is given by d= A250238(1)=469, the initial term of the disjoint sequence A250238.
PROG
(Magma)SetClassGroupBounds("GRH"); for n := 229 to 3000 do cnd := false; if (1 eq n mod 4) and IsSquarefree(n) then cnd := true; end if; if (0 eq n mod 4) then r := n div 4; if IsSquarefree(r) and ((2 eq r mod 4) or (3 eq r mod 4)) then cnd := true; end if; end if; if (true eq cnd) then R := QuadraticField(n); E := QuadraticField(-3); K := Compositum(R, E); C, mC := ClassGroup(K); if ([3, 3] eq pPrimaryInvariants(C, 3)) then s := Subgroups(C: Quot := [3]); a := [AbelianExtension(Inverse(mq)*mC) where _, mq := quo<C|x`subgroup> : x in s]; b := [NumberField(x) : x in a]; d := [MaximalOrder(x) : x in a]; b := [AbsoluteField(x) : x in b]; c := [MaximalOrder(x) : x in b]; c := [OptimizedRepresentation(x) : x in b]; b := [NumberField(DefiningPolynomial(x)) : x in c]; a := [Simplify(LLL(MaximalOrder(x))) : x in b]; if IsNormal(b[2]) then H := Compositum(NumberField(a[1]), NumberField(a[2])); else H := Compositum(NumberField(a[1]), NumberField(a[3])); end if; O := MaximalOrder(H); CH := ClassGroup(LLL(O)); if (0 eq #pPrimaryInvariants(CH, 3)) then n, ", "; end if; end if; end if; end for;
Fundamental discriminants d uniquely characterizing all complex biquadratic fields Q(sqrt(-3),sqrt(d)) which have 3-class group of type (3,3) and second 3-class group isomorphic to SmallGroup(81,9).
+10
7
469, 1489, 1708, 1937, 2557, 2941, 2981, 3021, 3173, 3305, 3580, 3592, 3736
COMMENTS
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).
REFERENCES
H. U. Besche, B. Eick, and E. A. O'Brien, The SmallGroups Library - a Library of Groups of Small Order, 2005, an accepted and refereed GAP 4 package, available also in MAGMA.
PROG
(Magma)SetClassGroupBounds("GRH"); for n := 469 to 10000 do cnd := false; if (1 eq n mod 4) and IsSquarefree(n) then cnd := true; end if; if (0 eq n mod 4) then r := n div 4; if IsSquarefree(r) and ((2 eq r mod 4) or (3 eq r mod 4)) then cnd := true; end if; end if; if (true eq cnd) then R := QuadraticField(n); E := QuadraticField(-3); K := Compositum(R, E); C, mC := ClassGroup(K); if ([3, 3] eq pPrimaryInvariants(C, 3)) then s := Subgroups(C: Quot := [3]); a := [AbelianExtension(Inverse(mq)*mC) where _, mq := quo<C|x`subgroup> : x in s]; b := [NumberField(x) : x in a]; d := [MaximalOrder(x) : x in a]; b := [AbsoluteField(x) : x in b]; c := [MaximalOrder(x) : x in b]; c := [OptimizedRepresentation(x) : x in b]; b := [NumberField(DefiningPolynomial(x)) : x in c]; a := [Simplify(LLL(MaximalOrder(x))) : x in b]; if IsNormal(b[2]) then H := Compositum(NumberField(a[1]), NumberField(a[2])); else H := Compositum(NumberField(a[1]), NumberField(a[3])); end if; O := MaximalOrder(H); CH := ClassGroup(LLL(O)); if ([3, 3] eq pPrimaryInvariants(CH, 3)) then n, ", "; end if; end if; end if; end for;
Fundamental discriminants d uniquely characterizing all complex biquadratic fields Q(sqrt(-3),sqrt(d)) which have 3-class group of type (3,3) and second 3-class group isomorphic to SmallGroup(729,95).
+10
7
7453, 8905, 9937, 10069, 14089, 15757, 16737, 17889, 18977, 19869, 20329
COMMENTS
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.
REFERENCES
H. U. Besche, B. Eick, and E. A. O'Brien, The SmallGroups Library - a Library of Groups of Small Order, 2005, an accepted and refereed GAP 4 package, available also in MAGMA.
PROG
(Magma) SetClassGroupBounds("GRH"); for n := 7453 to 20000 do cnd := false; if (1 eq n mod 4) and IsSquarefree(n) then cnd := true; end if; if (0 eq n mod 4) then r := n div 4; if IsSquarefree(r) and ((2 eq r mod 4) or (3 eq r mod 4)) then cnd := true; end if; end if; if (true eq cnd) then R := QuadraticField(n); E := QuadraticField(-3); K := Compositum(R, E); C, mC := ClassGroup(K); if ([3, 3] eq pPrimaryInvariants(C, 3)) then s := Subgroups(C: Quot := [3]); a := [AbelianExtension(Inverse(mq)*mC) where _, mq := quo<C|x`subgroup> : x in s]; b := [NumberField(x) : x in a]; d := [MaximalOrder(x) : x in a]; b := [AbsoluteField(x) : x in b]; c := [MaximalOrder(x) : x in b]; c := [OptimizedRepresentation(x) : x in b]; b := [NumberField(DefiningPolynomial(x)) : x in c]; a := [Simplify(LLL(MaximalOrder(x))) : x in b]; if IsNormal(b[2]) then H := Compositum(NumberField(a[1]), NumberField(a[2])); else H := Compositum(NumberField(a[1]), NumberField(a[3])); end if; O := MaximalOrder(H); CH := ClassGroup(LLL(O)); if ([9, 9] eq pPrimaryInvariants(CH, 3)) then n, ", "; end if; end if; end if; end for;
Fundamental discriminants d uniquely characterizing all complex biquadratic fields Q(sqrt(-3),sqrt(d)) which have 3-class group of type (3,3) and second 3-class group isomorphic to SmallGroup(729,37).
+10
7
2177, 2677, 4841, 6289, 6940, 6997, 8789, 9869, 11324, 17448, 17581, 23192, 23417, 24433, 25741, 26933, 30273, 33765, 34253, 34412, 34968, 35537, 36376, 38037, 38057, 40773, 41224, 42152, 42649, 43176, 43349, 44617, 45529, 47528
COMMENTS
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
REFERENCES
H. U. Besche, B. Eick, and E. A. O'Brien, The SmallGroups Library - a Library of Groups of Small Order, 2005, an accepted and refereed GAP 4 package, available also in MAGMA.
I. R. Shafarevich, Extensions with prescribed ramification points, Publ. Math., Inst. Hautes Études Sci. 18 (1964), 71-95 (Russian). English transl. by J. W. S. Cassels: Am. Math. Soc. Transl., II. Ser., 59 (1966), 128-149. - Daniel Constantin Mayer, Sep 24 2015
PROG
(Magma)SetClassGroupBounds("GRH"); for n := 2177 to 10000 do cnd := false; if (1 eq n mod 4) and IsSquarefree(n) then cnd := true; end if; if (0 eq n mod 4) then r := n div 4; if IsSquarefree(r) and ((2 eq r mod 4) or (3 eq r mod 4)) then cnd := true; end if; end if; if (true eq cnd) then R := QuadraticField(n); E := QuadraticField(-3); K := Compositum(R, E); C, mC := ClassGroup(K); if ([3, 3] eq pPrimaryInvariants(C, 3)) then s := Subgroups(C: Quot := [3]); a := [AbelianExtension(Inverse(mq)*mC) where _, mq := quo<C|x`subgroup> : x in s]; b := [NumberField(x) : x in a]; d := [MaximalOrder(x) : x in a]; b := [AbsoluteField(x) : x in b]; c := [MaximalOrder(x) : x in b]; c := [OptimizedRepresentation(x) : x in b]; b := [NumberField(DefiningPolynomial(x)) : x in c]; a := [Simplify(LLL(MaximalOrder(x))) : x in b]; if IsNormal(b[2]) then H := Compositum(NumberField(a[1]), NumberField(a[2])); else H := Compositum(NumberField(a[1]), NumberField(a[3])); end if; O := MaximalOrder(H); CH := ClassGroup(LLL(O)); if ([3, 3, 9] eq pPrimaryInvariants(CH, 3)) then n, ", "; end if; end if; end if; end for;
Fundamental discriminants d uniquely characterizing all complex biquadratic fields Q(sqrt(-3),sqrt(d)) which have 3-class group of type (3,3) and second 3-class group isomorphic to either SmallGroup(2187,247)-#1;5 or SmallGroup(2187,247)-#1;9.
+10
7
11608, 14056, 20521, 21109, 25949, 27245, 27329, 31065, 32421, 32765, 38085, 38285, 39853, 40156, 43257, 45541, 46489, 48481
COMMENTS
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
REFERENCES
H. U. Besche, B. Eick, and E. A. O'Brien, The SmallGroups Library - a Library of Groups of Small Order, 2005, an accepted and refereed GAP 4 package, available also in MAGMA.
I. R. Shafarevich, Extensions with prescribed ramification points, Publ. Math., Inst. Hautes Études Sci. 18 (1964), 71-95 (Russian). English transl. by J. W. S. Cassels: Am. Math. Soc. Transl., II. Ser., 59 (1966), 128-149. - Daniel Constantin Mayer, Sep 24 2015
EXAMPLE
Up to 50000, the discriminants 20521 and 40156 are the only two terms which show a twisted bipolarization. All the other discriminants, starting with 11608, 14056, 21109, etc., reveal the (usual) parallel bipolarization among the four unramified cyclic cubic extensions. In the twisted case, the Hilbert 3-class field of the complex quadratic subfield Q(sqrt(-3d)) gives rise to the distinguished extension of type (9,27) (contained in the transfer target type), whereas in the parallel case the Hilbert 3-class field of the real quadratic subfield Q(sqrt(d)) is responsible for (9,27).
PROG
(Magma) SetClassGroupBounds("GRH"); for n := 11608 to 50000 do cnd := false; if (1 eq n mod 4) and IsSquarefree(n) then cnd := true; end if; if (0 eq n mod 4) then r := n div 4; if IsSquarefree(r) and ((2 eq r mod 4) or (3 eq r mod 4)) then cnd := true; end if; end if; if (true eq cnd) then R := QuadraticField(n); E := QuadraticField(-3); K := Compositum(R, E); C, mC := ClassGroup(K); if ([3, 3] eq pPrimaryInvariants(C, 3)) then s := Subgroups(C: Quot := [3]); a := [AbelianExtension(Inverse(mq)*mC) where _, mq := quo<C|x`subgroup> : x in s]; b := [NumberField(x) : x in a]; d := [MaximalOrder(x) : x in a]; b := [AbsoluteField(x) : x in b]; c := [MaximalOrder(x) : x in b]; c := [OptimizedRepresentation(x) : x in b]; b := [NumberField(DefiningPolynomial(x)) : x in c]; a := [Simplify(LLL(MaximalOrder(x))) : x in b]; if IsNormal(b[2]) then H := Compositum(NumberField(a[1]), NumberField(a[2])); else H := Compositum(NumberField(a[1]), NumberField(a[3])); end if; O := MaximalOrder(H); CH := ClassGroup(LLL(O)); if ([3, 9, 27] eq pPrimaryInvariants(CH, 3)) then n, ", "; end if; end if; end if; end for;
Search completed in 0.009 seconds
|