sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 273*x^10 + 28665*x^8 - 1444716*x^6 + 35395542*x^4 - 371653191*x^2 + 1114959573)
gp: K = bnfinit(y^12 - 273*y^10 + 28665*y^8 - 1444716*y^6 + 35395542*y^4 - 371653191*y^2 + 1114959573, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 273*x^10 + 28665*x^8 - 1444716*x^6 + 35395542*x^4 - 371653191*x^2 + 1114959573);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 273*x^10 + 28665*x^8 - 1444716*x^6 + 35395542*x^4 - 371653191*x^2 + 1114959573)
x 12 − 273 x 10 + 28665 x 8 − 1444716 x 6 + 35395542 x 4 − 371653191 x 2 + 1114959573 x^{12} - 273x^{10} + 28665x^{8} - 1444716x^{6} + 35395542x^{4} - 371653191x^{2} + 1114959573 x 1 2 − 2 7 3 x 1 0 + 2 8 6 6 5 x 8 − 1 4 4 4 7 1 6 x 6 + 3 5 3 9 5 5 4 2 x 4 − 3 7 1 6 5 3 1 9 1 x 2 + 1 1 1 4 9 5 9 5 7 3
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Degree : 12 12 1 2
sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
Signature : [ 12 , 0 ] [12, 0] [ 1 2 , 0 ]
sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
Discriminant :
629582418765353043873792 629582418765353043873792 6 2 9 5 8 2 4 1 8 7 6 5 3 5 3 0 4 3 8 7 3 7 9 2
= 2 12 ⋅ 3 6 ⋅ 7 6 ⋅ 1 3 11 \medspace = 2^{12}\cdot 3^{6}\cdot 7^{6}\cdot 13^{11} = 2 1 2 ⋅ 3 6 ⋅ 7 6 ⋅ 1 3 1 1
sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant : 96.22 96.22 9 6 . 2 2
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
Galois root discriminant : 2 ⋅ 3 1 / 2 7 1 / 2 1 3 11 / 12 ≈ 96.21756959272508 2\cdot 3^{1/2}7^{1/2}13^{11/12}\approx 96.21756959272508 2 ⋅ 3 1 / 2 7 1 / 2 1 3 1 1 / 1 2 ≈ 9 6 . 2 1 7 5 6 9 5 9 2 7 2 5 0 8
Ramified primes :
2 2 2 , 3 3 3 , 7 7 7 , 13 13 1 3
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
Discriminant root field : Q ( 13 ) \Q(\sqrt{13}) Q ( 1 3 )
Aut ( K / Q ) \Aut(K/\Q) A u t ( K / Q )
= = =
Gal ( K / Q ) \Gal(K/\Q) G a l ( K / Q ) :
C 12 C_{12} C 1 2
sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
This field is Galois and abelian over Q \Q Q .
Conductor : 1092 = 2 2 ⋅ 3 ⋅ 7 ⋅ 13 1092=2^{2}\cdot 3\cdot 7\cdot 13 1 0 9 2 = 2 2 ⋅ 3 ⋅ 7 ⋅ 1 3
Dirichlet character group :
{ \lbrace { χ 1092 ( 1 , ⋅ ) \chi_{1092}(1,·) χ 1 0 9 2 ( 1 , ⋅ ) , χ 1092 ( 839 , ⋅ ) \chi_{1092}(839,·) χ 1 0 9 2 ( 8 3 9 , ⋅ ) , χ 1092 ( 841 , ⋅ ) \chi_{1092}(841,·) χ 1 0 9 2 ( 8 4 1 , ⋅ ) , χ 1092 ( 587 , ⋅ ) \chi_{1092}(587,·) χ 1 0 9 2 ( 5 8 7 , ⋅ ) , χ 1092 ( 589 , ⋅ ) \chi_{1092}(589,·) χ 1 0 9 2 ( 5 8 9 , ⋅ ) , χ 1092 ( 1007 , ⋅ ) \chi_{1092}(1007,·) χ 1 0 9 2 ( 1 0 0 7 , ⋅ ) , χ 1092 ( 337 , ⋅ ) \chi_{1092}(337,·) χ 1 0 9 2 ( 3 3 7 , ⋅ ) , χ 1092 ( 83 , ⋅ ) \chi_{1092}(83,·) χ 1 0 9 2 ( 8 3 , ⋅ ) , χ 1092 ( 757 , ⋅ ) \chi_{1092}(757,·) χ 1 0 9 2 ( 7 5 7 , ⋅ ) , χ 1092 ( 673 , ⋅ ) \chi_{1092}(673,·) χ 1 0 9 2 ( 6 7 3 , ⋅ ) , χ 1092 ( 167 , ⋅ ) \chi_{1092}(167,·) χ 1 0 9 2 ( 1 6 7 , ⋅ ) , χ 1092 ( 671 , ⋅ ) \chi_{1092}(671,·) χ 1 0 9 2 ( 6 7 1 , ⋅ ) } \rbrace }
This is not a CM field .
1 1 1 , a a a , 1 21 a 2 \frac{1}{21}a^{2} 2 1 1 a 2 , 1 21 a 3 \frac{1}{21}a^{3} 2 1 1 a 3 , 1 441 a 4 \frac{1}{441}a^{4} 4 4 1 1 a 4 , 1 441 a 5 \frac{1}{441}a^{5} 4 4 1 1 a 5 , 1 9261 a 6 \frac{1}{9261}a^{6} 9 2 6 1 1 a 6 , 1 9261 a 7 \frac{1}{9261}a^{7} 9 2 6 1 1 a 7 , 1 194481 a 8 \frac{1}{194481}a^{8} 1 9 4 4 8 1 1 a 8 , 1 194481 a 9 \frac{1}{194481}a^{9} 1 9 4 4 8 1 1 a 9 , 1 4084101 a 10 \frac{1}{4084101}a^{10} 4 0 8 4 1 0 1 1 a 1 0 , 1 4084101 a 11 \frac{1}{4084101}a^{11} 4 0 8 4 1 0 1 1 a 1 1
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
Ideal class group : C 2 × C 2 C_{2}\times C_{2} C 2 × C 2 , which has order 4 4 4 (assuming GRH )
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Narrow class group : C 2 × C 2 × C 2 × C 2 C_{2}\times C_{2}\times C_{2}\times C_{2} C 2 × C 2 × C 2 × C 2 , which has order 16 16 1 6 (assuming GRH )
sage: K.narrow_class_group().invariants()
gp: bnfnarrow(K)
magma: NarrowClassGroup(K);
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
Rank : 11 11 1 1
sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
Torsion generator :
− 1 -1 − 1
(order 2 2 2 )
sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
Fundamental units :
1 9261 a 6 − 2 147 a 4 + 3 7 a 2 − 2 \frac{1}{9261}a^{6}-\frac{2}{147}a^{4}+\frac{3}{7}a^{2}-2 9 2 6 1 1 a 6 − 1 4 7 2 a 4 + 7 3 a 2 − 2 , 1 4084101 a 10 − 11 194481 a 8 + 44 9261 a 6 − 11 63 a 4 + 55 21 a 2 − 11 \frac{1}{4084101}a^{10}-\frac{11}{194481}a^{8}+\frac{44}{9261}a^{6}-\frac{11}{63}a^{4}+\frac{55}{21}a^{2}-11 4 0 8 4 1 0 1 1 a 1 0 − 1 9 4 4 8 1 1 1 a 8 + 9 2 6 1 4 4 a 6 − 6 3 1 1 a 4 + 2 1 5 5 a 2 − 1 1 , 1 194481 a 8 − 1 1029 a 6 + 3 49 a 4 − 29 21 a 2 + 6 \frac{1}{194481}a^{8}-\frac{1}{1029}a^{6}+\frac{3}{49}a^{4}-\frac{29}{21}a^{2}+6 1 9 4 4 8 1 1 a 8 − 1 0 2 9 1 a 6 + 4 9 3 a 4 − 2 1 2 9 a 2 + 6 , 1 194481 a 8 − 8 9261 a 6 + 20 441 a 4 − 16 21 a 2 + 3 \frac{1}{194481}a^{8}-\frac{8}{9261}a^{6}+\frac{20}{441}a^{4}-\frac{16}{21}a^{2}+3 1 9 4 4 8 1 1 a 8 − 9 2 6 1 8 a 6 + 4 4 1 2 0 a 4 − 2 1 1 6 a 2 + 3 , 1 4084101 a 10 − 11 194481 a 8 + 44 9261 a 6 − 11 63 a 4 + 55 21 a 2 − 10 \frac{1}{4084101}a^{10}-\frac{11}{194481}a^{8}+\frac{44}{9261}a^{6}-\frac{11}{63}a^{4}+\frac{55}{21}a^{2}-10 4 0 8 4 1 0 1 1 a 1 0 − 1 9 4 4 8 1 1 1 a 8 + 9 2 6 1 4 4 a 6 − 6 3 1 1 a 4 + 2 1 5 5 a 2 − 1 0 , 1 4084101 a 11 − 1 453789 a 10 − 13 194481 a 9 + 11 21609 a 8 + 61 9261 a 7 − 407 9261 a 6 − 124 441 a 5 + 751 441 a 4 + 97 21 a 3 − 562 21 a 2 − 11 a + 82 \frac{1}{4084101}a^{11}-\frac{1}{453789}a^{10}-\frac{13}{194481}a^{9}+\frac{11}{21609}a^{8}+\frac{61}{9261}a^{7}-\frac{407}{9261}a^{6}-\frac{124}{441}a^{5}+\frac{751}{441}a^{4}+\frac{97}{21}a^{3}-\frac{562}{21}a^{2}-11a+82 4 0 8 4 1 0 1 1 a 1 1 − 4 5 3 7 8 9 1 a 1 0 − 1 9 4 4 8 1 1 3 a 9 + 2 1 6 0 9 1 1 a 8 + 9 2 6 1 6 1 a 7 − 9 2 6 1 4 0 7 a 6 − 4 4 1 1 2 4 a 5 + 4 4 1 7 5 1 a 4 + 2 1 9 7 a 3 − 2 1 5 6 2 a 2 − 1 1 a + 8 2 , 1 1361367 a 11 − 32 194481 a 9 + 2 27783 a 8 + 2 147 a 7 − 2 147 a 6 − 74 147 a 5 + 52 63 a 4 + 23 3 a 3 − 52 3 a 2 − 26 a + 64 \frac{1}{1361367}a^{11}-\frac{32}{194481}a^{9}+\frac{2}{27783}a^{8}+\frac{2}{147}a^{7}-\frac{2}{147}a^{6}-\frac{74}{147}a^{5}+\frac{52}{63}a^{4}+\frac{23}{3}a^{3}-\frac{52}{3}a^{2}-26a+64 1 3 6 1 3 6 7 1 a 1 1 − 1 9 4 4 8 1 3 2 a 9 + 2 7 7 8 3 2 a 8 + 1 4 7 2 a 7 − 1 4 7 2 a 6 − 1 4 7 7 4 a 5 + 6 3 5 2 a 4 + 3 2 3 a 3 − 3 5 2 a 2 − 2 6 a + 6 4 , 1 4084101 a 10 − 4 194481 a 9 + 1 21609 a 8 + 5 1323 a 7 − 125 9261 a 6 − 100 441 a 5 + 386 441 a 4 + 100 21 a 3 − 391 21 a 2 − 19 a + 69 \frac{1}{4084101}a^{10}-\frac{4}{194481}a^{9}+\frac{1}{21609}a^{8}+\frac{5}{1323}a^{7}-\frac{125}{9261}a^{6}-\frac{100}{441}a^{5}+\frac{386}{441}a^{4}+\frac{100}{21}a^{3}-\frac{391}{21}a^{2}-19a+69 4 0 8 4 1 0 1 1 a 1 0 − 1 9 4 4 8 1 4 a 9 + 2 1 6 0 9 1 a 8 + 1 3 2 3 5 a 7 − 9 2 6 1 1 2 5 a 6 − 4 4 1 1 0 0 a 5 + 4 4 1 3 8 6 a 4 + 2 1 1 0 0 a 3 − 2 1 3 9 1 a 2 − 1 9 a + 6 9 , 1 4084101 a 11 + 8 4084101 a 10 − 10 194481 a 9 − 80 194481 a 8 + 5 1323 a 7 + 40 1323 a 6 − 17 147 a 5 − 409 441 a 4 + 4 3 a 3 + 75 7 a 2 − 5 a − 35 \frac{1}{4084101}a^{11}+\frac{8}{4084101}a^{10}-\frac{10}{194481}a^{9}-\frac{80}{194481}a^{8}+\frac{5}{1323}a^{7}+\frac{40}{1323}a^{6}-\frac{17}{147}a^{5}-\frac{409}{441}a^{4}+\frac{4}{3}a^{3}+\frac{75}{7}a^{2}-5a-35 4 0 8 4 1 0 1 1 a 1 1 + 4 0 8 4 1 0 1 8 a 1 0 − 1 9 4 4 8 1 1 0 a 9 − 1 9 4 4 8 1 8 0 a 8 + 1 3 2 3 5 a 7 + 1 3 2 3 4 0 a 6 − 1 4 7 1 7 a 5 − 4 4 1 4 0 9 a 4 + 3 4 a 3 + 7 7 5 a 2 − 5 a − 3 5 , 2 4084101 a 11 + 11 4084101 a 10 − 22 194481 a 9 − 110 194481 a 8 + 89 9261 a 7 + 376 9261 a 6 − 55 147 a 5 − 496 441 a 4 + 145 21 a 3 + 62 7 a 2 − 51 a + 35 \frac{2}{4084101}a^{11}+\frac{11}{4084101}a^{10}-\frac{22}{194481}a^{9}-\frac{110}{194481}a^{8}+\frac{89}{9261}a^{7}+\frac{376}{9261}a^{6}-\frac{55}{147}a^{5}-\frac{496}{441}a^{4}+\frac{145}{21}a^{3}+\frac{62}{7}a^{2}-51a+35 4 0 8 4 1 0 1 2 a 1 1 + 4 0 8 4 1 0 1 1 1 a 1 0 − 1 9 4 4 8 1 2 2 a 9 − 1 9 4 4 8 1 1 1 0 a 8 + 9 2 6 1 8 9 a 7 + 9 2 6 1 3 7 6 a 6 − 1 4 7 5 5 a 5 − 4 4 1 4 9 6 a 4 + 2 1 1 4 5 a 3 + 7 6 2 a 2 − 5 1 a + 3 5 , 1 4084101 a 11 − 11 4084101 a 10 − 10 194481 a 9 + 43 64827 a 8 + 31 9261 a 7 − 548 9261 a 6 − 20 441 a 5 + 1007 441 a 4 − 41 21 a 3 − 106 3 a 2 + 35 a + 178 \frac{1}{4084101}a^{11}-\frac{11}{4084101}a^{10}-\frac{10}{194481}a^{9}+\frac{43}{64827}a^{8}+\frac{31}{9261}a^{7}-\frac{548}{9261}a^{6}-\frac{20}{441}a^{5}+\frac{1007}{441}a^{4}-\frac{41}{21}a^{3}-\frac{106}{3}a^{2}+35a+178 4 0 8 4 1 0 1 1 a 1 1 − 4 0 8 4 1 0 1 1 1 a 1 0 − 1 9 4 4 8 1 1 0 a 9 + 6 4 8 2 7 4 3 a 8 + 9 2 6 1 3 1 a 7 − 9 2 6 1 5 4 8 a 6 − 4 4 1 2 0 a 5 + 4 4 1 1 0 0 7 a 4 − 2 1 4 1 a 3 − 3 1 0 6 a 2 + 3 5 a + 1 7 8
(assuming GRH )
sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
Regulator : 13423777.933 13423777.933 1 3 4 2 3 7 7 7 . 9 3 3
(assuming GRH )
sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
lim s → 1 ( s − 1 ) ζ K ( s ) = ( 2 r 1 ⋅ ( 2 π ) r 2 ⋅ R ⋅ h w ⋅ ∣ D ∣ ≈ ( 2 12 ⋅ ( 2 π ) 0 ⋅ 13423777.933 ⋅ 4 2 ⋅ 629582418765353043873792 ≈ ( 0.13859207812
\begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 13423777.933 \cdot 4}{2\cdot\sqrt{629582418765353043873792}}\cr\approx \mathstrut & 0.13859207812
\end{aligned} s → 1 lim ( s − 1 ) ζ K ( s ) = ( ≈ ( ≈ ( w ⋅ ∣ D ∣ 2 r 1 ⋅ ( 2 π ) r 2 ⋅ R ⋅ h 2 ⋅ 6 2 9 5 8 2 4 1 8 7 6 5 3 5 3 0 4 3 8 7 3 7 9 2 2 1 2 ⋅ ( 2 π ) 0 ⋅ 1 3 4 2 3 7 7 7 . 9 3 3 ⋅ 4 0 . 1 3 8 5 9 2 0 7 8 1 2
(assuming GRH )
sage: # self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 273*x^10 + 28665*x^8 - 1444716*x^6 + 35395542*x^4 - 371653191*x^2 + 1114959573)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
gp: \\ self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^12 - 273*x^10 + 28665*x^8 - 1444716*x^6 + 35395542*x^4 - 371653191*x^2 + 1114959573, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
magma: /* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 273*x^10 + 28665*x^8 - 1444716*x^6 + 35395542*x^4 - 371653191*x^2 + 1114959573);
OK := Integers(K); DK := Discriminant(OK);
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
hK := #clK; wK := #TorsionSubgroup(UK);
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
oscar: # self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 273*x^10 + 28665*x^8 - 1444716*x^6 + 35395542*x^4 - 371653191*x^2 + 1114959573);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
C 12 C_{12} C 1 2 (as 12T1 ):
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
Fields in the database are given up to isomorphism. Isomorphic
intermediate fields are shown with their multiplicities.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
p p p
2 2 2
3 3 3
5 5 5
7 7 7
11 11 1 1
13 13 1 3
17 17 1 7
19 19 1 9
23 23 2 3
29 29 2 9
31 31 3 1
37 37 3 7
41 41 4 1
43 43 4 3
47 47 4 7
53 53 5 3
59 59 5 9
Cycle type
R
R
4 3 {\href{/padicField/5.4.0.1}{4} }^{3} 4 3
R
12 {\href{/padicField/11.12.0.1}{12} } 1 2
R
6 2 {\href{/padicField/17.6.0.1}{6} }^{2} 6 2
12 {\href{/padicField/19.12.0.1}{12} } 1 2
6 2 {\href{/padicField/23.6.0.1}{6} }^{2} 6 2
6 2 {\href{/padicField/29.6.0.1}{6} }^{2} 6 2
4 3 {\href{/padicField/31.4.0.1}{4} }^{3} 4 3
12 {\href{/padicField/37.12.0.1}{12} } 1 2
12 {\href{/padicField/41.12.0.1}{12} } 1 2
3 4 {\href{/padicField/43.3.0.1}{3} }^{4} 3 4
4 3 {\href{/padicField/47.4.0.1}{4} }^{3} 4 3
2 6 {\href{/padicField/53.2.0.1}{2} }^{6} 2 6
12 {\href{/padicField/59.12.0.1}{12} } 1 2
In the table, R denotes a ramified prime.
Cycle lengths which are repeated in a cycle type are indicated by
exponents.
sage: # to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
gp: \\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
magma: // to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
oscar: # to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
p p p Label Polynomial
e e e
f f f
c c c
Galois group
Slope content
2 2 2
2.6.2.12a1.2 x 12 + 2 x 10 + 2 x 9 + x 8 + 4 x 7 + 5 x 6 + 2 x 5 + 6 x 4 + 8 x 3 + x 2 + 4 x + 5 x^{12} + 2 x^{10} + 2 x^{9} + x^{8} + 4 x^{7} + 5 x^{6} + 2 x^{5} + 6 x^{4} + 8 x^{3} + x^{2} + 4 x + 5 x 1 2 + 2 x 1 0 + 2 x 9 + x 8 + 4 x 7 + 5 x 6 + 2 x 5 + 6 x 4 + 8 x 3 + x 2 + 4 x + 5 2 2 2 6 6 6 12 12 1 2 C 12 C_{12} C 1 2 [ 2 ] 6 [2]^{6} [ 2 ] 6
3 3 3
3.3.2.3a1.2 x 6 + 4 x 4 + 2 x 3 + 4 x 2 + 4 x + 4 x^{6} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 4 x + 4 x 6 + 4 x 4 + 2 x 3 + 4 x 2 + 4 x + 4 2 2 2 3 3 3 3 3 3 C 6 C_6 C 6 [ ] 2 3 [\ ]_{2}^{3} [ ] 2 3 3.3.2.3a1.2 x 6 + 4 x 4 + 2 x 3 + 4 x 2 + 4 x + 4 x^{6} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 4 x + 4 x 6 + 4 x 4 + 2 x 3 + 4 x 2 + 4 x + 4 2 2 2 3 3 3 3 3 3 C 6 C_6 C 6 [ ] 2 3 [\ ]_{2}^{3} [ ] 2 3
7 7 7
7.6.2.6a1.1 x 12 + 2 x 10 + 10 x 9 + 9 x 8 + 22 x 7 + 39 x 6 + 52 x 5 + 82 x 4 + 78 x 3 + 60 x 2 + 43 x + 9 x^{12} + 2 x^{10} + 10 x^{9} + 9 x^{8} + 22 x^{7} + 39 x^{6} + 52 x^{5} + 82 x^{4} + 78 x^{3} + 60 x^{2} + 43 x + 9 x 1 2 + 2 x 1 0 + 1 0 x 9 + 9 x 8 + 2 2 x 7 + 3 9 x 6 + 5 2 x 5 + 8 2 x 4 + 7 8 x 3 + 6 0 x 2 + 4 3 x + 9 2 2 2 6 6 6 6 6 6 C 12 C_{12} C 1 2 [ ] 2 6 [\ ]_{2}^{6} [ ] 2 6
13 13 1 3
13.1.12.11a1.12 x 12 + 156 x^{12} + 156 x 1 2 + 1 5 6 12 12 1 2 1 1 1 11 11 1 1 C 12 C_{12} C 1 2 [ ] 12 [\ ]_{12} [ ] 1 2
(0) (0) (2) (3) (5) (7) (11) (13) (17) (19) (23) (29) (31) (37) (41) (43) (47) (53) (59)