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Properties

Label 16.0.328683126924509184.2
Degree $16$
Signature $[0, 8]$
Discriminant $3.287\times 10^{17}$
Root discriminant \(12.44\)
Ramified primes $2,3$
Class number $1$
Class group trivial
Galois group $D_8:C_2$ (as 16T35)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^16 + 6*x^12 + 87*x^8 + 54*x^4 + 9)
 
Copy content gp:K = bnfinit(y^16 + 6*y^12 + 87*y^8 + 54*y^4 + 9, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 6*x^12 + 87*x^8 + 54*x^4 + 9);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 + 6*x^12 + 87*x^8 + 54*x^4 + 9)
 

\( x^{16} + 6x^{12} + 87x^{8} + 54x^{4} + 9 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $16$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[0, 8]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(328683126924509184\) \(\medspace = 2^{36}\cdot 3^{14}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(12.44\)
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{21/8}3^{7/8}\approx 16.131874566820663$
Ramified primes:   \(2\), \(3\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\Aut(K/\Q)$:   $D_4$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
Maximal CM subfield:  \(\Q(\zeta_{12})\)

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{6}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{6}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{6}a^{10}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{6}a^{11}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{276}a^{12}-\frac{1}{12}a^{10}+\frac{2}{69}a^{8}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{1}{23}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{13}{92}$, $\frac{1}{276}a^{13}-\frac{1}{12}a^{11}+\frac{2}{69}a^{9}-\frac{1}{4}a^{7}+\frac{21}{46}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}+\frac{33}{92}a-\frac{1}{2}$, $\frac{1}{828}a^{14}-\frac{5}{276}a^{10}-\frac{1}{12}a^{8}+\frac{19}{276}a^{6}+\frac{1}{4}a^{4}-\frac{3}{23}a^{2}+\frac{1}{4}$, $\frac{1}{828}a^{15}-\frac{5}{276}a^{11}-\frac{1}{12}a^{9}+\frac{19}{276}a^{7}+\frac{1}{4}a^{5}-\frac{3}{23}a^{3}+\frac{1}{4}a$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  Trivial group, which has order $1$
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  Trivial group, which has order $1$
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $7$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( \frac{97}{414} a^{14} + \frac{91}{69} a^{10} + \frac{1370}{69} a^{6} + \frac{239}{46} a^{2} \)  (order $12$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{37}{828}a^{14}+\frac{3}{92}a^{12}+\frac{17}{69}a^{10}+\frac{49}{276}a^{8}+\frac{262}{69}a^{6}+\frac{263}{92}a^{4}+\frac{39}{92}a^{2}+\frac{11}{23}$, $\frac{121}{828}a^{14}-\frac{1}{23}a^{12}+\frac{223}{276}a^{10}-\frac{73}{276}a^{8}+\frac{3403}{276}a^{6}-\frac{343}{92}a^{4}+\frac{51}{23}a^{2}-\frac{143}{92}$, $\frac{55}{207}a^{14}-\frac{10}{69}a^{12}+\frac{209}{138}a^{10}-\frac{19}{23}a^{8}+\frac{3125}{138}a^{6}-\frac{282}{23}a^{4}+\frac{359}{46}a^{2}-\frac{77}{23}$, $\frac{41}{138}a^{15}-\frac{13}{828}a^{14}-\frac{5}{69}a^{13}+\frac{1}{23}a^{12}+\frac{118}{69}a^{11}-\frac{9}{92}a^{10}-\frac{19}{46}a^{9}+\frac{73}{276}a^{8}+\frac{585}{23}a^{7}-\frac{385}{276}a^{6}-\frac{141}{23}a^{5}+\frac{343}{92}a^{4}+\frac{228}{23}a^{3}-\frac{30}{23}a^{2}-\frac{50}{23}a+\frac{189}{92}$, $\frac{41}{276}a^{15}-\frac{40}{207}a^{14}-\frac{49}{276}a^{13}+\frac{5}{69}a^{12}+\frac{59}{69}a^{11}-\frac{76}{69}a^{10}-\frac{277}{276}a^{9}+\frac{19}{46}a^{8}+\frac{585}{46}a^{7}-\frac{2279}{138}a^{6}-\frac{1391}{92}a^{5}+\frac{141}{23}a^{4}+\frac{433}{92}a^{3}-\frac{259}{46}a^{2}-\frac{199}{46}a+\frac{77}{46}$, $\frac{3}{23}a^{15}-\frac{2}{69}a^{14}+\frac{41}{276}a^{12}+\frac{49}{69}a^{11}-\frac{41}{276}a^{10}+\frac{59}{69}a^{8}+\frac{503}{46}a^{7}-\frac{221}{92}a^{6}+\frac{585}{46}a^{4}+\frac{21}{23}a^{3}+\frac{35}{92}a^{2}-\frac{1}{2}a+\frac{433}{92}$, $\frac{61}{207}a^{15}+\frac{2}{69}a^{14}-\frac{7}{92}a^{13}-\frac{1}{276}a^{12}+\frac{153}{92}a^{11}+\frac{41}{276}a^{10}-\frac{61}{138}a^{9}-\frac{2}{69}a^{8}+\frac{6913}{276}a^{7}+\frac{221}{92}a^{6}-\frac{303}{46}a^{5}-\frac{21}{46}a^{4}+\frac{637}{92}a^{3}-\frac{35}{92}a^{2}-\frac{233}{92}a-\frac{33}{92}$ Copy content Toggle raw display
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 763.555572776 \)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 763.555572776 \cdot 1}{12\cdot\sqrt{328683126924509184}}\cr\approx \mathstrut & 0.269593642787 \end{aligned}\]

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^16 + 6*x^12 + 87*x^8 + 54*x^4 + 9) 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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^16 + 6*x^12 + 87*x^8 + 54*x^4 + 9, 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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 6*x^12 + 87*x^8 + 54*x^4 + 9); 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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 6*x^12 + 87*x^8 + 54*x^4 + 9); 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))))
 

Galois group

$D_8:C_2$ (as 16T35):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 32
The 11 conjugacy class representatives for $D_8:C_2$
Character table for $D_8:C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1}) \), 4.0.432.1 x2, 4.2.1728.1 x2, \(\Q(\zeta_{12})\), 8.0.143327232.2 x2, 8.0.143327232.1 x2, 8.0.2985984.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Galois closure: deg 32
Degree 8 siblings: 8.0.143327232.2, 8.0.143327232.1
Degree 16 siblings: 16.4.21035720123168587776.1, 16.0.5258930030792146944.1, 16.0.5258930030792146944.2
Minimal sibling: 8.0.143327232.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.8.0.1}{8} }^{2}$ ${\href{/padicField/7.2.0.1}{2} }^{8}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{8}$ ${\href{/padicField/41.8.0.1}{8} }^{2}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content 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)]
 
Copy content 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]])
 
Copy content 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))];
 
Copy content 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]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.2.8.36b2.32$x^{16} + 8 x^{15} + 36 x^{14} + 112 x^{13} + 268 x^{12} + 520 x^{11} + 846 x^{10} + 1176 x^{9} + 1407 x^{8} + 1452 x^{7} + 1282 x^{6} + 960 x^{5} + 594 x^{4} + 296 x^{3} + 110 x^{2} + 28 x + 17$$8$$2$$36$16T35$$[2, 2, 3, 3]^{2}$$
\(3\) Copy content Toggle raw display 3.2.8.14a1.2$x^{16} + 16 x^{15} + 128 x^{14} + 672 x^{13} + 2576 x^{12} + 7616 x^{11} + 17920 x^{10} + 34176 x^{9} + 53344 x^{8} + 68352 x^{7} + 71680 x^{6} + 60928 x^{5} + 41216 x^{4} + 21504 x^{3} + 8192 x^{2} + 2048 x + 259$$8$$2$$14$$QD_{16}$$$[\ ]_{8}^{2}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)