Properties

Label 16.0.149...296.38
Degree $16$
Signature $(0, 8)$
Discriminant $1.496\times 10^{20}$
Root discriminant \(18.24\)
Ramified primes $2,3$
Class number $1$
Class group trivial
Galois group $D_8:C_4$ (as 16T114)

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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 - 12*x^14 + 78*x^12 - 324*x^10 + 738*x^8 - 756*x^6 + 162*x^4 + 324*x^2 + 81)
 
Copy content gp:K = bnfinit(y^16 - 12*y^14 + 78*y^12 - 324*y^10 + 738*y^8 - 756*y^6 + 162*y^4 + 324*y^2 + 81, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 12*x^14 + 78*x^12 - 324*x^10 + 738*x^8 - 756*x^6 + 162*x^4 + 324*x^2 + 81);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 12*x^14 + 78*x^12 - 324*x^10 + 738*x^8 - 756*x^6 + 162*x^4 + 324*x^2 + 81)
 

\( x^{16} - 12x^{14} + 78x^{12} - 324x^{10} + 738x^{8} - 756x^{6} + 162x^{4} + 324x^{2} + 81 \) 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:   \(149587343098087735296\) \(\medspace = 2^{48}\cdot 3^{12}\) 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:  \(18.24\)
Copy content comment:Root discriminant
 
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:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $2^{29/8}3^{3/4}\approx 28.12384367863563$
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:  \(\Q\)
$\Aut(K/\Q)$:   $C_8$
Copy content comment:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3}a^{4}$, $\frac{1}{3}a^{5}$, $\frac{1}{3}a^{6}$, $\frac{1}{3}a^{7}$, $\frac{1}{18}a^{8}-\frac{1}{2}$, $\frac{1}{18}a^{9}-\frac{1}{2}a$, $\frac{1}{18}a^{10}-\frac{1}{2}a^{2}$, $\frac{1}{18}a^{11}-\frac{1}{2}a^{3}$, $\frac{1}{54}a^{12}-\frac{1}{6}a^{4}$, $\frac{1}{54}a^{13}-\frac{1}{6}a^{5}$, $\frac{1}{81918}a^{14}+\frac{223}{40959}a^{12}+\frac{178}{13653}a^{10}+\frac{69}{3034}a^{8}-\frac{683}{9102}a^{6}-\frac{66}{1517}a^{4}+\frac{115}{1517}a^{2}+\frac{679}{3034}$, $\frac{1}{81918}a^{15}+\frac{223}{40959}a^{13}+\frac{178}{13653}a^{11}+\frac{69}{3034}a^{9}-\frac{683}{9102}a^{7}-\frac{66}{1517}a^{5}+\frac{115}{1517}a^{3}+\frac{679}{3034}a$ 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:   \( -\frac{292}{40959} a^{14} + \frac{1088}{13653} a^{12} - \frac{2288}{4551} a^{10} + \frac{54511}{27306} a^{8} - \frac{19012}{4551} a^{6} + \frac{17027}{4551} a^{4} - \frac{412}{1517} a^{2} - \frac{3633}{3034} \)  (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:   $\frac{1}{4551}a^{14}+\frac{443}{81918}a^{12}-\frac{1177}{27306}a^{10}+\frac{2209}{9102}a^{8}-\frac{3113}{4551}a^{6}+\frac{457}{9102}a^{4}+\frac{2623}{3034}a^{2}-\frac{1431}{3034}$, $\frac{59}{27306}a^{14}-\frac{1459}{81918}a^{12}+\frac{1166}{13653}a^{10}-\frac{5375}{27306}a^{8}-\frac{855}{3034}a^{6}+\frac{5459}{3034}a^{4}-\frac{2400}{1517}a^{2}-\frac{1177}{3034}$, $\frac{1}{81918}a^{14}-\frac{223}{40959}a^{12}+\frac{129}{3034}a^{10}-\frac{6689}{27306}a^{8}+\frac{6751}{9102}a^{6}-\frac{1451}{1517}a^{4}+\frac{1287}{3034}a^{2}+\frac{2355}{3034}$, $\frac{1}{1107}a^{15}-\frac{1}{999}a^{14}-\frac{17}{738}a^{13}-\frac{2}{999}a^{12}+\frac{23}{123}a^{11}+\frac{14}{333}a^{10}-\frac{121}{123}a^{9}-\frac{27}{74}a^{8}+\frac{383}{123}a^{7}+\frac{55}{37}a^{6}-\frac{387}{82}a^{5}-\frac{233}{111}a^{4}+\frac{107}{41}a^{3}+\frac{29}{37}a^{2}+\frac{23}{41}a+\frac{85}{74}$, $\frac{403}{27306}a^{15}-\frac{49}{40959}a^{14}+\frac{14491}{81918}a^{13}+\frac{95}{27306}a^{12}-\frac{31433}{27306}a^{11}+\frac{1}{9102}a^{10}+\frac{65294}{13653}a^{9}-\frac{1606}{13653}a^{8}-\frac{99623}{9102}a^{7}+\frac{3127}{4551}a^{6}+\frac{104065}{9102}a^{5}-\frac{5185}{9102}a^{4}-\frac{9561}{3034}a^{3}-\frac{2819}{3034}a^{2}-\frac{4658}{1517}a+\frac{103}{1517}$, $\frac{877}{40959}a^{15}-\frac{115}{40959}a^{14}-\frac{1207}{4551}a^{13}+\frac{1805}{40959}a^{12}+\frac{7994}{4551}a^{11}-\frac{4532}{13653}a^{10}-\frac{204767}{27306}a^{9}+\frac{14587}{9102}a^{8}+\frac{82142}{4551}a^{7}-\frac{21577}{4551}a^{6}-\frac{32329}{1517}a^{5}+\frac{33404}{4551}a^{4}+\frac{15119}{1517}a^{3}-\frac{6729}{1517}a^{2}+\frac{15291}{3034}a+\frac{81}{3034}$, $\frac{196}{40959}a^{15}+\frac{97}{27306}a^{14}+\frac{2087}{40959}a^{13}-\frac{5227}{81918}a^{12}-\frac{505}{1517}a^{11}+\frac{2096}{4551}a^{10}+\frac{37213}{27306}a^{9}-\frac{10082}{4551}a^{8}-\frac{14798}{4551}a^{7}+\frac{18701}{3034}a^{6}+\frac{21487}{4551}a^{5}-\frac{71243}{9102}a^{4}-\frac{5638}{1517}a^{3}+\frac{6159}{1517}a^{2}+\frac{5375}{3034}a+\frac{3982}{1517}$ 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:  \( 10409.609584098896 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 10409.609584098896 \cdot 1}{12\cdot\sqrt{149587343098087735296}}\cr\approx \mathstrut & 0.172283909389787 \end{aligned}\]

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 - 12*x^14 + 78*x^12 - 324*x^10 + 738*x^8 - 756*x^6 + 162*x^4 + 324*x^2 + 81) 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 - 12*x^14 + 78*x^12 - 324*x^10 + 738*x^8 - 756*x^6 + 162*x^4 + 324*x^2 + 81, 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 - 12*x^14 + 78*x^12 - 324*x^10 + 738*x^8 - 756*x^6 + 162*x^4 + 324*x^2 + 81); 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 = polynomial_ring(QQ); K, a = number_field(x^16 - 12*x^14 + 78*x^12 - 324*x^10 + 738*x^8 - 756*x^6 + 162*x^4 + 324*x^2 + 81); 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

$D_8:C_4$ (as 16T114):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 64
The 28 conjugacy class representatives for $D_8:C_4$
Character table for $D_8:C_4$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{12})\), 8.0.5308416.2

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(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 16 sibling: 16.0.149587343098087735296.34
Degree 32 siblings: deg 32, deg 32, deg 32
Minimal sibling: 16.0.149587343098087735296.34

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 ${\href{/padicField/5.8.0.1}{8} }^{2}$ ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.8.0.1}{8} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.8.0.1}{8} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.8.0.1}{8} }^{2}$

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 2.2.8.48c3.24$x^{16} + 16 x^{15} + 100 x^{14} + 392 x^{13} + 1106 x^{12} + 2416 x^{11} + 4240 x^{10} + 6120 x^{9} + 7349 x^{8} + 7384 x^{7} + 6208 x^{6} + 4352 x^{5} + 2528 x^{4} + 1200 x^{3} + 464 x^{2} + 136 x + 49$$8$$2$$48$16T114$$[2, 3, 3, 4, 4]^{2}$$
\(3\) Copy content Toggle raw display 3.4.4.12a1.1$x^{16} + 8 x^{15} + 24 x^{14} + 32 x^{13} + 24 x^{12} + 48 x^{11} + 96 x^{10} + 64 x^{9} + 24 x^{8} + 96 x^{7} + 96 x^{6} + 32 x^{4} + 64 x^{3} + 3 x^{2} + 16$$4$$4$$12$$C_8: C_2$$$[\ ]_{4}^{4}$$

Spectrum of ring of integers

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