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Properties

Label 16.8.484...304.46
Degree $16$
Signature $[8, 4]$
Discriminant $4.841\times 10^{23}$
Root discriminant \(30.22\)
Ramified primes $2,3$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^2 : C_8$ (as 16T24)

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Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^16 - 24*x^14 + 108*x^12 - 216*x^10 + 3402*x^8 - 17496*x^6 + 26244*x^4 - 17496*x^2 + 6561)
 
Copy content gp:K = bnfinit(y^16 - 24*y^14 + 108*y^12 - 216*y^10 + 3402*y^8 - 17496*y^6 + 26244*y^4 - 17496*y^2 + 6561, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 24*x^14 + 108*x^12 - 216*x^10 + 3402*x^8 - 17496*x^6 + 26244*x^4 - 17496*x^2 + 6561);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 24*x^14 + 108*x^12 - 216*x^10 + 3402*x^8 - 17496*x^6 + 26244*x^4 - 17496*x^2 + 6561)
 

\( x^{16} - 24x^{14} + 108x^{12} - 216x^{10} + 3402x^{8} - 17496x^{6} + 26244x^{4} - 17496x^{2} + 6561 \) 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:  $[8, 4]$
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:   \(484116351470433472610304\) \(\medspace = 2^{66}\cdot 3^{8}\) 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:  \(30.22\)
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:  $2^{67/16}3^{1/2}\approx 31.559036391689393$
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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2\times C_4$
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.
This field has no CM subfields.

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

$1$, $a$, $\frac{1}{3}a^{2}$, $\frac{1}{3}a^{3}$, $\frac{1}{9}a^{4}$, $\frac{1}{9}a^{5}$, $\frac{1}{27}a^{6}$, $\frac{1}{27}a^{7}$, $\frac{1}{162}a^{8}-\frac{1}{2}$, $\frac{1}{324}a^{9}-\frac{1}{324}a^{8}-\frac{1}{54}a^{7}-\frac{1}{54}a^{6}-\frac{1}{18}a^{5}-\frac{1}{18}a^{4}-\frac{1}{6}a^{3}-\frac{1}{6}a^{2}-\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{972}a^{10}-\frac{1}{324}a^{8}+\frac{1}{12}a^{2}-\frac{1}{4}$, $\frac{1}{972}a^{11}-\frac{1}{324}a^{8}-\frac{1}{54}a^{7}-\frac{1}{54}a^{6}-\frac{1}{18}a^{5}-\frac{1}{18}a^{4}-\frac{1}{12}a^{3}-\frac{1}{6}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{2916}a^{12}-\frac{1}{324}a^{8}+\frac{1}{36}a^{4}-\frac{1}{4}$, $\frac{1}{2916}a^{13}-\frac{1}{324}a^{8}-\frac{1}{54}a^{7}-\frac{1}{54}a^{6}-\frac{1}{36}a^{5}-\frac{1}{18}a^{4}-\frac{1}{6}a^{3}-\frac{1}{6}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{2248236}a^{14}-\frac{25}{749412}a^{12}-\frac{77}{249804}a^{10}-\frac{241}{83268}a^{8}-\frac{487}{27756}a^{6}-\frac{17}{9252}a^{4}-\frac{63}{1028}a^{2}-\frac{393}{1028}$, $\frac{1}{2248236}a^{15}-\frac{25}{749412}a^{13}-\frac{77}{249804}a^{11}+\frac{4}{20817}a^{9}-\frac{1}{324}a^{8}+\frac{1}{1028}a^{7}-\frac{1}{54}a^{6}+\frac{497}{9252}a^{5}-\frac{1}{18}a^{4}+\frac{325}{3084}a^{3}-\frac{1}{6}a^{2}+\frac{189}{514}a+\frac{1}{4}$ 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 $1$ (assuming GRH)
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:  $C_{2}$, which has order $2$ (assuming GRH)
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:  $11$
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:   \( -1 \)  (order $2$) 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:   $\frac{140}{562059}a^{14}-\frac{499}{83268}a^{12}+\frac{1123}{41634}a^{10}-\frac{4147}{83268}a^{8}+\frac{5836}{6939}a^{6}-\frac{40103}{9252}a^{4}+\frac{9017}{1542}a^{2}-\frac{2915}{1028}$, $\frac{19}{62451}a^{14}-\frac{5021}{749412}a^{12}+\frac{1225}{62451}a^{10}-\frac{721}{27756}a^{8}+\frac{6673}{6939}a^{6}-\frac{31417}{9252}a^{4}+\frac{278}{257}a^{2}-\frac{247}{1028}$, $\frac{311}{1124118}a^{14}+\frac{1189}{187353}a^{12}-\frac{5819}{249804}a^{10}+\frac{3155}{83268}a^{8}-\frac{12509}{13878}a^{6}+\frac{2980}{771}a^{4}-\frac{10171}{3084}a^{2}+\frac{1067}{1028}$, $\frac{40}{562059}a^{14}-\frac{715}{374706}a^{12}+\frac{2843}{249804}a^{10}-\frac{1295}{83268}a^{8}+\frac{617}{2313}a^{6}-\frac{8299}{4626}a^{4}+\frac{6511}{3084}a^{2}-\frac{943}{1028}$, $\frac{479}{2248236}a^{14}-\frac{3751}{749412}a^{12}+\frac{65}{3084}a^{10}-\frac{3901}{83268}a^{8}+\frac{19615}{27756}a^{6}-\frac{31787}{9252}a^{4}+\frac{5803}{1028}a^{2}-\frac{2693}{1028}$, $\frac{1}{2187}a^{15}+\frac{5}{20817}a^{14}+\frac{8}{729}a^{13}-\frac{118}{20817}a^{12}-\frac{4}{81}a^{11}+\frac{5965}{249804}a^{10}+\frac{8}{81}a^{9}-\frac{3953}{83268}a^{8}-\frac{14}{9}a^{7}+\frac{5701}{6939}a^{6}+8a^{5}-\frac{8977}{2313}a^{4}-12a^{3}+\frac{5815}{1028}a^{2}+8a-\frac{5849}{1028}$, $\frac{140}{562059}a^{15}-\frac{5}{124902}a^{14}-\frac{499}{83268}a^{13}+\frac{59}{62451}a^{12}+\frac{1123}{41634}a^{11}-\frac{1037}{249804}a^{10}-\frac{4147}{83268}a^{9}+\frac{97}{9252}a^{8}+\frac{5836}{6939}a^{7}-\frac{1729}{13878}a^{6}-\frac{40103}{9252}a^{5}+\frac{171}{257}a^{4}+\frac{9017}{1542}a^{3}-\frac{1783}{1028}a^{2}-\frac{2915}{1028}a+\frac{1189}{1028}$, $\frac{311}{1124118}a^{14}-\frac{1189}{187353}a^{12}+\frac{5819}{249804}a^{10}-\frac{3155}{83268}a^{8}+\frac{12509}{13878}a^{6}-\frac{2980}{771}a^{4}+\frac{3733}{1028}a^{2}+a-\frac{1067}{1028}$, $\frac{1261}{2248236}a^{15}-\frac{17}{62451}a^{14}-\frac{9937}{749412}a^{13}+\frac{751}{124902}a^{12}+\frac{13927}{249804}a^{11}-\frac{1511}{83268}a^{10}-\frac{8351}{83268}a^{9}+\frac{2287}{83268}a^{8}+\frac{52037}{27756}a^{7}-\frac{5930}{6939}a^{6}-\frac{84145}{9252}a^{5}+\frac{14711}{4626}a^{4}+\frac{35119}{3084}a^{3}-\frac{4865}{3084}a^{2}-\frac{5731}{1028}a+\frac{735}{1028}$, $\frac{697}{1124118}a^{15}-\frac{1783}{2248236}a^{14}+\frac{11}{771}a^{13}+\frac{13735}{749412}a^{12}-\frac{13195}{249804}a^{11}-\frac{17423}{249804}a^{10}+\frac{6737}{83268}a^{9}+\frac{1085}{9252}a^{8}-\frac{3119}{1542}a^{7}-\frac{72299}{27756}a^{6}+\frac{6815}{771}a^{5}+\frac{107411}{9252}a^{4}-\frac{7525}{1028}a^{3}-\frac{35663}{3084}a^{2}+\frac{2745}{1028}a+\frac{5277}{1028}$, $\frac{2}{62451}a^{15}+\frac{463}{2248236}a^{14}+\frac{515}{749412}a^{13}-\frac{902}{187353}a^{12}-\frac{367}{249804}a^{11}+\frac{29}{1542}a^{10}-\frac{31}{20817}a^{9}-\frac{2101}{83268}a^{8}-\frac{743}{6939}a^{7}+\frac{19183}{27756}a^{6}+\frac{665}{3084}a^{5}-\frac{7172}{2313}a^{4}+\frac{1529}{3084}a^{3}+\frac{4691}{1542}a^{2}+\frac{135}{257}a-\frac{1545}{1028}$ Copy content Toggle raw display (assuming GRH)
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:  \( 983030.5219427672 \) (assuming GRH)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 983030.5219427672 \cdot 1}{2\cdot\sqrt{484116351470433472610304}}\cr\approx \mathstrut & 0.281852291251777 \end{aligned}\] (assuming GRH)

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 - 24*x^14 + 108*x^12 - 216*x^10 + 3402*x^8 - 17496*x^6 + 26244*x^4 - 17496*x^2 + 6561) 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 - 24*x^14 + 108*x^12 - 216*x^10 + 3402*x^8 - 17496*x^6 + 26244*x^4 - 17496*x^2 + 6561, 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 - 24*x^14 + 108*x^12 - 216*x^10 + 3402*x^8 - 17496*x^6 + 26244*x^4 - 17496*x^2 + 6561); 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 - 24*x^14 + 108*x^12 - 216*x^10 + 3402*x^8 - 17496*x^6 + 26244*x^4 - 17496*x^2 + 6561); 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

$C_2^2:C_8$ (as 16T24):

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); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2^2 : C_8$
Character table for $C_2^2 : C_8$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 4.2.1024.1, 4.2.2048.1, 8.4.173946175488.1, 8.8.173946175488.1, 8.4.67108864.1

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

Sibling fields

Galois closure: deg 32
Degree 16 sibling: 16.0.484116351470433472610304.300
Minimal sibling: This field is its own minimal sibling

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 ${\href{/padicField/5.8.0.1}{8} }^{2}$ ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.8.0.1}{8} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.8.0.1}{8} }^{2}$

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 2.1.16.66h1.268$x^{16} + 16 x^{15} + 8 x^{14} + 4 x^{12} + 8 x^{10} + 8 x^{6} + 16 x^{5} + 16 x^{3} + 34$$16$$1$$66$$C_2^2 : C_8$$$[2, 3, \frac{7}{2}, 4, 5]$$
\(3\) Copy content Toggle raw display 3.4.2.4a1.1$x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{4} + 8 x^{3} + 3 x + 4$$2$$4$$4$$C_8$$$[\ ]_{2}^{4}$$
3.4.2.4a1.1$x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{4} + 8 x^{3} + 3 x + 4$$2$$4$$4$$C_8$$$[\ ]_{2}^{4}$$

Spectrum of ring of integers

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