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Properties

Label 16.0.169561224228515625.2
Degree $16$
Signature $[0, 8]$
Discriminant $1.696\times 10^{17}$
Root discriminant \(11.94\)
Ramified primes $3,5,11$
Class number $1$
Class group trivial
Galois group $C_4\wr C_2$ (as 16T42)

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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 - 4*x^15 + 9*x^14 - 10*x^13 + 5*x^12 - 6*x^11 + 5*x^10 + 19*x^9 - 33*x^8 + 6*x^7 + 14*x^6 - 9*x^5 + 21*x^4 - 31*x^3 + 18*x^2 - 5*x + 1)
 
Copy content gp:K = bnfinit(y^16 - 4*y^15 + 9*y^14 - 10*y^13 + 5*y^12 - 6*y^11 + 5*y^10 + 19*y^9 - 33*y^8 + 6*y^7 + 14*y^6 - 9*y^5 + 21*y^4 - 31*y^3 + 18*y^2 - 5*y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 + 9*x^14 - 10*x^13 + 5*x^12 - 6*x^11 + 5*x^10 + 19*x^9 - 33*x^8 + 6*x^7 + 14*x^6 - 9*x^5 + 21*x^4 - 31*x^3 + 18*x^2 - 5*x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 4*x^15 + 9*x^14 - 10*x^13 + 5*x^12 - 6*x^11 + 5*x^10 + 19*x^9 - 33*x^8 + 6*x^7 + 14*x^6 - 9*x^5 + 21*x^4 - 31*x^3 + 18*x^2 - 5*x + 1)
 

\( x^{16} - 4 x^{15} + 9 x^{14} - 10 x^{13} + 5 x^{12} - 6 x^{11} + 5 x^{10} + 19 x^{9} - 33 x^{8} + \cdots + 1 \) 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:   \(169561224228515625\) \(\medspace = 3^{4}\cdot 5^{10}\cdot 11^{8}\) 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:  \(11.94\)
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:  $3^{1/2}5^{3/4}11^{1/2}\approx 19.208102881010017$
Ramified primes:   \(3\), \(5\), \(11\) 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.
Maximal CM subfield:  \(\Q(\sqrt{5}, \sqrt{-11})\)

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{9}-\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{9}-\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{9}a^{14}-\frac{1}{9}a^{12}+\frac{1}{9}a^{11}+\frac{1}{9}a^{10}-\frac{1}{3}a^{9}+\frac{1}{9}a^{8}+\frac{2}{9}a^{7}+\frac{4}{9}a^{6}-\frac{1}{9}a^{5}+\frac{1}{3}a^{4}+\frac{4}{9}a^{3}+\frac{4}{9}a^{2}-\frac{4}{9}a+\frac{4}{9}$, $\frac{1}{5235327}a^{15}-\frac{252806}{5235327}a^{14}+\frac{480623}{5235327}a^{13}+\frac{252823}{1745109}a^{12}-\frac{373867}{5235327}a^{11}-\frac{778112}{5235327}a^{10}-\frac{2161760}{5235327}a^{9}-\frac{360967}{1745109}a^{8}-\frac{315196}{1745109}a^{7}-\frac{373}{8193}a^{6}+\frac{1219865}{5235327}a^{5}-\frac{865022}{5235327}a^{4}-\frac{317125}{5235327}a^{3}+\frac{423956}{1745109}a^{2}-\frac{473768}{1745109}a-\frac{723752}{5235327}$ 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:  No
Index:  Not computed
Inessential primes:  $3$

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:   \( -1 \)  (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:   $\frac{906400}{5235327}a^{15}-\frac{3566264}{5235327}a^{14}+\frac{6872639}{5235327}a^{13}-\frac{1966244}{1745109}a^{12}-\frac{802744}{5235327}a^{11}-\frac{3639995}{5235327}a^{10}+\frac{4827181}{5235327}a^{9}+\frac{7670798}{1745109}a^{8}-\frac{8258743}{1745109}a^{7}-\frac{8301}{2731}a^{6}+\frac{15985562}{5235327}a^{5}+\frac{4081810}{5235327}a^{4}+\frac{20980025}{5235327}a^{3}-\frac{7427242}{1745109}a^{2}-\frac{1435055}{1745109}a-\frac{3143501}{5235327}$, $\frac{3748139}{5235327}a^{15}-\frac{12521116}{5235327}a^{14}+\frac{25123603}{5235327}a^{13}-\frac{6750149}{1745109}a^{12}+\frac{4246021}{5235327}a^{11}-\frac{20177836}{5235327}a^{10}+\frac{6004976}{5235327}a^{9}+\frac{25414871}{1745109}a^{8}-\frac{2602297}{193901}a^{7}-\frac{140800}{24579}a^{6}+\frac{31602502}{5235327}a^{5}-\frac{7369375}{5235327}a^{4}+\frac{72774580}{5235327}a^{3}-\frac{21346267}{1745109}a^{2}+\frac{6153595}{1745109}a-\frac{7582219}{5235327}$, $\frac{860995}{5235327}a^{15}-\frac{1328321}{5235327}a^{14}+\frac{1538042}{5235327}a^{13}+\frac{344953}{581703}a^{12}-\frac{2373664}{5235327}a^{11}-\frac{5523152}{5235327}a^{10}-\frac{6331487}{5235327}a^{9}+\frac{1597664}{581703}a^{8}+\frac{5149798}{1745109}a^{7}-\frac{82231}{24579}a^{6}-\frac{14800471}{5235327}a^{5}-\frac{252761}{5235327}a^{4}+\frac{13584152}{5235327}a^{3}+\frac{639058}{193901}a^{2}-\frac{394018}{581703}a-\frac{3913223}{5235327}$, $\frac{1231802}{5235327}a^{15}-\frac{4451125}{5235327}a^{14}+\frac{9379723}{5235327}a^{13}-\frac{3111688}{1745109}a^{12}+\frac{4146766}{5235327}a^{11}-\frac{8466427}{5235327}a^{10}+\frac{3276689}{5235327}a^{9}+\frac{8665339}{1745109}a^{8}-\frac{9539684}{1745109}a^{7}-\frac{479}{2731}a^{6}+\frac{4297171}{5235327}a^{5}-\frac{5431315}{5235327}a^{4}+\frac{30126817}{5235327}a^{3}-\frac{9725519}{1745109}a^{2}+\frac{4001408}{1745109}a-\frac{7542037}{5235327}$, $\frac{3132173}{5235327}a^{15}-\frac{10441699}{5235327}a^{14}+\frac{21477763}{5235327}a^{13}-\frac{5921819}{1745109}a^{12}+\frac{5532685}{5235327}a^{11}-\frac{16848058}{5235327}a^{10}+\frac{6144866}{5235327}a^{9}+\frac{20404382}{1745109}a^{8}-\frac{2316612}{193901}a^{7}-\frac{86392}{24579}a^{6}+\frac{27820042}{5235327}a^{5}-\frac{8633548}{5235327}a^{4}+\frac{54765325}{5235327}a^{3}-\frac{19899973}{1745109}a^{2}+\frac{8771176}{1745109}a-\frac{4757818}{5235327}$, $\frac{1286407}{1745109}a^{15}-\frac{4722763}{1745109}a^{14}+\frac{10267793}{1745109}a^{13}-\frac{10195018}{1745109}a^{12}+\frac{488202}{193901}a^{11}-\frac{6970453}{1745109}a^{10}+\frac{4198273}{1745109}a^{9}+\frac{24102961}{1745109}a^{8}-\frac{33789829}{1745109}a^{7}+\frac{18305}{24579}a^{6}+\frac{15461224}{1745109}a^{5}-\frac{10327649}{1745109}a^{4}+\frac{8267857}{581703}a^{3}-\frac{30284597}{1745109}a^{2}+\frac{17361089}{1745109}a-\frac{4403563}{1745109}$, $\frac{1039367}{5235327}a^{15}-\frac{4713811}{5235327}a^{14}+\frac{11469718}{5235327}a^{13}-\frac{4763090}{1745109}a^{12}+\frac{8287792}{5235327}a^{11}-\frac{5162719}{5235327}a^{10}+\frac{7714196}{5235327}a^{9}+\frac{5928932}{1745109}a^{8}-\frac{5445824}{581703}a^{7}+\frac{97613}{24579}a^{6}+\frac{26181151}{5235327}a^{5}-\frac{17850691}{5235327}a^{4}+\frac{18562105}{5235327}a^{3}-\frac{14700154}{1745109}a^{2}+\frac{10991338}{1745109}a-\frac{4821583}{5235327}$ 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:  \( 93.9208233904 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 93.9208233904 \cdot 1}{2\cdot\sqrt{169561224228515625}}\cr\approx \mathstrut & 0.277017732940 \end{aligned}\]

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 - 4*x^15 + 9*x^14 - 10*x^13 + 5*x^12 - 6*x^11 + 5*x^10 + 19*x^9 - 33*x^8 + 6*x^7 + 14*x^6 - 9*x^5 + 21*x^4 - 31*x^3 + 18*x^2 - 5*x + 1) 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 - 4*x^15 + 9*x^14 - 10*x^13 + 5*x^12 - 6*x^11 + 5*x^10 + 19*x^9 - 33*x^8 + 6*x^7 + 14*x^6 - 9*x^5 + 21*x^4 - 31*x^3 + 18*x^2 - 5*x + 1, 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 - 4*x^15 + 9*x^14 - 10*x^13 + 5*x^12 - 6*x^11 + 5*x^10 + 19*x^9 - 33*x^8 + 6*x^7 + 14*x^6 - 9*x^5 + 21*x^4 - 31*x^3 + 18*x^2 - 5*x + 1); 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 - 4*x^15 + 9*x^14 - 10*x^13 + 5*x^12 - 6*x^11 + 5*x^10 + 19*x^9 - 33*x^8 + 6*x^7 + 14*x^6 - 9*x^5 + 21*x^4 - 31*x^3 + 18*x^2 - 5*x + 1); 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

$C_4\wr C_2$ (as 16T42):

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 14 conjugacy class representatives for $C_4\wr C_2$
Character table for $C_4\wr C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-55}) \), 4.2.275.1 x2, 4.0.605.1 x2, \(\Q(\sqrt{5}, \sqrt{-11})\), 8.0.16471125.2, 8.0.411778125.2, 8.0.9150625.1

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 8 siblings: 8.0.16471125.2, 8.0.411778125.2
Degree 16 sibling: 16.4.2837698174072265625.2
Minimal sibling: 8.0.16471125.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.8.0.1}{8} }^{2}$ R R ${\href{/padicField/7.8.0.1}{8} }^{2}$ R ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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
\(3\) Copy content Toggle raw display 3.2.1.0a1.1$x^{2} + 2 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
3.2.1.0a1.1$x^{2} + 2 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
3.2.1.0a1.1$x^{2} + 2 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
3.2.1.0a1.1$x^{2} + 2 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
3.2.2.2a1.1$x^{4} + 4 x^{3} + 8 x^{2} + 11 x + 4$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
3.2.2.2a1.1$x^{4} + 4 x^{3} + 8 x^{2} + 11 x + 4$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
\(5\) Copy content Toggle raw display 5.4.2.4a1.2$x^{8} + 8 x^{6} + 8 x^{5} + 20 x^{4} + 32 x^{3} + 32 x^{2} + 16 x + 9$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
5.2.4.6a1.3$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 664 x^{4} + 704 x^{3} + 416 x^{2} + 133 x + 31$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$
\(11\) Copy content Toggle raw display 11.4.2.4a1.2$x^{8} + 16 x^{6} + 20 x^{5} + 68 x^{4} + 160 x^{3} + 132 x^{2} + 40 x + 15$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
11.4.2.4a1.2$x^{8} + 16 x^{6} + 20 x^{5} + 68 x^{4} + 160 x^{3} + 132 x^{2} + 40 x + 15$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$

Spectrum of ring of integers

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