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Properties

Label 16.8.127...496.2
Degree $16$
Signature $[8, 4]$
Discriminant $1.273\times 10^{31}$
Root discriminant \(87.91\)
Ramified primes $2,47$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_4^2.(C_2\times C_4)$ (as 16T362)

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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 - 16*x^14 - 116*x^12 + 1888*x^10 - 98*x^8 - 26832*x^6 - 16516*x^4 + 3008*x^2 + 2209)
 
Copy content gp:K = bnfinit(y^16 - 16*y^14 - 116*y^12 + 1888*y^10 - 98*y^8 - 26832*y^6 - 16516*y^4 + 3008*y^2 + 2209, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 16*x^14 - 116*x^12 + 1888*x^10 - 98*x^8 - 26832*x^6 - 16516*x^4 + 3008*x^2 + 2209);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 16*x^14 - 116*x^12 + 1888*x^10 - 98*x^8 - 26832*x^6 - 16516*x^4 + 3008*x^2 + 2209)
 

\( x^{16} - 16x^{14} - 116x^{12} + 1888x^{10} - 98x^{8} - 26832x^{6} - 16516x^{4} + 3008x^{2} + 2209 \) 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:   \(12725851295326073899065850986496\) \(\medspace = 2^{70}\cdot 47^{6}\) 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:  \(87.91\)
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:  $2^{305/64}47^{1/2}\approx 186.48547112066305$
Ramified primes:   \(2\), \(47\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}+\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{188}a^{12}+\frac{2}{47}a^{10}+\frac{29}{188}a^{8}+\frac{23}{94}a^{6}-\frac{75}{188}a^{4}+\frac{19}{94}a^{2}-\frac{1}{4}$, $\frac{1}{188}a^{13}+\frac{2}{47}a^{11}+\frac{29}{188}a^{9}+\frac{23}{94}a^{7}-\frac{75}{188}a^{5}+\frac{19}{94}a^{3}-\frac{1}{4}a$, $\frac{1}{28407576008164}a^{14}+\frac{97945945}{151104127703}a^{12}-\frac{3834055528039}{28407576008164}a^{10}-\frac{603399376270}{7101894002041}a^{8}+\frac{1634445620901}{28407576008164}a^{6}+\frac{2545193430783}{14203788004082}a^{4}+\frac{12957544923297}{28407576008164}a^{2}-\frac{14019893447}{302208255406}$, $\frac{1}{28407576008164}a^{15}+\frac{97945945}{151104127703}a^{13}+\frac{1633919237001}{14203788004082}a^{11}-\frac{1}{4}a^{10}+\frac{4688296496961}{28407576008164}a^{9}-\frac{1}{4}a^{8}+\frac{1634445620901}{28407576008164}a^{7}+\frac{2545193430783}{14203788004082}a^{5}+\frac{1463912730314}{7101894002041}a^{3}+\frac{1}{4}a^{2}-\frac{179143914597}{604416510812}a+\frac{1}{4}$ 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:  $C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:   $\frac{1146972}{7727849839}a^{14}-\frac{18025867}{7727849839}a^{12}-\frac{138468992}{7727849839}a^{10}+\frac{4260356393}{15455699678}a^{8}+\frac{542260568}{7727849839}a^{6}-\frac{31101201656}{7727849839}a^{4}-\frac{30265222288}{7727849839}a^{2}+\frac{485786665}{328844674}$, $\frac{7929915033}{14203788004082}a^{14}-\frac{259114732713}{28407576008164}a^{12}-\frac{874546419297}{14203788004082}a^{10}+\frac{30482267166475}{28407576008164}a^{8}-\frac{6104169418183}{14203788004082}a^{6}-\frac{415551497269639}{28407576008164}a^{4}-\frac{59719349553961}{14203788004082}a^{2}+\frac{1507333469879}{604416510812}$, $\frac{7561748043}{7101894002041}a^{14}-\frac{500276323933}{28407576008164}a^{12}-\frac{809655859980}{7101894002041}a^{10}+\frac{58848886274823}{28407576008164}a^{8}-\frac{8653045501004}{7101894002041}a^{6}-\frac{792756105294791}{28407576008164}a^{4}-\frac{20652792146347}{7101894002041}a^{2}+\frac{2624128559431}{604416510812}$, $\frac{4500718925}{14203788004082}a^{14}-\frac{139032506339}{28407576008164}a^{12}-\frac{565532408231}{14203788004082}a^{10}+\frac{16523888574647}{28407576008164}a^{8}+\frac{2298363455718}{7101894002041}a^{6}-\frac{251861151680195}{28407576008164}a^{4}-\frac{64799708269950}{7101894002041}a^{2}-\frac{1472984171879}{604416510812}$, $\frac{9937450108}{7101894002041}a^{14}-\frac{161964366185}{7101894002041}a^{12}-\frac{2202785837963}{14203788004082}a^{10}+\frac{19062716879145}{7101894002041}a^{8}-\frac{7195276504820}{7101894002041}a^{6}-\frac{262190213922503}{7101894002041}a^{4}-\frac{153386006050819}{14203788004082}a^{2}+\frac{893344708837}{151104127703}$, $\frac{106034709499}{28407576008164}a^{14}-\frac{438634994719}{7101894002041}a^{12}-\frac{11331057173077}{28407576008164}a^{10}+\frac{103138086467773}{14203788004082}a^{8}-\frac{124615023044357}{28407576008164}a^{6}-\frac{13\cdots 53}{14203788004082}a^{4}-\frac{219948419521165}{28407576008164}a^{2}+\frac{1704908162957}{151104127703}$, $\frac{13490354199}{28407576008164}a^{14}-\frac{202221326165}{28407576008164}a^{12}-\frac{1790723308719}{28407576008164}a^{10}+\frac{23953462047503}{28407576008164}a^{8}+\frac{25942777618683}{28407576008164}a^{6}-\frac{367727340568025}{28407576008164}a^{4}-\frac{638229939831391}{28407576008164}a^{2}-\frac{5052080000751}{604416510812}$, $\frac{4733851224}{7101894002041}a^{15}+\frac{2834620719}{7101894002041}a^{14}-\frac{238908665985}{28407576008164}a^{13}-\frac{191476164799}{14203788004082}a^{12}-\frac{685726024760}{7101894002041}a^{11}+\frac{318926532521}{14203788004082}a^{10}+\frac{24130411608397}{28407576008164}a^{9}+\frac{27832710178959}{14203788004082}a^{8}+\frac{16904489611883}{14203788004082}a^{7}-\frac{38266329771623}{7101894002041}a^{6}-\frac{233356165589561}{28407576008164}a^{5}-\frac{494857746735145}{14203788004082}a^{4}-\frac{70932772383857}{14203788004082}a^{3}+\frac{42991457847367}{14203788004082}a^{2}+\frac{152408463287}{604416510812}a+\frac{1945080515549}{302208255406}$, $\frac{202045620007}{14203788004082}a^{15}+\frac{449173218911}{14203788004082}a^{14}-\frac{2113634207803}{14203788004082}a^{13}-\frac{4430434149499}{14203788004082}a^{12}-\frac{34561938028211}{14203788004082}a^{11}-\frac{39779849350723}{7101894002041}a^{10}+\frac{184105676994779}{14203788004082}a^{9}+\frac{181271076132007}{7101894002041}a^{8}+\frac{898673618994349}{14203788004082}a^{7}+\frac{22\cdots 67}{14203788004082}a^{6}+\frac{56742326390953}{14203788004082}a^{5}+\frac{13\cdots 15}{14203788004082}a^{4}-\frac{303355418704273}{14203788004082}a^{3}-\frac{76832376734136}{7101894002041}a^{2}-\frac{1175306507903}{302208255406}a-\frac{1670221383674}{151104127703}$, $\frac{3634802672675}{28407576008164}a^{15}+\frac{2252510645541}{28407576008164}a^{14}-\frac{57046381790581}{28407576008164}a^{13}-\frac{34782800920015}{28407576008164}a^{12}-\frac{440095459079677}{28407576008164}a^{11}-\frac{278335238586901}{28407576008164}a^{10}+\frac{67\cdots 47}{28407576008164}a^{9}+\frac{40\cdots 63}{28407576008164}a^{8}+\frac{18\cdots 85}{28407576008164}a^{7}+\frac{16\cdots 59}{28407576008164}a^{6}-\frac{97\cdots 15}{28407576008164}a^{5}-\frac{57\cdots 89}{28407576008164}a^{4}-\frac{95\cdots 35}{28407576008164}a^{3}-\frac{12\cdots 69}{604416510812}a^{2}-\frac{485187290312833}{604416510812}a-\frac{291401934972753}{604416510812}$, $\frac{7757395686610}{7101894002041}a^{15}+\frac{4631226061310}{7101894002041}a^{14}-\frac{242717643472859}{14203788004082}a^{13}-\frac{144960283330669}{14203788004082}a^{12}-\frac{40133322971387}{302208255406}a^{11}-\frac{562972136889804}{7101894002041}a^{10}+\frac{28\cdots 11}{14203788004082}a^{9}+\frac{85\cdots 43}{7101894002041}a^{8}+\frac{86\cdots 23}{14203788004082}a^{7}+\frac{25\cdots 87}{7101894002041}a^{6}-\frac{20\cdots 00}{7101894002041}a^{5}-\frac{24\cdots 39}{14203788004082}a^{4}-\frac{20\cdots 41}{7101894002041}a^{3}-\frac{12\cdots 63}{7101894002041}a^{2}-\frac{10\cdots 12}{151104127703}a-\frac{612351106728126}{151104127703}$ 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:  \( 7025756974.37 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 7025756974.37 \cdot 1}{2\cdot\sqrt{12725851295326073899065850986496}}\cr\approx \mathstrut & 0.392897390354 \end{aligned}\] (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 - 16*x^14 - 116*x^12 + 1888*x^10 - 98*x^8 - 26832*x^6 - 16516*x^4 + 3008*x^2 + 2209) 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 - 16*x^14 - 116*x^12 + 1888*x^10 - 98*x^8 - 26832*x^6 - 16516*x^4 + 3008*x^2 + 2209, 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 - 16*x^14 - 116*x^12 + 1888*x^10 - 98*x^8 - 26832*x^6 - 16516*x^4 + 3008*x^2 + 2209); 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 - 16*x^14 - 116*x^12 + 1888*x^10 - 98*x^8 - 26832*x^6 - 16516*x^4 + 3008*x^2 + 2209); 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

$C_4^2.(C_2\times C_4)$ (as 16T362):

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 128
The 14 conjugacy class representatives for $C_4^2.(C_2\times C_4)$
Character table for $C_4^2.(C_2\times C_4)$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.8.592973922304.1

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

Degree 32 siblings: data not computed
Arithmetically equivalent siblings: data not computed
Minimal sibling: 16.8.12725851295326073899065850986496.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.4.0.1}{4} }^{4}$ ${\href{/padicField/5.4.0.1}{4} }^{4}$ ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.8.0.1}{8} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.8.0.1}{8} }^{2}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ R ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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 2.1.16.70e1.3585$x^{16} + 16 x^{15} + 16 x^{13} + 8 x^{12} + 8 x^{10} + 16 x^{9} + 4 x^{8} + 16 x^{7} + 56 x^{4} + 2$$16$$1$$70$16T362$$[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, \frac{21}{4}]$$
\(47\) Copy content Toggle raw display $\Q_{47}$$x + 42$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{47}$$x + 42$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{47}$$x + 42$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{47}$$x + 42$$1$$1$$0$Trivial$$[\ ]$$
47.1.2.1a1.2$x^{2} + 235$$2$$1$$1$$C_2$$$[\ ]_{2}$$
47.1.2.1a1.2$x^{2} + 235$$2$$1$$1$$C_2$$$[\ ]_{2}$$
47.1.2.1a1.2$x^{2} + 235$$2$$1$$1$$C_2$$$[\ ]_{2}$$
47.1.2.1a1.2$x^{2} + 235$$2$$1$$1$$C_2$$$[\ ]_{2}$$
47.1.2.1a1.2$x^{2} + 235$$2$$1$$1$$C_2$$$[\ ]_{2}$$
47.1.2.1a1.2$x^{2} + 235$$2$$1$$1$$C_2$$$[\ ]_{2}$$

Spectrum of ring of integers

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