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Properties

Label 16.0.42832588164...5409.4
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 97^{14}$
Root discriminant $94.84$
Ramified primes $3, 97$
Class number $68$ (GRH)
Class group $[2, 34]$ (GRH)
Galois group $C_2^2 : C_8$ (as 16T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![27618832, -7607472, -11939112, 8808112, -1662839, -2049201, 1527356, -452696, 182354, -46309, 1617, 2322, -825, 139, -3, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 3*x^14 + 139*x^13 - 825*x^12 + 2322*x^11 + 1617*x^10 - 46309*x^9 + 182354*x^8 - 452696*x^7 + 1527356*x^6 - 2049201*x^5 - 1662839*x^4 + 8808112*x^3 - 11939112*x^2 - 7607472*x + 27618832)
 
gp: K = bnfinit(x^16 - 5*x^15 - 3*x^14 + 139*x^13 - 825*x^12 + 2322*x^11 + 1617*x^10 - 46309*x^9 + 182354*x^8 - 452696*x^7 + 1527356*x^6 - 2049201*x^5 - 1662839*x^4 + 8808112*x^3 - 11939112*x^2 - 7607472*x + 27618832, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} - 3 x^{14} + 139 x^{13} - 825 x^{12} + 2322 x^{11} + 1617 x^{10} - 46309 x^{9} + 182354 x^{8} - 452696 x^{7} + 1527356 x^{6} - 2049201 x^{5} - 1662839 x^{4} + 8808112 x^{3} - 11939112 x^{2} - 7607472 x + 27618832 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(42832588164190465585875603485409=3^{8}\cdot 97^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $94.84$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $3, 97$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{1648} a^{14} + \frac{11}{1648} a^{13} - \frac{295}{1648} a^{12} + \frac{159}{1648} a^{11} - \frac{301}{1648} a^{10} - \frac{137}{824} a^{9} - \frac{331}{1648} a^{8} + \frac{819}{1648} a^{7} + \frac{83}{824} a^{6} - \frac{67}{412} a^{5} - \frac{185}{412} a^{4} + \frac{783}{1648} a^{3} - \frac{423}{1648} a^{2} - \frac{97}{412} a + \frac{1}{4}$, $\frac{1}{103070248336715173006210173273961046666448108276823616} a^{15} + \frac{11298887615838152807613687678309851518022619650509}{103070248336715173006210173273961046666448108276823616} a^{14} + \frac{10820245823384553848959866672882764040484178349283607}{103070248336715173006210173273961046666448108276823616} a^{13} + \frac{2388776446681551280510147941076218583532057406821625}{103070248336715173006210173273961046666448108276823616} a^{12} - \frac{17907865499293899593782552154298941446928926260548391}{103070248336715173006210173273961046666448108276823616} a^{11} + \frac{497079966407640011629941602525668136418410615611993}{25767562084178793251552543318490261666612027069205904} a^{10} - \frac{15610498938514816243838214429746490898224037534477847}{103070248336715173006210173273961046666448108276823616} a^{9} + \frac{16773192437718307260482703140999373367467461986837757}{103070248336715173006210173273961046666448108276823616} a^{8} + \frac{6398624836540560502094829069534110631909693554000251}{25767562084178793251552543318490261666612027069205904} a^{7} - \frac{7131022731199374531009734098983205037814677109639}{62542626417909692358137241064296751617990356964092} a^{6} + \frac{1049978189600145134249653156424777324948037561127583}{25767562084178793251552543318490261666612027069205904} a^{5} - \frac{1183563936771195711251402497496095161519336013825593}{103070248336715173006210173273961046666448108276823616} a^{4} + \frac{26439239128774113166573243904831793024996349894863975}{103070248336715173006210173273961046666448108276823616} a^{3} + \frac{21508505760717960105768564991139971074186673448913487}{51535124168357586503105086636980523333224054138411808} a^{2} + \frac{69352683602068725813361771033862810705883236945079}{548246001791038154288351985499792801417277171685232} a - \frac{34850810695955387251132115088866536747483400755011}{125085252835819384716274482128593503235980713928184}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{34}$, which has order $68$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 22316334.7724 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

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

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
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

\(\Q(\sqrt{97}) \), 4.2.2738019.1, 4.4.912673.1, 4.2.28227.1, 8.4.727184560303017.1, 8.0.6544661042727153.1, 8.4.7496748044361.1

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

Sibling fields

Galois closure: data not computed
Degree 16 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
97Data not computed