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Properties

Label 16.0.44157307385...8097.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 97^{13}$
Root discriminant $71.25$
Ramified primes $3, 97$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T289)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![87721, 79985, 157928, -93725, 337044, 222822, 59197, 91739, 62090, -13264, -13574, 1148, 1730, 35, -72, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 72*x^14 + 35*x^13 + 1730*x^12 + 1148*x^11 - 13574*x^10 - 13264*x^9 + 62090*x^8 + 91739*x^7 + 59197*x^6 + 222822*x^5 + 337044*x^4 - 93725*x^3 + 157928*x^2 + 79985*x + 87721)
 
gp: K = bnfinit(x^16 - 2*x^15 - 72*x^14 + 35*x^13 + 1730*x^12 + 1148*x^11 - 13574*x^10 - 13264*x^9 + 62090*x^8 + 91739*x^7 + 59197*x^6 + 222822*x^5 + 337044*x^4 - 93725*x^3 + 157928*x^2 + 79985*x + 87721, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 72 x^{14} + 35 x^{13} + 1730 x^{12} + 1148 x^{11} - 13574 x^{10} - 13264 x^{9} + 62090 x^{8} + 91739 x^{7} + 59197 x^{6} + 222822 x^{5} + 337044 x^{4} - 93725 x^{3} + 157928 x^{2} + 79985 x + 87721 \)

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:  \(441573073857633665833769108097=3^{8}\cdot 97^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.25$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{962983920835710704386594201628098159861367715773} a^{15} + \frac{151661597968671546894837480237339088548066843159}{962983920835710704386594201628098159861367715773} a^{14} + \frac{474825016579997423504101681954468256809786297360}{962983920835710704386594201628098159861367715773} a^{13} - \frac{252478898715124893002352306276203895012641460254}{962983920835710704386594201628098159861367715773} a^{12} - \frac{319341840780444890575577329902556095041971366947}{962983920835710704386594201628098159861367715773} a^{11} - \frac{108264278800289011808576998414582427787808527923}{962983920835710704386594201628098159861367715773} a^{10} + \frac{177037298257555752254606938531248382337855039102}{962983920835710704386594201628098159861367715773} a^{9} - \frac{96497359861604888538605026024280978290730645835}{962983920835710704386594201628098159861367715773} a^{8} - \frac{18419656116876753980178660913904306007216243498}{962983920835710704386594201628098159861367715773} a^{7} + \frac{382127055915696022020118353751196006914293694927}{962983920835710704386594201628098159861367715773} a^{6} - \frac{128140171508256801156945601633562375298221863327}{962983920835710704386594201628098159861367715773} a^{5} - \frac{158302012761801407131399693983276060041165102785}{962983920835710704386594201628098159861367715773} a^{4} + \frac{135656674561513538928328853955610692279988948695}{962983920835710704386594201628098159861367715773} a^{3} + \frac{141960535734200598164324042832944172467218606277}{962983920835710704386594201628098159861367715773} a^{2} - \frac{113310692292001139857700598220795337332289747783}{962983920835710704386594201628098159861367715773} a - \frac{117387518655688097609603961710165665565018945180}{962983920835710704386594201628098159861367715773}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( \frac{2475638524556353805067203}{725586800318428447068314216129} a^{15} - \frac{7831173201742465246578600}{725586800318428447068314216129} a^{14} - \frac{172262284349322798908248744}{725586800318428447068314216129} a^{13} + \frac{292259433104916196866827210}{725586800318428447068314216129} a^{12} + \frac{4171342826626793490753196657}{725586800318428447068314216129} a^{11} - \frac{2049010912259052100256315358}{725586800318428447068314216129} a^{10} - \frac{36767717137095873364159817340}{725586800318428447068314216129} a^{9} + \frac{4712633925790957324158665750}{725586800318428447068314216129} a^{8} + \frac{192276702719408665524693806164}{725586800318428447068314216129} a^{7} + \frac{57038615102003250817917753016}{725586800318428447068314216129} a^{6} - \frac{129790372624164160805174898044}{725586800318428447068314216129} a^{5} + \frac{367239234383181361919430939211}{725586800318428447068314216129} a^{4} + \frac{271498001454160684966409591561}{725586800318428447068314216129} a^{3} - \frac{1360238530301925428731835758362}{725586800318428447068314216129} a^{2} + \frac{381412522245624557704659609374}{725586800318428447068314216129} a + \frac{296077233591924523970722728333}{725586800318428447068314216129} \) (order $6$)
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:  \( 56013234.4844 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T289):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 44 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.873.1, 8.0.695574560817.1

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

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
3Data not computed
$97$97.8.7.7$x^{8} + 303125$$8$$1$$7$$C_8$$[\ ]_{8}$
97.8.6.3$x^{8} - 97 x^{4} + 47045$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$