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Properties

Label 16.0.97316019886...9849.4
Degree $16$
Signature $[0, 8]$
Discriminant $37^{6}\cdot 41^{14}$
Root discriminant $99.83$
Ramified primes $37, 41$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^5.C_2.C_2$ (as 16T258)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![170589721, 194830937, 43882564, -16647515, 1248320, 5169794, 632479, -509316, -36006, 37333, -5306, -2506, 973, 158, -46, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 46*x^14 + 158*x^13 + 973*x^12 - 2506*x^11 - 5306*x^10 + 37333*x^9 - 36006*x^8 - 509316*x^7 + 632479*x^6 + 5169794*x^5 + 1248320*x^4 - 16647515*x^3 + 43882564*x^2 + 194830937*x + 170589721)
 
gp: K = bnfinit(x^16 - 4*x^15 - 46*x^14 + 158*x^13 + 973*x^12 - 2506*x^11 - 5306*x^10 + 37333*x^9 - 36006*x^8 - 509316*x^7 + 632479*x^6 + 5169794*x^5 + 1248320*x^4 - 16647515*x^3 + 43882564*x^2 + 194830937*x + 170589721, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 46 x^{14} + 158 x^{13} + 973 x^{12} - 2506 x^{11} - 5306 x^{10} + 37333 x^{9} - 36006 x^{8} - 509316 x^{7} + 632479 x^{6} + 5169794 x^{5} + 1248320 x^{4} - 16647515 x^{3} + 43882564 x^{2} + 194830937 x + 170589721 \)

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:  \(97316019886955839743696883969849=37^{6}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $99.83$
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:  $37, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{21964356943030702395869358698783691119895795433099590859426} a^{15} + \frac{5203467295806995763625237168087166483680474464810793796521}{21964356943030702395869358698783691119895795433099590859426} a^{14} + \frac{4837772267453824263609929933313062044603262178059024474221}{21964356943030702395869358698783691119895795433099590859426} a^{13} - \frac{2625602986068160417451439460893623655421243358695988104277}{10982178471515351197934679349391845559947897716549795429713} a^{12} + \frac{2777172172896542591725110358818816197591641380151095006333}{21964356943030702395869358698783691119895795433099590859426} a^{11} - \frac{10857186619487594024652099806828490063776289635950356304215}{21964356943030702395869358698783691119895795433099590859426} a^{10} - \frac{780957020150081957451682507263897350657645211270214549931}{10982178471515351197934679349391845559947897716549795429713} a^{9} - \frac{258530477052763344232046359660020476426172313046760458779}{593631268730559524212685370237397057294480957651340293498} a^{8} + \frac{13543005183143786929754419940723351561695354819000042777}{62221974342863179591697899996554365778741630122095158242} a^{7} - \frac{448925047917618159845442385590112663421564663794085730236}{10982178471515351197934679349391845559947897716549795429713} a^{6} + \frac{10145938241307720348222476663470882920215250204031646285575}{21964356943030702395869358698783691119895795433099590859426} a^{5} - \frac{4425155668028453655006630284799690669672204976565959333069}{21964356943030702395869358698783691119895795433099590859426} a^{4} + \frac{23747326741843156733912506910733571578234711317766122475}{10982178471515351197934679349391845559947897716549795429713} a^{3} - \frac{4900583027677626628536009451280556910791294522482921738863}{21964356943030702395869358698783691119895795433099590859426} a^{2} - \frac{6351780804527348826695182480790949172091580024888748193695}{21964356943030702395869358698783691119895795433099590859426} a + \frac{34935144951264453347322662502954751483955731663646239}{1681674982239545394370213513420388264290314327624193466}$

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

Class group and class number

$C_{2}\times C_{2}$, 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:  \( -1 \) (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:  \( 421643255.118 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^5.C_2.C_2$ (as 16T258):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^5.C_2.C_2$
Character table for $C_2^5.C_2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.0.194754273881.1, 8.4.9864888234894293.1, 8.4.240607030119373.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ R R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{12}$

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
37Data not computed
$41$41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$