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Properties

Label 16.8.14014641336...5089.1
Degree $16$
Signature $[8, 4]$
Discriminant $37^{14}\cdot 41^{15}$
Root discriminant $765.88$
Ramified primes $37, 41$
Class number $8$ (GRH)
Class group $[8]$ (GRH)
Galois group $(C_2\times C_8).D_4$ (as 16T306)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-265429377926851, -105017935194008, 19640886189987, 7056647500815, 103874593488, -96196910865, -25151684794, -2252123415, 363015846, 52941213, 1274883, -67631, -39678, -2156, 94, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 94*x^14 - 2156*x^13 - 39678*x^12 - 67631*x^11 + 1274883*x^10 + 52941213*x^9 + 363015846*x^8 - 2252123415*x^7 - 25151684794*x^6 - 96196910865*x^5 + 103874593488*x^4 + 7056647500815*x^3 + 19640886189987*x^2 - 105017935194008*x - 265429377926851)
 
gp: K = bnfinit(x^16 - x^15 + 94*x^14 - 2156*x^13 - 39678*x^12 - 67631*x^11 + 1274883*x^10 + 52941213*x^9 + 363015846*x^8 - 2252123415*x^7 - 25151684794*x^6 - 96196910865*x^5 + 103874593488*x^4 + 7056647500815*x^3 + 19640886189987*x^2 - 105017935194008*x - 265429377926851, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 94 x^{14} - 2156 x^{13} - 39678 x^{12} - 67631 x^{11} + 1274883 x^{10} + 52941213 x^{9} + 363015846 x^{8} - 2252123415 x^{7} - 25151684794 x^{6} - 96196910865 x^{5} + 103874593488 x^{4} + 7056647500815 x^{3} + 19640886189987 x^{2} - 105017935194008 x - 265429377926851 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(14014641336002292789493279203934745080101945089=37^{14}\cdot 41^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $765.88$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{524} a^{14} - \frac{41}{524} a^{13} - \frac{107}{524} a^{12} + \frac{53}{524} a^{11} + \frac{85}{524} a^{10} + \frac{31}{131} a^{9} + \frac{50}{131} a^{8} + \frac{223}{524} a^{7} + \frac{18}{131} a^{6} - \frac{111}{262} a^{5} + \frac{3}{131} a^{4} + \frac{45}{524} a^{3} - \frac{15}{262} a^{2} - \frac{71}{262} a - \frac{241}{524}$, $\frac{1}{37203473845996429103285850931889590294182562017449346102992837032932975500352541113167393138872} a^{15} + \frac{2585575104464894343551549167227768597234325151374294331603642774601912338353733869702462249}{4650434230749553637910731366486198786772820252181168262874104629116621937544067639145924142359} a^{14} - \frac{2223242354198064260817989604576873552276048061407035048846467426797078654473177055657487296125}{18601736922998214551642925465944795147091281008724673051496418516466487750176270556583696569436} a^{13} - \frac{4212044318540461814585238115796722873606811408205930939093623351100427770845418399547606263325}{18601736922998214551642925465944795147091281008724673051496418516466487750176270556583696569436} a^{12} + \frac{50691747299289898387211019303109208737689602702282276070711252134324713908460263492197548177}{9300868461499107275821462732972397573545640504362336525748209258233243875088135278291848284718} a^{11} + \frac{2447519941059541859699811232780776709846821579096149417977335697806826300311237334067063455113}{37203473845996429103285850931889590294182562017449346102992837032932975500352541113167393138872} a^{10} - \frac{55432860714426421263720274155542332386866003825446723268126627722918415106862881949665165639}{4650434230749553637910731366486198786772820252181168262874104629116621937544067639145924142359} a^{9} + \frac{5061091525998388284481849416413371795683679789213993733043024154222157835896645486346641504117}{37203473845996429103285850931889590294182562017449346102992837032932975500352541113167393138872} a^{8} - \frac{16968324282883955116872076963270060954212986146408398378833560947936079533274642610956904771665}{37203473845996429103285850931889590294182562017449346102992837032932975500352541113167393138872} a^{7} + \frac{3240330909407779683927742712031747757070343305256228998639218813990395363623414370702852353895}{9300868461499107275821462732972397573545640504362336525748209258233243875088135278291848284718} a^{6} + \frac{4676023026182388613880930341340031953860214772048747878749105270122188207167644247939666066219}{18601736922998214551642925465944795147091281008724673051496418516466487750176270556583696569436} a^{5} - \frac{15922813652074822010127671727550984441596055897854001045233849126508273914291181561440460911431}{37203473845996429103285850931889590294182562017449346102992837032932975500352541113167393138872} a^{4} - \frac{11180273621181252422507041027354369095359003630853033238664838621055072766229925569825285331023}{37203473845996429103285850931889590294182562017449346102992837032932975500352541113167393138872} a^{3} + \frac{2326908802131385681793137801975713983990070615987363329489666301570611577988179840911234162175}{9300868461499107275821462732972397573545640504362336525748209258233243875088135278291848284718} a^{2} + \frac{14946942416875850373001359524156036984021937062916089057523801197066787971128552378155543152155}{37203473845996429103285850931889590294182562017449346102992837032932975500352541113167393138872} a - \frac{461974329809573691755425576828902040491504355568514824586167600994670475084324535989829439}{361198775203848826245493698367860099943520019586886855368862495465368694178180010807450418824}$

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

Class group and class number

$C_{8}$, which has order $8$ (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:  $11$
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:  \( 14382371380200000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_8).D_4$ (as 16T306):

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 $(C_2\times C_8).D_4$
Character table for $(C_2\times C_8).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.94352849.1, 8.8.499686183762100623329.1

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

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ $16$ $16$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$37$37.8.7.2$x^{8} - 148$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
37.8.7.2$x^{8} - 148$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
41Data not computed