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Properties

Label 16.0.22138542742...0625.4
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 41^{6}\cdot 661^{4}$
Root discriminant $68.25$
Ramified primes $5, 41, 661$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T864

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8851257136, -2622967800, -2369999976, 11780888, 432811443, 139657834, -75731926, -16608204, 6911639, 630514, -254277, -17750, 5257, 362, -81, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 81*x^14 + 362*x^13 + 5257*x^12 - 17750*x^11 - 254277*x^10 + 630514*x^9 + 6911639*x^8 - 16608204*x^7 - 75731926*x^6 + 139657834*x^5 + 432811443*x^4 + 11780888*x^3 - 2369999976*x^2 - 2622967800*x + 8851257136)
 
gp: K = bnfinit(x^16 - 4*x^15 - 81*x^14 + 362*x^13 + 5257*x^12 - 17750*x^11 - 254277*x^10 + 630514*x^9 + 6911639*x^8 - 16608204*x^7 - 75731926*x^6 + 139657834*x^5 + 432811443*x^4 + 11780888*x^3 - 2369999976*x^2 - 2622967800*x + 8851257136, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 81 x^{14} + 362 x^{13} + 5257 x^{12} - 17750 x^{11} - 254277 x^{10} + 630514 x^{9} + 6911639 x^{8} - 16608204 x^{7} - 75731926 x^{6} + 139657834 x^{5} + 432811443 x^{4} + 11780888 x^{3} - 2369999976 x^{2} - 2622967800 x + 8851257136 \)

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:  \(221385427428590205586181640625=5^{12}\cdot 41^{6}\cdot 661^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68.25$
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:  $5, 41, 661$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{492} a^{12} - \frac{11}{123} a^{11} - \frac{55}{246} a^{10} + \frac{17}{82} a^{9} - \frac{31}{492} a^{8} - \frac{20}{123} a^{7} - \frac{23}{246} a^{6} - \frac{17}{123} a^{5} + \frac{81}{164} a^{4} + \frac{40}{123} a^{3} + \frac{95}{492} a^{2} + \frac{31}{82} a + \frac{40}{123}$, $\frac{1}{492} a^{13} - \frac{13}{82} a^{11} - \frac{16}{123} a^{10} + \frac{29}{492} a^{9} + \frac{8}{123} a^{8} - \frac{61}{246} a^{7} + \frac{61}{246} a^{6} - \frac{43}{492} a^{5} + \frac{7}{123} a^{4} + \frac{1}{492} a^{3} + \frac{46}{123} a^{2} + \frac{113}{246} a + \frac{38}{123}$, $\frac{1}{984} a^{14} - \frac{1}{984} a^{13} + \frac{13}{492} a^{11} + \frac{1}{8} a^{10} + \frac{29}{328} a^{9} - \frac{14}{123} a^{8} + \frac{77}{492} a^{7} - \frac{21}{328} a^{6} - \frac{313}{984} a^{5} + \frac{77}{328} a^{4} - \frac{43}{328} a^{3} + \frac{3}{41} a^{2} + \frac{103}{246} a - \frac{58}{123}$, $\frac{1}{68469361983680629133613744791870625113053830739028695019256} a^{15} - \frac{1140022926131408803282580972286480729566084582722388177}{22823120661226876377871248263956875037684610246342898339752} a^{14} - \frac{3306321513336755179084044580175432991137317085876782117}{34234680991840314566806872395935312556526915369514347509628} a^{13} + \frac{406028081668802413830644489082300447443679185570544452}{2852890082653359547233906032994609379710576280792862292469} a^{12} + \frac{3194394297657684064657645804400830287642921532540736235041}{22823120661226876377871248263956875037684610246342898339752} a^{11} - \frac{12821342356359405717183804917430528667127764295994725585431}{68469361983680629133613744791870625113053830739028695019256} a^{10} + \frac{51615526538805122737701989454190004469940255236850114685}{368114849374627038352762068773497984478784036231337069996} a^{9} - \frac{337016936131987201315667430572006884057244910318241861480}{2852890082653359547233906032994609379710576280792862292469} a^{8} - \frac{11919831150519327757390002640186401494601276678196415862095}{68469361983680629133613744791870625113053830739028695019256} a^{7} - \frac{146068867729052221720780578483251615073149010858861666997}{2208689096247762230116572412640987906872704217388022419976} a^{6} + \frac{7973218078076123132195078624901526884825762945541905081817}{68469361983680629133613744791870625113053830739028695019256} a^{5} + \frac{16449845665036828692432027648077432685108737002463712007163}{68469361983680629133613744791870625113053830739028695019256} a^{4} - \frac{3669334787032917032382251138904201044212578950267360212983}{11411560330613438188935624131978437518842305123171449169876} a^{3} + \frac{5331541737688472674428583546371817901779889036764470191425}{34234680991840314566806872395935312556526915369514347509628} a^{2} + \frac{6107333415036721664058305189807518184699431727418437286763}{17117340495920157283403436197967656278263457684757173754814} a + \frac{129644464187302028980118644386700998290562131142761574015}{2852890082653359547233906032994609379710576280792862292469}$

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:  \( \frac{3978400208909123013736137199800842732381777}{35271387997655475616847072916125510914217814142838} a^{15} - \frac{98147738157053727493465293200963509094950859}{423256655971865707402164874993506130970613769714056} a^{14} - \frac{1285745992357102937174742969982774683582660895}{141085551990621902467388291664502043656871256571352} a^{13} + \frac{2245676358115778297048564437138151588973221741}{105814163992966426850541218748376532742653442428514} a^{12} + \frac{126931828710628509668953230888925625403697018577}{211628327985932853701082437496753065485306884857028} a^{11} - \frac{288013204289883219539140629913169815669539499769}{423256655971865707402164874993506130970613769714056} a^{10} - \frac{11766487175331131687465341697573478959917559328433}{423256655971865707402164874993506130970613769714056} a^{9} + \frac{1055006890954234282671862064326171019483133262391}{105814163992966426850541218748376532742653442428514} a^{8} + \frac{147483214071184065909216838836147902627727123364289}{211628327985932853701082437496753065485306884857028} a^{7} - \frac{39394758697587511719243890152065842965021471116209}{141085551990621902467388291664502043656871256571352} a^{6} - \frac{940050742974543237378351165281472131935464730727859}{141085551990621902467388291664502043656871256571352} a^{5} - \frac{1305072962064520025317942298632309872892507408886357}{423256655971865707402164874993506130970613769714056} a^{4} + \frac{3830496426100566082315618384795595383309771016710393}{141085551990621902467388291664502043656871256571352} a^{3} + \frac{3794654651304602382052478559023484131055296857639498}{52907081996483213425270609374188266371326721214257} a^{2} - \frac{2826037450900485203142294054997774126854607688741189}{52907081996483213425270609374188266371326721214257} a - \frac{13262900754316954241640748518715783069277638301150498}{52907081996483213425270609374188266371326721214257} \) (order $10$)
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:  \( 148204050.915 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T864:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 41 conjugacy class representatives for t16n864
Character table for t16n864 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.26265625.2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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

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
5Data not computed
$41$41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.4.3.4$x^{4} + 8856$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.2$x^{4} - 1476$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
661Data not computed