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Properties

Label 16.8.48244291271...9984.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{48}\cdot 1889^{5}\cdot 8441569^{2}$
Root discriminant $620.46$
Ramified primes $2, 1889, 8441569$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1186

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![123315291818966506466, 11490145946673401496, -371206890725959692, -133813426495447008, -70735834258824534, -1755899218481624, -268140864788520, -3265494601808, 565392805539, 7159006480, 2771892296, 13075880, 384562, -6536, -3380, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3380*x^14 - 6536*x^13 + 384562*x^12 + 13075880*x^11 + 2771892296*x^10 + 7159006480*x^9 + 565392805539*x^8 - 3265494601808*x^7 - 268140864788520*x^6 - 1755899218481624*x^5 - 70735834258824534*x^4 - 133813426495447008*x^3 - 371206890725959692*x^2 + 11490145946673401496*x + 123315291818966506466)
 
gp: K = bnfinit(x^16 - 3380*x^14 - 6536*x^13 + 384562*x^12 + 13075880*x^11 + 2771892296*x^10 + 7159006480*x^9 + 565392805539*x^8 - 3265494601808*x^7 - 268140864788520*x^6 - 1755899218481624*x^5 - 70735834258824534*x^4 - 133813426495447008*x^3 - 371206890725959692*x^2 + 11490145946673401496*x + 123315291818966506466, 1)
 

Normalized defining polynomial

\( x^{16} - 3380 x^{14} - 6536 x^{13} + 384562 x^{12} + 13075880 x^{11} + 2771892296 x^{10} + 7159006480 x^{9} + 565392805539 x^{8} - 3265494601808 x^{7} - 268140864788520 x^{6} - 1755899218481624 x^{5} - 70735834258824534 x^{4} - 133813426495447008 x^{3} - 371206890725959692 x^{2} + 11490145946673401496 x + 123315291818966506466 \)

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:  \(482442912719743132475274701949659184820649984=2^{48}\cdot 1889^{5}\cdot 8441569^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $620.46$
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:  $2, 1889, 8441569$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{18951322405} a^{14} + \frac{1481022826}{18951322405} a^{13} + \frac{330165242}{3790264481} a^{12} - \frac{1108647857}{18951322405} a^{11} - \frac{1071741349}{3790264481} a^{10} + \frac{5138399267}{18951322405} a^{9} + \frac{7043142648}{18951322405} a^{8} - \frac{7088850874}{18951322405} a^{7} - \frac{6681733638}{18951322405} a^{6} + \frac{3989548958}{18951322405} a^{5} - \frac{2310219184}{18951322405} a^{4} - \frac{260708951}{18951322405} a^{3} - \frac{5897715806}{18951322405} a^{2} + \frac{1449109297}{18951322405} a + \frac{4816314871}{18951322405}$, $\frac{1}{196623263918665130627687352214570146988295821184360457008259625135881371279088091728949086828335068568923440640535928391922233179166249726085} a^{15} - \frac{166695694596035599030728054242517921247803555498542729161754589382589111121966352430966842190607618882492215860320436766781007218}{196623263918665130627687352214570146988295821184360457008259625135881371279088091728949086828335068568923440640535928391922233179166249726085} a^{14} + \frac{3194455898448645677904510844667662953565861422545440998810581354884696264751457247220594643901691132105834632930670809849017797599839874911}{196623263918665130627687352214570146988295821184360457008259625135881371279088091728949086828335068568923440640535928391922233179166249726085} a^{13} + \frac{50456516540861883852998534931416392388752800511267136985236357310344829053784739652452456863201378178628685814244775014135181451449697925913}{196623263918665130627687352214570146988295821184360457008259625135881371279088091728949086828335068568923440640535928391922233179166249726085} a^{12} + \frac{12807696644564097616000442883490930482885065681119351647192845230285799373427909111625460224236637477723788693640417389386393759165137125008}{196623263918665130627687352214570146988295821184360457008259625135881371279088091728949086828335068568923440640535928391922233179166249726085} a^{11} + \frac{15302781203578770742498091513369456668782042317502874632745400625142957595044184317641335089619581001380768677366627438114436077841099539677}{196623263918665130627687352214570146988295821184360457008259625135881371279088091728949086828335068568923440640535928391922233179166249726085} a^{10} - \frac{9314439022834361357955840267249618267209357190574500880476656374347732169349119519559976631632683147571259817882860487620568926421110773385}{39324652783733026125537470442914029397659164236872091401651925027176274255817618345789817365667013713784688128107185678384446635833249945217} a^{9} + \frac{35375399823528989037784528894864846918974287678060284655422340652067855919862439942697048073703416288357574855826793421057863541756186366474}{196623263918665130627687352214570146988295821184360457008259625135881371279088091728949086828335068568923440640535928391922233179166249726085} a^{8} - \frac{68652052003694342452002835145816051221435436515332886803958268999869661054737380722359375480395095160752758371407354106295979766266955809947}{196623263918665130627687352214570146988295821184360457008259625135881371279088091728949086828335068568923440640535928391922233179166249726085} a^{7} - \frac{18951661984205927268869316775234235314505940256600092754990465361890622500655366462423280125656164407926264198682616189802387450756309575991}{39324652783733026125537470442914029397659164236872091401651925027176274255817618345789817365667013713784688128107185678384446635833249945217} a^{6} + \frac{66599469333940086854805651225105526951204592295235847230933223829262548034030664185249537237841300995593321075503994424277606400583961499294}{196623263918665130627687352214570146988295821184360457008259625135881371279088091728949086828335068568923440640535928391922233179166249726085} a^{5} + \frac{1674231368249820314417465135712914619112841864951730132726919741049905420698400718013154305002514150304489170368104994322123380194185980831}{39324652783733026125537470442914029397659164236872091401651925027176274255817618345789817365667013713784688128107185678384446635833249945217} a^{4} + \frac{4425645423583536436055094958815627979015536907399866595561827468618975301400251548845144748839947425248685587308377522849062443159642137393}{196623263918665130627687352214570146988295821184360457008259625135881371279088091728949086828335068568923440640535928391922233179166249726085} a^{3} + \frac{37478513278745570852659732882126619762918074905958689003648669537249472086276507492908364140547982599162929191858653440272220534321173434651}{196623263918665130627687352214570146988295821184360457008259625135881371279088091728949086828335068568923440640535928391922233179166249726085} a^{2} + \frac{62366189435835493628453105499934770076103134000018380950298201737408049878923152601762844363288555288307170024448778271157310051824338046953}{196623263918665130627687352214570146988295821184360457008259625135881371279088091728949086828335068568923440640535928391922233179166249726085} a - \frac{27196181858808286534368512236238535200416830544403298474001207832671366590670588767810044686589230640510566308512124740426541971104856849764}{196623263918665130627687352214570146988295821184360457008259625135881371279088091728949086828335068568923440640535928391922233179166249726085}$

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:  $11$
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:  \( 13617365774000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1186:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 46 conjugacy class representatives for t16n1186
Character table for t16n1186 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.8.7923040256.1

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 R $16$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.24.10$x^{8} + 16$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
2.8.24.10$x^{8} + 16$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
1889Data not computed
8441569Data not computed