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Properties

Label 16.0.14247797437...7289.2
Degree $16$
Signature $[0, 8]$
Discriminant $67^{12}\cdot 89^{15}$
Root discriminant $1574.38$
Ramified primes $67, 89$
Class number $384$ (GRH)
Class group $[2, 4, 48]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T260)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![59350353865950707, -8100247836751296, 2159497799389466, -183630121999979, 42709216585313, -1054372576950, 681690587107, -33095688555, -3534340512, -866862815, 2204199, 2734522, 374025, 6249, 30, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 30*x^14 + 6249*x^13 + 374025*x^12 + 2734522*x^11 + 2204199*x^10 - 866862815*x^9 - 3534340512*x^8 - 33095688555*x^7 + 681690587107*x^6 - 1054372576950*x^5 + 42709216585313*x^4 - 183630121999979*x^3 + 2159497799389466*x^2 - 8100247836751296*x + 59350353865950707)
 
gp: K = bnfinit(x^16 - 8*x^15 + 30*x^14 + 6249*x^13 + 374025*x^12 + 2734522*x^11 + 2204199*x^10 - 866862815*x^9 - 3534340512*x^8 - 33095688555*x^7 + 681690587107*x^6 - 1054372576950*x^5 + 42709216585313*x^4 - 183630121999979*x^3 + 2159497799389466*x^2 - 8100247836751296*x + 59350353865950707, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 30 x^{14} + 6249 x^{13} + 374025 x^{12} + 2734522 x^{11} + 2204199 x^{10} - 866862815 x^{9} - 3534340512 x^{8} - 33095688555 x^{7} + 681690587107 x^{6} - 1054372576950 x^{5} + 42709216585313 x^{4} - 183630121999979 x^{3} + 2159497799389466 x^{2} - 8100247836751296 x + 59350353865950707 \)

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:  \(1424779743739660246627742650217447173810309649827289=67^{12}\cdot 89^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1574.38$
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:  $67, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{42} a^{13} - \frac{17}{42} a^{12} - \frac{2}{7} a^{11} + \frac{13}{42} a^{10} - \frac{5}{42} a^{9} + \frac{2}{21} a^{8} - \frac{17}{42} a^{7} + \frac{5}{42} a^{6} + \frac{8}{21} a^{5} + \frac{13}{42} a^{4} - \frac{1}{2} a^{3} - \frac{10}{21} a^{2} + \frac{3}{14} a + \frac{1}{6}$, $\frac{1}{1764} a^{14} + \frac{65}{252} a^{12} - \frac{695}{1764} a^{11} - \frac{73}{147} a^{10} - \frac{139}{588} a^{9} - \frac{235}{588} a^{8} - \frac{218}{441} a^{7} - \frac{361}{1764} a^{6} + \frac{13}{196} a^{5} - \frac{341}{882} a^{4} - \frac{461}{1764} a^{3} - \frac{793}{1764} a^{2} - \frac{191}{441} a + \frac{47}{252}$, $\frac{1}{21968463598442226823019487490094214751912529001675551247199048586227661735838212090661013371639882331365529655592} a^{15} - \frac{5151566058780484756349548406445148345368835864258282679575580732357104059401094515220828622237957528015115785}{21968463598442226823019487490094214751912529001675551247199048586227661735838212090661013371639882331365529655592} a^{14} + \frac{14840121864951238180896993813414821821238017984031274759522930032050880517577499607969450969723590521324850009}{3138351942634603831859926784299173535987504143096507321028435512318237390834030298665859053091411761623647093656} a^{13} + \frac{1406325720385164190085523477576686357348956916710238224657159770033523761515646655620627484237029341226542761801}{10984231799221113411509743745047107375956264500837775623599524293113830867919106045330506685819941165682764827796} a^{12} - \frac{285063790469659989568507570104210035764593121229109628274454768303512604285491559286422986485652156950355114335}{3138351942634603831859926784299173535987504143096507321028435512318237390834030298665859053091411761623647093656} a^{11} + \frac{486966560345519641670061539029196026883727750745847392038053316374258539865716943316861560531752887512034557}{1059895961713814195157016813339808691654003425564507707203119051779208845266474265000290122624590260595625496} a^{10} - \frac{10605770121187461903473272688512705505129025286423099966244417065195592498449187803200220398740967547185246143}{610235099956728522861652430280394854219792472268765312422195794061879492662172558073917038101107842537931379322} a^{9} - \frac{2821885705566309499777288063761016459588308184545164086739469509928206203183466065539358715334050800869719005499}{21968463598442226823019487490094214751912529001675551247199048586227661735838212090661013371639882331365529655592} a^{8} + \frac{4965635286173945437857553796315042103034663718359541832278222045634015114097336449120176031309372671916689477851}{21968463598442226823019487490094214751912529001675551247199048586227661735838212090661013371639882331365529655592} a^{7} + \frac{340141237665815101367273943648577801108585289771890366331800874549321953518676357433111412937317083002118306831}{1569175971317301915929963392149586767993752071548253660514217756159118695417015149332929526545705880811823546828} a^{6} - \frac{5625526680460504653287173562398095837150328782057405233834963881328306267884339657488330567522577710864570309319}{21968463598442226823019487490094214751912529001675551247199048586227661735838212090661013371639882331365529655592} a^{5} + \frac{7534320465252198977295338025337772103538058968735096065984785684614248258875912870999605548032389309376277130841}{21968463598442226823019487490094214751912529001675551247199048586227661735838212090661013371639882331365529655592} a^{4} + \frac{832105379884701353966555974990893641654358992563769293078214744710996376125219019467581411795278631473175472118}{2746057949805278352877435936261776843989066125209443905899881073278457716979776511332626671454985291420691206949} a^{3} + \frac{554538699398920024107155532599912829010441653176192727020790772707935873335384708029246845492934367356524723243}{3138351942634603831859926784299173535987504143096507321028435512318237390834030298665859053091411761623647093656} a^{2} + \frac{6126686280655644149241554479272322619089016071686834139793323674313656408683882452661280328059414867554316430429}{21968463598442226823019487490094214751912529001675551247199048586227661735838212090661013371639882331365529655592} a - \frac{235066234539380119037706243885791059723526350727385991363606759375926058784336428090009089809502082024842935267}{3138351942634603831859926784299173535987504143096507321028435512318237390834030298665859053091411761623647093656}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{48}$, which has order $384$ (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:  \( 62393778709200000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T260):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-5963}) \), 4.0.3164605841.1, 8.0.891310981471327238009.2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ $16$ $16$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

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
$67$67.8.6.2$x^{8} + 1541 x^{4} + 646416$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
67.8.6.2$x^{8} + 1541 x^{4} + 646416$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
89Data not computed