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Properties

Label 16.0.14247797437...7289.4
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![683884486037949879, -372235254486644566, 68539999241660465, -3212027220574933, 70407651610811, -21501492709695, 1048437250287, 143085855695, -22022514513, 1033057887, 73569971, -4116907, 219047, 5113, -417, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 417*x^14 + 5113*x^13 + 219047*x^12 - 4116907*x^11 + 73569971*x^10 + 1033057887*x^9 - 22022514513*x^8 + 143085855695*x^7 + 1048437250287*x^6 - 21501492709695*x^5 + 70407651610811*x^4 - 3212027220574933*x^3 + 68539999241660465*x^2 - 372235254486644566*x + 683884486037949879)
 
gp: K = bnfinit(x^16 - 6*x^15 - 417*x^14 + 5113*x^13 + 219047*x^12 - 4116907*x^11 + 73569971*x^10 + 1033057887*x^9 - 22022514513*x^8 + 143085855695*x^7 + 1048437250287*x^6 - 21501492709695*x^5 + 70407651610811*x^4 - 3212027220574933*x^3 + 68539999241660465*x^2 - 372235254486644566*x + 683884486037949879, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 417 x^{14} + 5113 x^{13} + 219047 x^{12} - 4116907 x^{11} + 73569971 x^{10} + 1033057887 x^{9} - 22022514513 x^{8} + 143085855695 x^{7} + 1048437250287 x^{6} - 21501492709695 x^{5} + 70407651610811 x^{4} - 3212027220574933 x^{3} + 68539999241660465 x^{2} - 372235254486644566 x + 683884486037949879 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{18} a^{14} + \frac{1}{18} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{7}{18} a^{10} + \frac{1}{18} a^{9} + \frac{7}{18} a^{8} - \frac{1}{2} a^{7} + \frac{5}{18} a^{6} - \frac{5}{18} a^{5} + \frac{5}{18} a^{4} + \frac{1}{18} a^{3} + \frac{7}{18} a^{2} - \frac{1}{18} a + \frac{1}{6}$, $\frac{1}{7307037332047068502621691316124469656029603183917612891749581459629530282495884499117885186454182883077726001057625523676713086} a^{15} + \frac{629954259281407833956216851113343805898997971984532761718251802157836410331333532268055570034506662500067436931771999319971}{119787497246673254141339201903679830426714806293731358881140679666057873483539090149473527646789883329143049197665992191421526} a^{14} - \frac{116226225091153756543959753111991651161051720480132553160187478828767820514282377141606800672364942713349152815604799646544911}{7307037332047068502621691316124469656029603183917612891749581459629530282495884499117885186454182883077726001057625523676713086} a^{13} - \frac{885631736061726804831078140306905848430380379534729746998475858165731695016785657414196118967467738571168919943033832860953593}{2435679110682356167540563772041489885343201061305870963916527153209843427498628166372628395484727627692575333685875174558904362} a^{12} - \frac{3093017960786675782666221643818267663013103083363325546812202865633781757614289712740267154355328698617591490803708815693254729}{7307037332047068502621691316124469656029603183917612891749581459629530282495884499117885186454182883077726001057625523676713086} a^{11} + \frac{129264157425957260470209036109196326648067018435842727152862988170967875493940956246929267958099662697237041469454550553546329}{2435679110682356167540563772041489885343201061305870963916527153209843427498628166372628395484727627692575333685875174558904362} a^{10} + \frac{709937390391962615270394316167502540587364921843403326426830037815232710237958754689602771108622815974204902621869996681351023}{7307037332047068502621691316124469656029603183917612891749581459629530282495884499117885186454182883077726001057625523676713086} a^{9} - \frac{2688628312874298445274832880524709684063946321442109249602006641672372099043142083684234371055793908801363708624851004304564553}{7307037332047068502621691316124469656029603183917612891749581459629530282495884499117885186454182883077726001057625523676713086} a^{8} + \frac{3470680754609839197640667074880481885606384061890761730591083257305657991707233085744960413682178423929615600767573523302782169}{7307037332047068502621691316124469656029603183917612891749581459629530282495884499117885186454182883077726001057625523676713086} a^{7} - \frac{478105910032749577325487144700446384624418027499058271368290187296221108307031406595539519287611813577538692254223770805046399}{2435679110682356167540563772041489885343201061305870963916527153209843427498628166372628395484727627692575333685875174558904362} a^{6} - \frac{332614035816201935375308255523572396129120309288351523573327603349529761285995397072441675188305225148477479704941172298959611}{811893036894118722513521257347163295114400353768623654638842384403281142499542722124209465161575875897525111228625058186301454} a^{5} - \frac{1066013036461883228738677160347936645497584552273280374293793679605205980710762497188366899447576034513994814020735142063103243}{2435679110682356167540563772041489885343201061305870963916527153209843427498628166372628395484727627692575333685875174558904362} a^{4} + \frac{2964859652198373990408538278543612528801669615694163048921909343196657703686370431765989946535047023052287475734984598718081231}{7307037332047068502621691316124469656029603183917612891749581459629530282495884499117885186454182883077726001057625523676713086} a^{3} + \frac{1268629559050510388753883588025071385276409051140928883021164927256787751010260994619200169100721504131462017821072352698353}{13309721916297028237926577989297758936301645143747928764571186629561985942615454461052614182976653703238116577518443576824614} a^{2} + \frac{801260430677776230935516681018809987413577286164174114958112186675389820304373563487895572901939798319282198470843602523830561}{7307037332047068502621691316124469656029603183917612891749581459629530282495884499117885186454182883077726001057625523676713086} a + \frac{112424584845438119776394941625851971703184603711791885641796587376781703861526234047246533962217084264505563864185262506457651}{1217839555341178083770281886020744942671600530652935481958263576604921713749314083186314197742363813846287666842937587279452181}$

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:  \( 121032964497000000 \) (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, 8.0.891310981471327238009.3

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ $16$ $16$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

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