↑

Properties

Label 16.0.27668269071...3481.5
Degree $16$
Signature $[0, 8]$
Discriminant $41^{15}\cdot 59^{12}$
Root discriminant $692.03$
Ramified primes $41, 59$
Class number $256$ (GRH)
Class group $[2, 2, 2, 2, 16]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T260)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![36644678113511, 22845585920160, 15997195291737, -142328996665, -2336821044932, 237616125806, 87603516131, -23486633086, 3194261138, -304305287, 18522881, -644417, -15576, 2442, -19, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 19*x^14 + 2442*x^13 - 15576*x^12 - 644417*x^11 + 18522881*x^10 - 304305287*x^9 + 3194261138*x^8 - 23486633086*x^7 + 87603516131*x^6 + 237616125806*x^5 - 2336821044932*x^4 - 142328996665*x^3 + 15997195291737*x^2 + 22845585920160*x + 36644678113511)
 
gp: K = bnfinit(x^16 - 6*x^15 - 19*x^14 + 2442*x^13 - 15576*x^12 - 644417*x^11 + 18522881*x^10 - 304305287*x^9 + 3194261138*x^8 - 23486633086*x^7 + 87603516131*x^6 + 237616125806*x^5 - 2336821044932*x^4 - 142328996665*x^3 + 15997195291737*x^2 + 22845585920160*x + 36644678113511, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 19 x^{14} + 2442 x^{13} - 15576 x^{12} - 644417 x^{11} + 18522881 x^{10} - 304305287 x^{9} + 3194261138 x^{8} - 23486633086 x^{7} + 87603516131 x^{6} + 237616125806 x^{5} - 2336821044932 x^{4} - 142328996665 x^{3} + 15997195291737 x^{2} + 22845585920160 x + 36644678113511 \)

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:  \(2766826907141591867309101193151875470463173481=41^{15}\cdot 59^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $692.03$
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:  $41, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{11} a^{14} - \frac{5}{11} a^{13} - \frac{4}{11} a^{12} - \frac{5}{11} a^{11} + \frac{3}{11} a^{10} - \frac{2}{11} a^{9} - \frac{5}{11} a^{8} - \frac{1}{11} a^{7} - \frac{2}{11} a^{5} - \frac{3}{11} a^{4} + \frac{5}{11} a^{3} + \frac{3}{11} a + \frac{3}{11}$, $\frac{1}{86263145734179615861283807147704676820889400892628790101246152021966594405478659553998995868892201670615} a^{15} - \frac{134371241648470867162252490928590238806804340326283622489941014934422786937957987015304247356872457787}{7842104157652692351025800649791334256444490990238980918295104729269690400498059959454454169899291060965} a^{14} - \frac{18770644015850099279635536371706562900847265628085982890281139355655134366596607383904161378801995078897}{86263145734179615861283807147704676820889400892628790101246152021966594405478659553998995868892201670615} a^{13} - \frac{41666189929798361046470207522152895667202977393104381386513091394822432840520547137207163151823584978246}{86263145734179615861283807147704676820889400892628790101246152021966594405478659553998995868892201670615} a^{12} - \frac{467608313175879992167759520727912883937409677324875351821267366174013448665721591502044130647871540088}{1568420831530538470205160129958266851288898198047796183659020945853938080099611991890890833979858212193} a^{11} + \frac{37074394809575486500520957522954841117402747786115397149008366732965732251028647252790931120974268324373}{86263145734179615861283807147704676820889400892628790101246152021966594405478659553998995868892201670615} a^{10} - \frac{39624092589159321315071166226274551077032832706447725850507850859234774188586492190023044367296621334962}{86263145734179615861283807147704676820889400892628790101246152021966594405478659553998995868892201670615} a^{9} + \frac{6927300219759257784635686665254341091839499384322126526063780958542548056079848354360137984332772473723}{17252629146835923172256761429540935364177880178525758020249230404393318881095731910799799173778440334123} a^{8} + \frac{4659919797491229669104774478091080806386746213468976292718020234479554772635140734360236318326442026788}{86263145734179615861283807147704676820889400892628790101246152021966594405478659553998995868892201670615} a^{7} - \frac{9168237221072650841739188413947553020569461729806455495940916997546414628397544667757829341657963660849}{86263145734179615861283807147704676820889400892628790101246152021966594405478659553998995868892201670615} a^{6} - \frac{6476161996418767455218738691234512671379064036691652065867195653366502858179752422042846543433005784714}{17252629146835923172256761429540935364177880178525758020249230404393318881095731910799799173778440334123} a^{5} - \frac{36276744725086928847210688439647231539580147421001761540458664376978809506602653146231449975088014353424}{86263145734179615861283807147704676820889400892628790101246152021966594405478659553998995868892201670615} a^{4} - \frac{4991663855247653731578744827183096605824855708474682553745445221702978045255601822661181370768552379563}{86263145734179615861283807147704676820889400892628790101246152021966594405478659553998995868892201670615} a^{3} + \frac{23573285547235348991829680242126536280011124611684062453681570987330212118668065843980143220401685934363}{86263145734179615861283807147704676820889400892628790101246152021966594405478659553998995868892201670615} a^{2} - \frac{33922827150542665638837008687202344026121429852499429178994412313366959984077786981592945592871138133491}{86263145734179615861283807147704676820889400892628790101246152021966594405478659553998995868892201670615} a + \frac{3180207195751861663038409888641320047093381053574142003682981413734165960628629809803746230296837508157}{6635626594936893527791062088284975140068415453279137700095857847843584185036819965692230451453246282355}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{16}$, which has order $256$ (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:  \( 65918524003600 \) (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

\(\Q(\sqrt{-2419}) \), 4.0.239914001.1, 8.0.2359907842908948041.4

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ $16$ $16$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ $16$ R

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
41Data not computed
$59$59.8.6.2$x^{8} + 177 x^{4} + 13924$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
59.8.6.2$x^{8} + 177 x^{4} + 13924$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$