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Properties

Label 16.0.65240483238...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 83^{12}$
Root discriminant $112.44$
Ramified primes $5, 83$
Class number $15$ (GRH)
Class group $[15]$ (GRH)
Galois group $C_8: C_2$ (as 16T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![276590161, -2417066385, 15754050266, 6091897560, 3240856012, 641154675, 210234688, 22638150, 7746025, 460820, 255252, 4535, 6012, -90, 104, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 104*x^14 - 90*x^13 + 6012*x^12 + 4535*x^11 + 255252*x^10 + 460820*x^9 + 7746025*x^8 + 22638150*x^7 + 210234688*x^6 + 641154675*x^5 + 3240856012*x^4 + 6091897560*x^3 + 15754050266*x^2 - 2417066385*x + 276590161)
 
gp: K = bnfinit(x^16 - 5*x^15 + 104*x^14 - 90*x^13 + 6012*x^12 + 4535*x^11 + 255252*x^10 + 460820*x^9 + 7746025*x^8 + 22638150*x^7 + 210234688*x^6 + 641154675*x^5 + 3240856012*x^4 + 6091897560*x^3 + 15754050266*x^2 - 2417066385*x + 276590161, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 104 x^{14} - 90 x^{13} + 6012 x^{12} + 4535 x^{11} + 255252 x^{10} + 460820 x^{9} + 7746025 x^{8} + 22638150 x^{7} + 210234688 x^{6} + 641154675 x^{5} + 3240856012 x^{4} + 6091897560 x^{3} + 15754050266 x^{2} - 2417066385 x + 276590161 \)

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:  \(652404832389289089417486572265625=5^{14}\cdot 83^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $112.44$
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:  $5, 83$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{11} a^{9} - \frac{4}{11} a^{8} + \frac{5}{11} a^{7} + \frac{2}{11} a^{6} + \frac{3}{11} a^{5} - \frac{1}{11} a^{4} + \frac{4}{11} a^{3} - \frac{5}{11} a^{2} - \frac{2}{11} a - \frac{3}{11}$, $\frac{1}{11} a^{10} - \frac{1}{11}$, $\frac{1}{11} a^{11} - \frac{1}{11} a$, $\frac{1}{11} a^{12} - \frac{1}{11} a^{2}$, $\frac{1}{11} a^{13} - \frac{1}{11} a^{3}$, $\frac{1}{566401} a^{14} - \frac{16656}{566401} a^{13} - \frac{2742}{566401} a^{12} - \frac{23789}{566401} a^{11} - \frac{14166}{566401} a^{10} - \frac{10702}{566401} a^{9} - \frac{79457}{566401} a^{8} - \frac{4109}{566401} a^{7} - \frac{63369}{566401} a^{6} + \frac{4206}{18271} a^{5} - \frac{239538}{566401} a^{4} + \frac{4023}{18271} a^{3} - \frac{77057}{566401} a^{2} - \frac{240301}{566401} a - \frac{26141}{566401}$, $\frac{1}{4583034990193475404095975927511616554067495569901423812514049971} a^{15} + \frac{1138387496833303656224924561915783632066712847208246141317}{4583034990193475404095975927511616554067495569901423812514049971} a^{14} - \frac{5161514302100353240796892863700957281136504315456303004507954}{416639544563043218554179629773783323097045051809220346592186361} a^{13} - \frac{1357573851721693055003092667501471903170231077242756426260556}{147839838393337916261160513790697308195725663545207219758517741} a^{12} - \frac{140647082398574657340678577787492552006977724614927707862087284}{4583034990193475404095975927511616554067495569901423812514049971} a^{11} - \frac{80880433380561162793370125304187865679114247804608243131803786}{4583034990193475404095975927511616554067495569901423812514049971} a^{10} - \frac{3554673210017531251367833221482032316435234637922165581425659}{416639544563043218554179629773783323097045051809220346592186361} a^{9} + \frac{87636500894460786897372714687399870302689311724985412728658613}{416639544563043218554179629773783323097045051809220346592186361} a^{8} - \frac{157937541076295244380040316030654453150533244284663664045085956}{416639544563043218554179629773783323097045051809220346592186361} a^{7} + \frac{157471562127794384760792134817676105681543069641409404179213595}{416639544563043218554179629773783323097045051809220346592186361} a^{6} - \frac{385477412394946037013752576154870846597134433462184374050821107}{4583034990193475404095975927511616554067495569901423812514049971} a^{5} - \frac{490184797004918826193460479431712197696380233970289085408113989}{4583034990193475404095975927511616554067495569901423812514049971} a^{4} - \frac{33207436965807101478794515785343569204019721069568751608617234}{416639544563043218554179629773783323097045051809220346592186361} a^{3} - \frac{141461736085308743665803043633378969431830113122910883725669068}{4583034990193475404095975927511616554067495569901423812514049971} a^{2} + \frac{1612200587189555990079178354445660951256143354640543679193276811}{4583034990193475404095975927511616554067495569901423812514049971} a - \frac{76372535340618206458183802457612303125646845273796586072420}{275571823113070495105283863117769018944591159274933787055141}$

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

Class group and class number

$C_{15}$, which has order $15$ (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{523984486099651013517038609596612096131851762502645}{979071777439323948749407376097333166859110354604021322904091} a^{15} - \frac{1039212966909683875110419342014293046522930769253334}{979071777439323948749407376097333166859110354604021322904091} a^{14} + \frac{38932285208742420071658306828556415764167809910066842}{979071777439323948749407376097333166859110354604021322904091} a^{13} + \frac{167228019045304536236139142893003528304060929011894199}{979071777439323948749407376097333166859110354604021322904091} a^{12} + \frac{2223490518680505169379942860250237313341836272799897677}{979071777439323948749407376097333166859110354604021322904091} a^{11} + \frac{12857233077560231768547120612931050958732611938704061078}{979071777439323948749407376097333166859110354604021322904091} a^{10} + \frac{105898814797813932755048153843493592076771437120124240187}{979071777439323948749407376097333166859110354604021322904091} a^{9} + \frac{605541794125866312782460858813813444885142764608610261853}{979071777439323948749407376097333166859110354604021322904091} a^{8} + \frac{3513919932940063902110548975099493685931981339647382338140}{979071777439323948749407376097333166859110354604021322904091} a^{7} + \frac{20683278194243331935544221963553069761323307445038181803069}{979071777439323948749407376097333166859110354604021322904091} a^{6} + \frac{113335227689452874037833972343184995576568569522307953059883}{979071777439323948749407376097333166859110354604021322904091} a^{5} + \frac{45948176749832396863562168682150606823194000653602940895962}{89006525221756722613582488736121196987191850418547392991281} a^{4} + \frac{1867148487804552260943980912767707696189227088484456202133530}{979071777439323948749407376097333166859110354604021322904091} a^{3} + \frac{4686897303920390217706582858403374594574508885497060655651917}{979071777439323948749407376097333166859110354604021322904091} a^{2} + \frac{8221968366145718639393366728047314932224341281047537185128884}{979071777439323948749407376097333166859110354604021322904091} a - \frac{53916996156431065782366260197204105242352094367390555893}{58870289064958448003692344182390305264813321784860881661} \) (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:  \( 6305600871.78 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}$ (as 16T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_8: C_2$
Character table for $C_8: C_2$

Intermediate fields

\(\Q(\sqrt{-415}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-83}) \), \(\Q(\sqrt{5}, \sqrt{-83})\), 4.4.861125.1, \(\Q(\zeta_{5})\), 8.0.741536265625.1, 8.4.25542216669453125.1 x2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
83Data not computed