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Properties

Label 16.0.83680629612...2769.5
Degree $16$
Signature $[0, 8]$
Discriminant $17^{14}\cdot 89^{6}$
Root discriminant $64.22$
Ramified primes $17, 89$
Class number $20$ (GRH)
Class group $[2, 10]$ (GRH)
Galois group $C_2^5.C_2.C_2$ (as 16T257)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2660159, -5374135, 7719080, -7740329, 5538260, -3303540, 1574231, -628712, 236402, -70205, 22994, -4670, 1377, -160, 52, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 52*x^14 - 160*x^13 + 1377*x^12 - 4670*x^11 + 22994*x^10 - 70205*x^9 + 236402*x^8 - 628712*x^7 + 1574231*x^6 - 3303540*x^5 + 5538260*x^4 - 7740329*x^3 + 7719080*x^2 - 5374135*x + 2660159)
 
gp: K = bnfinit(x^16 - 2*x^15 + 52*x^14 - 160*x^13 + 1377*x^12 - 4670*x^11 + 22994*x^10 - 70205*x^9 + 236402*x^8 - 628712*x^7 + 1574231*x^6 - 3303540*x^5 + 5538260*x^4 - 7740329*x^3 + 7719080*x^2 - 5374135*x + 2660159, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 52 x^{14} - 160 x^{13} + 1377 x^{12} - 4670 x^{11} + 22994 x^{10} - 70205 x^{9} + 236402 x^{8} - 628712 x^{7} + 1574231 x^{6} - 3303540 x^{5} + 5538260 x^{4} - 7740329 x^{3} + 7719080 x^{2} - 5374135 x + 2660159 \)

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:  \(83680629612698426645202702769=17^{14}\cdot 89^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.22$
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:  $17, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{1742} a^{14} + \frac{72}{871} a^{13} - \frac{10}{871} a^{12} - \frac{243}{1742} a^{11} + \frac{112}{871} a^{10} - \frac{108}{871} a^{9} - \frac{161}{1742} a^{8} - \frac{425}{871} a^{7} + \frac{185}{871} a^{6} + \frac{491}{1742} a^{5} - \frac{381}{871} a^{4} - \frac{26}{67} a^{3} + \frac{145}{1742} a^{2} - \frac{172}{871} a + \frac{302}{871}$, $\frac{1}{263666683339880233293287611703924981199314086} a^{15} + \frac{23253129357075535590333751632812108566347}{263666683339880233293287611703924981199314086} a^{14} + \frac{10159926541492710699161457774316919107919195}{131833341669940116646643805851962490599657043} a^{13} - \frac{24187000509884619236381784372023661961743}{20282052564606171791791354746455767784562622} a^{12} - \frac{11617168569564562643939740260414062319998047}{263666683339880233293287611703924981199314086} a^{11} - \frac{25471598238912300151772229810594702448946377}{131833341669940116646643805851962490599657043} a^{10} - \frac{113706155356527955627298142471331081554889791}{263666683339880233293287611703924981199314086} a^{9} + \frac{48144547729802376065580691353737611343990635}{263666683339880233293287611703924981199314086} a^{8} + \frac{52433776201424930480329797215229336855845881}{131833341669940116646643805851962490599657043} a^{7} - \frac{1871844664980246160258859233027591518557769}{3935323631938510944675934503043656435810658} a^{6} - \frac{126872267781937879546525089190839703893742663}{263666683339880233293287611703924981199314086} a^{5} - \frac{27545635085171551494233885434918546180781755}{131833341669940116646643805851962490599657043} a^{4} - \frac{69167388056731706559509818244703736974743373}{263666683339880233293287611703924981199314086} a^{3} - \frac{105754712554621666855674176988511688514586483}{263666683339880233293287611703924981199314086} a^{2} + \frac{15612371852408136158286980669417286260092420}{131833341669940116646643805851962490599657043} a + \frac{58272125066733218811944547522155417112648856}{131833341669940116646643805851962490599657043}$

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

Class group and class number

$C_{2}\times C_{10}$, which has order $20$ (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:  \( 2253287.49487 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^5.C_2.C_2$ (as 16T257):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^5.C_2.C_2$
Character table for $C_2^5.C_2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 8.0.3250292628833.2, 8.4.289276043966137.1, 8.4.2148243641.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
17Data not computed
89Data not computed