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Properties

Label 16.0.67886842328...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{14}\cdot 61^{8}\cdot 97^{4}$
Root discriminant $200.44$
Ramified primes $2, 5, 61, 97$
Class number $16754880$ (GRH)
Class group $[2, 2, 2, 2, 1047180]$ (GRH)
Galois group 16T852

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1203481647881, -523452247142, 863745327204, -218586158402, 145236326283, -27067446942, 9453260051, -1407931996, 321152208, -34976022, 6492584, -427776, 76128, -2524, 451, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 451*x^14 - 2524*x^13 + 76128*x^12 - 427776*x^11 + 6492584*x^10 - 34976022*x^9 + 321152208*x^8 - 1407931996*x^7 + 9453260051*x^6 - 27067446942*x^5 + 145236326283*x^4 - 218586158402*x^3 + 863745327204*x^2 - 523452247142*x + 1203481647881)
 
gp: K = bnfinit(x^16 - 6*x^15 + 451*x^14 - 2524*x^13 + 76128*x^12 - 427776*x^11 + 6492584*x^10 - 34976022*x^9 + 321152208*x^8 - 1407931996*x^7 + 9453260051*x^6 - 27067446942*x^5 + 145236326283*x^4 - 218586158402*x^3 + 863745327204*x^2 - 523452247142*x + 1203481647881, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 451 x^{14} - 2524 x^{13} + 76128 x^{12} - 427776 x^{11} + 6492584 x^{10} - 34976022 x^{9} + 321152208 x^{8} - 1407931996 x^{7} + 9453260051 x^{6} - 27067446942 x^{5} + 145236326283 x^{4} - 218586158402 x^{3} + 863745327204 x^{2} - 523452247142 x + 1203481647881 \)

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:  \(6788684232836496753984400000000000000=2^{16}\cdot 5^{14}\cdot 61^{8}\cdot 97^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $200.44$
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:  $2, 5, 61, 97$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{346749853507749690947028468414836531463159826899668033288108768230488300436289055933199421} a^{15} + \frac{19937740233630881201064202704406034484202337278025296028251380124224083878444652683821611}{346749853507749690947028468414836531463159826899668033288108768230488300436289055933199421} a^{14} + \frac{139690294322420154704976930012823626725678267025000596302573394518010934942585779617965554}{346749853507749690947028468414836531463159826899668033288108768230488300436289055933199421} a^{13} + \frac{145585779868732168243613981080258031339056097225105253910311511852910072070866377768746456}{346749853507749690947028468414836531463159826899668033288108768230488300436289055933199421} a^{12} - \frac{28990908368428605523960060166860670127327084313577543208929676897361362681753427635300901}{346749853507749690947028468414836531463159826899668033288108768230488300436289055933199421} a^{11} + \frac{150279409036091310848152036425702633513450888789086807747429841378922030172805555682192072}{346749853507749690947028468414836531463159826899668033288108768230488300436289055933199421} a^{10} + \frac{32140166789904513910702849639849292351300333341161552862029605635548212899461390846624038}{346749853507749690947028468414836531463159826899668033288108768230488300436289055933199421} a^{9} - \frac{10026488091267041734063259211215678319144092307358110809539418215031136862035308315714156}{346749853507749690947028468414836531463159826899668033288108768230488300436289055933199421} a^{8} - \frac{156151879613997179314244441414102859657825968843270612288199425571787217308625216630888520}{346749853507749690947028468414836531463159826899668033288108768230488300436289055933199421} a^{7} + \frac{156500716297649423686338871331209432891431126897271308391609470234628933635150340238284901}{346749853507749690947028468414836531463159826899668033288108768230488300436289055933199421} a^{6} + \frac{165153227842349642655982258947846481968847760152144300297190545338363789214343606171993777}{346749853507749690947028468414836531463159826899668033288108768230488300436289055933199421} a^{5} - \frac{50010729784239198883738346834626036890773365427768166573173975548087300569473887354102989}{346749853507749690947028468414836531463159826899668033288108768230488300436289055933199421} a^{4} - \frac{42421551608345148104971159447612987190727389268935705606214009100989636605425102424947549}{346749853507749690947028468414836531463159826899668033288108768230488300436289055933199421} a^{3} + \frac{72391206907412781184774750169718485831380068427504858946326485985332393807959156144815895}{346749853507749690947028468414836531463159826899668033288108768230488300436289055933199421} a^{2} - \frac{29352414865230013976100852507948594519499282869338172008970028269357293395350858328317888}{346749853507749690947028468414836531463159826899668033288108768230488300436289055933199421} a - \frac{57513513181823237589809959399075141177736071832905898207494975731874569389598556898916442}{346749853507749690947028468414836531463159826899668033288108768230488300436289055933199421}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{1047180}$, which has order $16754880$ (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:  \( 26657.4092537 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T852:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 41 conjugacy class representatives for t16n852
Character table for t16n852 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 8.8.14884000000.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
2Data not computed
5Data not computed
61Data not computed
$97$97.8.0.1$x^{8} - x + 84$$1$$8$$0$$C_8$$[\ ]^{8}$
97.8.4.2$x^{8} - 912673 x^{2} + 2036173463$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$