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Properties

Label 16.0.44272035895...3296.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{62}\cdot 127^{4}\cdot 577^{6}$
Root discriminant $534.42$
Ramified primes $2, 127, 577$
Class number $5583312896$ (GRH)
Class group $[2, 2, 2, 2, 2, 4, 43619632]$ (GRH)
Galois group $C_4:Q_8.C_2^3$ (as 16T520)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5543020491743296, 0, 1588496435904000, 0, 103065013482464, 0, 2864580268288, 0, 40150581314, 0, 294264512, 0, 1082972, 0, 1776, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 1776*x^14 + 1082972*x^12 + 294264512*x^10 + 40150581314*x^8 + 2864580268288*x^6 + 103065013482464*x^4 + 1588496435904000*x^2 + 5543020491743296)
 
gp: K = bnfinit(x^16 + 1776*x^14 + 1082972*x^12 + 294264512*x^10 + 40150581314*x^8 + 2864580268288*x^6 + 103065013482464*x^4 + 1588496435904000*x^2 + 5543020491743296, 1)
 

Normalized defining polynomial

\( x^{16} + 1776 x^{14} + 1082972 x^{12} + 294264512 x^{10} + 40150581314 x^{8} + 2864580268288 x^{6} + 103065013482464 x^{4} + 1588496435904000 x^{2} + 5543020491743296 \)

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:  \(44272035895977179125717718174809351883063296=2^{62}\cdot 127^{4}\cdot 577^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $534.42$
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:  $2, 127, 577$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{508} a^{10} - \frac{1}{254} a^{8} - \frac{21}{127} a^{6} - \frac{19}{127} a^{4} + \frac{99}{254} a^{2}$, $\frac{1}{508} a^{11} - \frac{1}{254} a^{9} - \frac{21}{127} a^{7} - \frac{19}{127} a^{5} + \frac{99}{254} a^{3}$, $\frac{1}{129032} a^{12} - \frac{1}{64516} a^{10} - \frac{2561}{32258} a^{8} - \frac{4645}{16129} a^{6} - \frac{20983}{64516} a^{4} - \frac{63}{254} a^{2}$, $\frac{1}{258064} a^{13} + \frac{63}{64516} a^{11} + \frac{13441}{64516} a^{9} - \frac{3656}{16129} a^{7} + \frac{33881}{129032} a^{5} - \frac{109}{254} a^{3}$, $\frac{1}{1090839553459839056045946125698603672038009095408} a^{14} - \frac{586868868276092700564913131555130656284727}{272709888364959764011486531424650918009502273852} a^{12} - \frac{167208881234068692860144984835418303506202555}{272709888364959764011486531424650918009502273852} a^{10} + \frac{7335147553595710465899863295838712803555647906}{68177472091239941002871632856162729502375568463} a^{8} - \frac{269565211482876803882851068792907270555825871735}{545419776729919528022973062849301836019004547704} a^{6} - \frac{324975770379831974428731947852144284321071131}{1073660977814802220517663509545869755942922338} a^{4} - \frac{1016260676586321331931729533881937953107303}{4227011723680323702825446887975865180877647} a^{2} + \frac{6879833090394075349463156297523660059}{57683807416590342428600920972937201393}$, $\frac{1}{2181679106919678112091892251397207344076018190816} a^{15} - \frac{586868868276092700564913131555130656284727}{545419776729919528022973062849301836019004547704} a^{13} - \frac{167208881234068692860144984835418303506202555}{545419776729919528022973062849301836019004547704} a^{11} + \frac{3667573776797855232949931647919356401777823953}{68177472091239941002871632856162729502375568463} a^{9} + \frac{275854565247042724140121994056394565463178675969}{1090839553459839056045946125698603672038009095408} a^{7} + \frac{748685207434970246088931561693725471621851207}{2147321955629604441035327019091739511885844676} a^{5} - \frac{1016260676586321331931729533881937953107303}{8454023447360647405650893775951730361755294} a^{3} - \frac{25401987163098133539568882337706770667}{57683807416590342428600920972937201393} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{43619632}$, which has order $5583312896$ (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:  \( 994324.851518 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:Q_8.C_2^3$ (as 16T520):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 28 conjugacy class representatives for $C_4:Q_8.C_2^3$
Character table for $C_4:Q_8.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.8.1396405436416.1

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
Arithmetically equvalently 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.8.31.33$x^{8} + 12 x^{4} + 34$$8$$1$$31$$(C_8:C_2):C_2$$[2, 3, 7/2, 4, 5]$
2.8.31.33$x^{8} + 12 x^{4} + 34$$8$$1$$31$$(C_8:C_2):C_2$$[2, 3, 7/2, 4, 5]$
127Data not computed
577Data not computed