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Properties

Label 16.0.44272035895...3296.2
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 $3547708928$ (GRH)
Class group $[2, 2, 2, 2, 2, 4, 27716476]$ (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, 1833276172408064, 0, 168134672246112, 0, 6294328954624, 0, 101963947458, 0, 641999728, 0, 1816476, 0, 2288, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 2288*x^14 + 1816476*x^12 + 641999728*x^10 + 101963947458*x^8 + 6294328954624*x^6 + 168134672246112*x^4 + 1833276172408064*x^2 + 5543020491743296)
 
gp: K = bnfinit(x^16 + 2288*x^14 + 1816476*x^12 + 641999728*x^10 + 101963947458*x^8 + 6294328954624*x^6 + 168134672246112*x^4 + 1833276172408064*x^2 + 5543020491743296, 1)
 

Normalized defining polynomial

\( x^{16} + 2288 x^{14} + 1816476 x^{12} + 641999728 x^{10} + 101963947458 x^{8} + 6294328954624 x^{6} + 168134672246112 x^{4} + 1833276172408064 x^{2} + 5543020491743296 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{508} a^{10} + \frac{1}{254} a^{8} - \frac{33}{127} a^{6} - \frac{1}{127} a^{4} + \frac{1}{254} a^{2}$, $\frac{1}{508} a^{11} + \frac{1}{254} a^{9} - \frac{33}{127} a^{7} - \frac{1}{127} a^{5} + \frac{1}{254} a^{3}$, $\frac{1}{2193544} a^{12} - \frac{63}{548386} a^{10} - \frac{63279}{548386} a^{8} + \frac{17843}{274193} a^{6} + \frac{80773}{1096772} a^{4} + \frac{752}{2159} a^{2} - \frac{4}{17}$, $\frac{1}{4387088} a^{13} - \frac{63}{1096772} a^{11} - \frac{63279}{1096772} a^{9} - \frac{128175}{274193} a^{7} - \frac{1015999}{2193544} a^{5} - \frac{1407}{4318} a^{3} - \frac{2}{17} a$, $\frac{1}{4024960351781489090678232084406495242360780151568} a^{14} + \frac{50910872408695453469775876465693853289365}{1006240087945372272669558021101623810590195037892} a^{12} - \frac{173251175644568114405646848362787642852190267}{251560021986343068167389505275405952647548759473} a^{10} - \frac{51305757449075060826851479164064642484829099200}{251560021986343068167389505275405952647548759473} a^{8} - \frac{146200105648337477329654389491563421723890923015}{2012480175890744545339116042203247621180390075784} a^{6} + \frac{862896279610917643409556739473503705302316621}{3961575149391229419958889846856786655866909598} a^{4} - \frac{12728089938427151049271009421318148561493715}{31193505113316767086290471235092808313912674} a^{2} + \frac{43333782620231067188725629704736647420}{212840685007415269629023807878742943503}$, $\frac{1}{8049920703562978181356464168812990484721560303136} a^{15} + \frac{50910872408695453469775876465693853289365}{2012480175890744545339116042203247621180390075784} a^{13} + \frac{1287782872117342252356857529977242756524693731}{2012480175890744545339116042203247621180390075784} a^{11} + \frac{37732323665722140305916497967676265251456003968}{251560021986343068167389505275405952647548759473} a^{9} - \frac{669128025367979760764227849276659260298322989951}{4024960351781489090678232084406495242360780151568} a^{7} + \frac{831702774497600876323266268238410896988403947}{7923150298782458839917779693713573311733819196} a^{5} - \frac{3151320215794468109673765671043028470773121}{15596752556658383543145235617546404156956337} a^{3} + \frac{21666891310115533594362814852368323710}{212840685007415269629023807878742943503} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{27716476}$, which has order $3547708928$ (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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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