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Properties

Label 16.0.68356794856...3328.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 17^{15}\cdot 43691^{4}$
Root discriminant $411.79$
Ramified primes $2, 17, 43691$
Class number $33887823936$ (GRH)
Class group $[2, 2, 2, 2, 2, 1058994498]$ (GRH)
Galois group $(C_2\times OD_{16}).D_4$ (as 16T591)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![61946512496156595137, 0, 2486877913489246478, 0, 39051900730883446, 0, 318824584999742, 0, 1482245714409, 0, 3978693634, 0, 5886131, 0, 4199, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 4199*x^14 + 5886131*x^12 + 3978693634*x^10 + 1482245714409*x^8 + 318824584999742*x^6 + 39051900730883446*x^4 + 2486877913489246478*x^2 + 61946512496156595137)
 
gp: K = bnfinit(x^16 + 4199*x^14 + 5886131*x^12 + 3978693634*x^10 + 1482245714409*x^8 + 318824584999742*x^6 + 39051900730883446*x^4 + 2486877913489246478*x^2 + 61946512496156595137, 1)
 

Normalized defining polynomial

\( x^{16} + 4199 x^{14} + 5886131 x^{12} + 3978693634 x^{10} + 1482245714409 x^{8} + 318824584999742 x^{6} + 39051900730883446 x^{4} + 2486877913489246478 x^{2} + 61946512496156595137 \)

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:  \(683567948564897299391469349182811817443328=2^{16}\cdot 17^{15}\cdot 43691^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $411.79$
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:  $2, 17, 43691$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{43691} a^{10} + \frac{4199}{43691} a^{8} - \frac{12154}{43691} a^{6} + \frac{16410}{43691} a^{4} + \frac{9186}{43691} a^{2}$, $\frac{1}{43691} a^{11} + \frac{4199}{43691} a^{9} - \frac{12154}{43691} a^{7} + \frac{16410}{43691} a^{5} + \frac{9186}{43691} a^{3}$, $\frac{1}{1908903481} a^{12} + \frac{4199}{1908903481} a^{10} + \frac{5886131}{1908903481} a^{8} + \frac{160886672}{1908903481} a^{6} + \frac{936613153}{1908903481} a^{4} - \frac{10858}{43691} a^{2}$, $\frac{1}{1908903481} a^{13} + \frac{4199}{1908903481} a^{11} + \frac{5886131}{1908903481} a^{9} + \frac{160886672}{1908903481} a^{7} + \frac{936613153}{1908903481} a^{5} - \frac{10858}{43691} a^{3}$, $\frac{1}{64304204224453862009502791278548539159511631539233} a^{14} + \frac{1002407739929701527358805277758414152591}{64304204224453862009502791278548539159511631539233} a^{12} + \frac{151115938155732555601296502824576107009235202}{64304204224453862009502791278548539159511631539233} a^{10} - \frac{6525703014864899477273230994307063333051823074573}{64304204224453862009502791278548539159511631539233} a^{8} - \frac{27151864483337259390671199723630272113106618099125}{64304204224453862009502791278548539159511631539233} a^{6} + \frac{126435201286581271916405993297029891825689005}{1471795203233019661017207005528565131480433763} a^{4} + \frac{6149993590339042541122429965363458247588}{33686461816690386143993202387873134775593} a^{2} - \frac{360120796087521893319395917240229825}{771016040298697355153079636260857723}$, $\frac{1}{64304204224453862009502791278548539159511631539233} a^{15} + \frac{1002407739929701527358805277758414152591}{64304204224453862009502791278548539159511631539233} a^{13} + \frac{151115938155732555601296502824576107009235202}{64304204224453862009502791278548539159511631539233} a^{11} - \frac{6525703014864899477273230994307063333051823074573}{64304204224453862009502791278548539159511631539233} a^{9} - \frac{27151864483337259390671199723630272113106618099125}{64304204224453862009502791278548539159511631539233} a^{7} + \frac{126435201286581271916405993297029891825689005}{1471795203233019661017207005528565131480433763} a^{5} + \frac{6149993590339042541122429965363458247588}{33686461816690386143993202387873134775593} a^{3} - \frac{360120796087521893319395917240229825}{771016040298697355153079636260857723} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{1058994498}$, which has order $33887823936$ (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:  \( 3640.01221338 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times OD_{16}).D_4$ (as 16T591):

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 $(C_2\times OD_{16}).D_4$
Character table for $(C_2\times OD_{16}).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

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 $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$2.8.8.6$x^{8} + 2 x^{7} + 2 x^{6} + 16 x^{2} + 16$$2$$4$$8$$(C_8:C_2):C_2$$[2, 2, 2]^{4}$
2.8.8.6$x^{8} + 2 x^{7} + 2 x^{6} + 16 x^{2} + 16$$2$$4$$8$$(C_8:C_2):C_2$$[2, 2, 2]^{4}$
17Data not computed
43691Data not computed