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Properties

Label 16.0.37787193273...9856.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 83^{8}$
Root discriminant $25.77$
Ramified primes $2, 83$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^3:S_4.C_2$ (as 16T764)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![114577, -169936, 326644, -290252, 306639, -184906, 138428, -59336, 35612, -10874, 5724, -1188, 588, -74, 36, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 36*x^14 - 74*x^13 + 588*x^12 - 1188*x^11 + 5724*x^10 - 10874*x^9 + 35612*x^8 - 59336*x^7 + 138428*x^6 - 184906*x^5 + 306639*x^4 - 290252*x^3 + 326644*x^2 - 169936*x + 114577)
 
gp: K = bnfinit(x^16 - 2*x^15 + 36*x^14 - 74*x^13 + 588*x^12 - 1188*x^11 + 5724*x^10 - 10874*x^9 + 35612*x^8 - 59336*x^7 + 138428*x^6 - 184906*x^5 + 306639*x^4 - 290252*x^3 + 326644*x^2 - 169936*x + 114577, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 36 x^{14} - 74 x^{13} + 588 x^{12} - 1188 x^{11} + 5724 x^{10} - 10874 x^{9} + 35612 x^{8} - 59336 x^{7} + 138428 x^{6} - 184906 x^{5} + 306639 x^{4} - 290252 x^{3} + 326644 x^{2} - 169936 x + 114577 \)

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:  \(37787193273718832889856=2^{24}\cdot 83^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.77$
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:  $2, 83$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{289971328627635333027516647} a^{15} + \frac{143608130772397188019239758}{289971328627635333027516647} a^{14} - \frac{27878565113370517968637062}{289971328627635333027516647} a^{13} - \frac{123342215755573737562399900}{289971328627635333027516647} a^{12} + \frac{19216873119815125692897251}{289971328627635333027516647} a^{11} + \frac{102263434799783139311683331}{289971328627635333027516647} a^{10} + \frac{64852745936498376408049381}{289971328627635333027516647} a^{9} - \frac{58169529929743246192466455}{289971328627635333027516647} a^{8} + \frac{132371336598737628221274936}{289971328627635333027516647} a^{7} - \frac{135634837320559358109139141}{289971328627635333027516647} a^{6} - \frac{86521599192572598088826046}{289971328627635333027516647} a^{5} - \frac{9225587976984681039496588}{289971328627635333027516647} a^{4} - \frac{59123630851688917618436080}{289971328627635333027516647} a^{3} - \frac{50048030935434709825975990}{289971328627635333027516647} a^{2} + \frac{3423661284106212923284562}{289971328627635333027516647} a - \frac{134626357982186597489399327}{289971328627635333027516647}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( \frac{4944473640752}{684287501725903} a^{15} - \frac{1821871644945}{684287501725903} a^{14} + \frac{159740111461603}{684287501725903} a^{13} - \frac{89486260655878}{684287501725903} a^{12} + \frac{2256997087501000}{684287501725903} a^{11} - \frac{1580863062294933}{684287501725903} a^{10} + \frac{18339748668756864}{684287501725903} a^{9} - \frac{14077652113659103}{684287501725903} a^{8} + \frac{89939294721723229}{684287501725903} a^{7} - \frac{61482899490151477}{684287501725903} a^{6} + \frac{249724906531646531}{684287501725903} a^{5} - \frac{96960802653260533}{684287501725903} a^{4} + \frac{315499104328338364}{684287501725903} a^{3} + \frac{61953506061153671}{684287501725903} a^{2} + \frac{95851018417367252}{684287501725903} a + \frac{179115243170663793}{684287501725903} \) (order $4$)
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:  \( 56269.8079492 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3:S_4.C_2$ (as 16T764):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 384
The 23 conjugacy class representatives for $C_2^3:S_4.C_2$
Character table for $C_2^3:S_4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 4.2.1328.1, 8.0.1763584.1

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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }$

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
$83$83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.4.2.1$x^{4} + 249 x^{2} + 27556$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
83.8.6.1$x^{8} - 83 x^{4} + 110224$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$