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Properties

Label 16.0.28860419378...1376.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{46}\cdot 239^{4}\cdot 257^{10}$
Root discriminant $925.27$
Ramified primes $2, 239, 257$
Class number $57273205248$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 6, 149148972]$ (GRH)
Galois group 16T875

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3207580110181323556, 0, 388439643284578428, 0, 10768253589679158, 0, 127016546026864, 0, 744702238048, 0, 2296419422, 0, 3763147, 0, 3092, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 3092*x^14 + 3763147*x^12 + 2296419422*x^10 + 744702238048*x^8 + 127016546026864*x^6 + 10768253589679158*x^4 + 388439643284578428*x^2 + 3207580110181323556)
 
gp: K = bnfinit(x^16 + 3092*x^14 + 3763147*x^12 + 2296419422*x^10 + 744702238048*x^8 + 127016546026864*x^6 + 10768253589679158*x^4 + 388439643284578428*x^2 + 3207580110181323556, 1)
 

Normalized defining polynomial

\( x^{16} + 3092 x^{14} + 3763147 x^{12} + 2296419422 x^{10} + 744702238048 x^{8} + 127016546026864 x^{6} + 10768253589679158 x^{4} + 388439643284578428 x^{2} + 3207580110181323556 \)

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:  \(288604193782091750893321301214124833643876581376=2^{46}\cdot 239^{4}\cdot 257^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $925.27$
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:  $2, 239, 257$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{1434} a^{10} - \frac{493}{1434} a^{8} - \frac{193}{717} a^{6} - \frac{101}{239} a^{4} + \frac{72}{239} a^{2} - \frac{1}{3}$, $\frac{1}{1434} a^{11} - \frac{493}{1434} a^{9} - \frac{193}{717} a^{7} - \frac{101}{239} a^{5} + \frac{72}{239} a^{3} - \frac{1}{3} a$, $\frac{1}{1849692222} a^{12} - \frac{8333}{264241746} a^{10} - \frac{115098530}{924846111} a^{8} - \frac{14343410}{924846111} a^{6} - \frac{19141568}{44040291} a^{4} + \frac{2425}{15057} a^{2} + \frac{1}{63}$, $\frac{1}{1849692222} a^{13} - \frac{8333}{264241746} a^{11} - \frac{115098530}{924846111} a^{9} - \frac{14343410}{924846111} a^{7} - \frac{19141568}{44040291} a^{5} + \frac{2425}{15057} a^{3} + \frac{1}{63} a$, $\frac{1}{81144833946025990416380956111510031876206459093908} a^{14} - \frac{636249644161140402620176620191659669625}{9016092660669554490708995123501114652911828788212} a^{12} - \frac{1847820469316817486706207969184385392999729653}{40572416973012995208190478055755015938103229546954} a^{10} - \frac{1}{2} a^{9} - \frac{1491428960219537114063844127385854034541395698357}{13524138991004331736063492685251671979367743182318} a^{8} - \frac{5713233187329944186186700415539669975255985281409}{40572416973012995208190478055755015938103229546954} a^{6} + \frac{73189504559821357030686388576913987931277764437}{169759066832690356519625431195627681749385897686} a^{4} - \frac{7933414544732981678689889903198650958558}{20625151941563330538055708044075531256023} a^{2} + \frac{440478825819403250970057820161317325356}{5781946360187209816107667108590211691019}$, $\frac{1}{4949834870707585415399238322802111944448594004728388} a^{15} + \frac{28609997553399561101077864415851950788651}{549981652300842823933248702533567993827621556080932} a^{13} - \frac{415013404815542158544458702845291041567876269952}{1237458717676896353849809580700527986112148501182097} a^{11} + \frac{23293422148926887643204282459618916716182322394528}{58926605603661731135705218128596570767245166722957} a^{9} - \frac{1}{2} a^{8} + \frac{753521767639358634310026653329417032383500431134159}{2474917435353792707699619161401055972224297002364194} a^{7} + \frac{1201646656286907724607388959106725700672700256957}{10355303076794111747697151302933288586712539758846} a^{5} - \frac{518819948620663409218363864748290127603315}{1258134268435363162821398190688607406617403} a^{3} + \frac{85793020333344881488749905613635870859446}{352698727971419798782567693624002913152159} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{6}\times C_{149148972}$, which has order $57273205248$ (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:  \( 9996199.50284 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T875:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n875
Character table for t16n875 is not computed

Intermediate fields

\(\Q(\sqrt{257}) \), 4.4.528392.1, 8.8.285898860199936.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$2$2.4.9.1$x^{4} + 6 x^{2} + 2$$4$$1$$9$$D_{4}$$[2, 3, 7/2]$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.8.31.31$x^{8} + 4 x^{4} + 34$$8$$1$$31$$(C_8:C_2):C_2$$[2, 3, 7/2, 4, 5]$
239Data not computed
257Data not computed