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Properties

Label 16.8.13986292138...2672.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{59}\cdot 113^{6}\cdot 1039^{4}$
Root discriminant $430.63$
Ramified primes $2, 113, 1039$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T942

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![119044425692577032, 0, -9806891286708672, 0, -234046949464216, 0, 11094212289568, 0, 48177644922, 0, -108139328, 0, -516032, 0, 92, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 92*x^14 - 516032*x^12 - 108139328*x^10 + 48177644922*x^8 + 11094212289568*x^6 - 234046949464216*x^4 - 9806891286708672*x^2 + 119044425692577032)
 
gp: K = bnfinit(x^16 + 92*x^14 - 516032*x^12 - 108139328*x^10 + 48177644922*x^8 + 11094212289568*x^6 - 234046949464216*x^4 - 9806891286708672*x^2 + 119044425692577032, 1)
 

Normalized defining polynomial

\( x^{16} + 92 x^{14} - 516032 x^{12} - 108139328 x^{10} + 48177644922 x^{8} + 11094212289568 x^{6} - 234046949464216 x^{4} - 9806891286708672 x^{2} + 119044425692577032 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1398629213800614161925350535491003636252672=2^{59}\cdot 113^{6}\cdot 1039^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $430.63$
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:  $2, 113, 1039$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2078} a^{10} + \frac{46}{1039} a^{8} - \frac{344}{1039} a^{6} - \frac{104}{1039} a^{4} + \frac{203}{1039} a^{2}$, $\frac{1}{2078} a^{11} + \frac{46}{1039} a^{9} - \frac{344}{1039} a^{7} - \frac{104}{1039} a^{5} + \frac{203}{1039} a^{3}$, $\frac{1}{4318084} a^{12} + \frac{23}{1079521} a^{10} - \frac{129008}{1079521} a^{8} - \frac{46807}{1079521} a^{6} + \frac{390867}{2159042} a^{4} + \frac{212}{1039} a^{2}$, $\frac{1}{8636168} a^{13} - \frac{993}{4318084} a^{11} + \frac{725917}{4318084} a^{9} - \frac{384456}{1079521} a^{7} + \frac{606979}{4318084} a^{5} + \frac{9}{2078} a^{3} - \frac{1}{2} a$, $\frac{1}{9105011497661378862590704053993233151684323617706994776} a^{14} - \frac{193289421505172181537607322866565686606052751727}{2276252874415344715647676013498308287921080904426748694} a^{12} + \frac{968070581923893981911100398882696472389444303253293}{4552505748830689431295352026996616575842161808853497388} a^{10} - \frac{188694827509746923140583668104248932776774369533550108}{1138126437207672357823838006749154143960540452213374347} a^{8} + \frac{830769357869122379619365821942682667847301013651234179}{4552505748830689431295352026996616575842161808853497388} a^{6} - \frac{21753437457205279851361511916350629666281106257419}{99582328918336893676072972854068960010546894060143} a^{4} + \frac{104149310206502886335289679436172030085459754609}{2108576743217913051851400772655935630637181587414} a^{2} - \frac{3770132985644215266400370922998640553727118}{8979774388315488224089708333642524000430901}$, $\frac{1}{9105011497661378862590704053993233151684323617706994776} a^{15} + \frac{281130685588267799775271094861705068894379786799}{9105011497661378862590704053993233151684323617706994776} a^{13} - \frac{39418885541899924166560042370487784110958177448879}{2276252874415344715647676013498308287921080904426748694} a^{11} + \frac{10546541814271201868071974925043681485527995061203887}{4552505748830689431295352026996616575842161808853497388} a^{9} - \frac{790540602912049580905798448961738109777195555090439389}{4552505748830689431295352026996616575842161808853497388} a^{7} - \frac{31021895641155284848629554989493552301115069760319}{398329315673347574704291891416275840042187576240572} a^{5} + \frac{56640870379709868929594456405743238496948990463}{1054288371608956525925700386327967815318590793707} a^{3} + \frac{1439508417027057691288966487645242892976665}{17959548776630976448179416667285048000861802} a$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $11$
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:  \( 862186582842000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T942:

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 t16n942
Character table for t16n942 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.7232.1, 8.8.26778533888.1

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 $16$ $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ $16$

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$2.8.30.21$x^{8} + 8 x^{7} + 30$$8$$1$$30$$(((C_4 \times C_2): C_2):C_2):C_2$$[2, 3, 7/2, 4, 17/4, 19/4]$
2.8.29.76$x^{8} + 4 x^{6} + 12 x^{4} + 14$$8$$1$$29$$(((C_4 \times C_2): C_2):C_2):C_2$$[2, 3, 7/2, 4, 17/4, 19/4]$
$113$113.4.2.2$x^{4} - 113 x^{2} + 127690$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
113.4.2.1$x^{4} + 2147 x^{2} + 1276900$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
113.4.0.1$x^{4} - x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
113.4.2.1$x^{4} + 2147 x^{2} + 1276900$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
1039Data not computed