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Properties

Label 16.8.81464396628...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{32}\cdot 5^{10}\cdot 761^{6}$
Root discriminant $131.65$
Ramified primes $2, 5, 761$
Class number $8$ (GRH)
Class group $[2, 2, 2]$ (GRH)
Galois group 16T869

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5857289, 147258708, 44381336, -97021992, -12392604, -3305052, 2055924, 1730976, 148909, 49452, -15384, -5112, -346, -168, 40, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 40*x^14 - 168*x^13 - 346*x^12 - 5112*x^11 - 15384*x^10 + 49452*x^9 + 148909*x^8 + 1730976*x^7 + 2055924*x^6 - 3305052*x^5 - 12392604*x^4 - 97021992*x^3 + 44381336*x^2 + 147258708*x - 5857289)
 
gp: K = bnfinit(x^16 + 40*x^14 - 168*x^13 - 346*x^12 - 5112*x^11 - 15384*x^10 + 49452*x^9 + 148909*x^8 + 1730976*x^7 + 2055924*x^6 - 3305052*x^5 - 12392604*x^4 - 97021992*x^3 + 44381336*x^2 + 147258708*x - 5857289, 1)
 

Normalized defining polynomial

\( x^{16} + 40 x^{14} - 168 x^{13} - 346 x^{12} - 5112 x^{11} - 15384 x^{10} + 49452 x^{9} + 148909 x^{8} + 1730976 x^{7} + 2055924 x^{6} - 3305052 x^{5} - 12392604 x^{4} - 97021992 x^{3} + 44381336 x^{2} + 147258708 x - 5857289 \)

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:  \(8146439662885973461565440000000000=2^{32}\cdot 5^{10}\cdot 761^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $131.65$
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:  $2, 5, 761$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{1522} a^{12} + \frac{73}{761} a^{11} + \frac{45}{1522} a^{10} - \frac{51}{1522} a^{9} - \frac{123}{1522} a^{8} + \frac{249}{761} a^{7} - \frac{643}{1522} a^{6} - \frac{665}{1522} a^{5} + \frac{525}{1522} a^{4} + \frac{231}{761} a^{3} + \frac{643}{1522} a^{2} + \frac{633}{1522} a + \frac{256}{761}$, $\frac{1}{1522} a^{13} + \frac{37}{1522} a^{11} + \frac{114}{761} a^{10} - \frac{287}{1522} a^{9} + \frac{96}{761} a^{8} - \frac{295}{1522} a^{7} - \frac{195}{761} a^{6} + \frac{207}{1522} a^{5} - \frac{44}{761} a^{4} + \frac{159}{1522} a^{3} + \frac{179}{761} a^{2} - \frac{293}{761} a - \frac{87}{761}$, $\frac{1}{7610} a^{14} - \frac{1}{3805} a^{13} - \frac{1}{3805} a^{12} + \frac{274}{3805} a^{11} - \frac{43}{1522} a^{10} - \frac{105}{761} a^{9} - \frac{1209}{7610} a^{8} - \frac{248}{761} a^{7} + \frac{2473}{7610} a^{6} + \frac{32}{761} a^{5} + \frac{3451}{7610} a^{4} + \frac{143}{3805} a^{3} + \frac{889}{3805} a^{2} + \frac{1473}{3805} a - \frac{3639}{7610}$, $\frac{1}{8096778031556501885301911519874564238707257470} a^{15} + \frac{214157824622514698303538747160918205224658}{4048389015778250942650955759937282119353628735} a^{14} - \frac{1021240119260118165127773470509989473184099}{4048389015778250942650955759937282119353628735} a^{13} - \frac{1040678537929965937538879142268132778880213}{8096778031556501885301911519874564238707257470} a^{12} + \frac{1129037688457031912963513939547730013427204189}{8096778031556501885301911519874564238707257470} a^{11} + \frac{360503108614877326670408599458825144273158849}{1619355606311300377060382303974912847741451494} a^{10} - \frac{543835421602217267240808278772815829587290962}{4048389015778250942650955759937282119353628735} a^{9} + \frac{1576205223662668727913390603078691752826627193}{8096778031556501885301911519874564238707257470} a^{8} - \frac{125730279635853100620661852349441104854413807}{261186388114725867267803597415308523829266370} a^{7} + \frac{526133065678704209812188438840635063875815279}{8096778031556501885301911519874564238707257470} a^{6} + \frac{1051326210306496203541398033697815941947625038}{4048389015778250942650955759937282119353628735} a^{5} - \frac{1543861831422739589654691155261906364698711261}{8096778031556501885301911519874564238707257470} a^{4} - \frac{403805369274431444077782813591742273274252102}{4048389015778250942650955759937282119353628735} a^{3} + \frac{605165394662622260723839237675300790014639677}{1619355606311300377060382303974912847741451494} a^{2} + \frac{364974030114084138422676945121594914007864082}{4048389015778250942650955759937282119353628735} a + \frac{1314653590856962935694612511147091139306557984}{4048389015778250942650955759937282119353628735}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:  \( 24813244748.9 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T869:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 41 conjugacy class representatives for t16n869
Character table for t16n869 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), 4.4.1217600.3, 4.4.48704.1, \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.1482549760000.4

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.8.16.16$x^{8} + 2 x^{6} + 4 x^{5} + 6 x^{4} + 8 x^{3} + 4$$4$$2$$16$$D_4\times C_2$$[2, 2, 3]^{2}$
2.8.16.13$x^{8} + 6 x^{6} + 4 x^{5} + 2 x^{4} + 4$$4$$2$$16$$D_4\times C_2$$[2, 2, 3]^{2}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
761Data not computed