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Properties

Label 16.8.13854815685...5057.1
Degree $16$
Signature $[8, 4]$
Discriminant $41^{15}\cdot 73^{15}$
Root discriminant $1814.89$
Ramified primes $41, 73$
Class number $48$ (GRH)
Class group $[48]$ (GRH)
Galois group $C_2^3.C_8$ (as 16T104)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-261511764516720983, 210034438044426092, 83906088161932922, 5872021418457196, -848384823179655, -124062279034770, -1039599254791, 640346059654, 37182839562, -1081392193, -196778428, -280415, 668694, 4314, -1380, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 - 1380*x^14 + 4314*x^13 + 668694*x^12 - 280415*x^11 - 196778428*x^10 - 1081392193*x^9 + 37182839562*x^8 + 640346059654*x^7 - 1039599254791*x^6 - 124062279034770*x^5 - 848384823179655*x^4 + 5872021418457196*x^3 + 83906088161932922*x^2 + 210034438044426092*x - 261511764516720983)
 
gp: K = bnfinit(x^16 - 7*x^15 - 1380*x^14 + 4314*x^13 + 668694*x^12 - 280415*x^11 - 196778428*x^10 - 1081392193*x^9 + 37182839562*x^8 + 640346059654*x^7 - 1039599254791*x^6 - 124062279034770*x^5 - 848384823179655*x^4 + 5872021418457196*x^3 + 83906088161932922*x^2 + 210034438044426092*x - 261511764516720983, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} - 1380 x^{14} + 4314 x^{13} + 668694 x^{12} - 280415 x^{11} - 196778428 x^{10} - 1081392193 x^{9} + 37182839562 x^{8} + 640346059654 x^{7} - 1039599254791 x^{6} - 124062279034770 x^{5} - 848384823179655 x^{4} + 5872021418457196 x^{3} + 83906088161932922 x^{2} + 210034438044426092 x - 261511764516720983 \)

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:  \(13854815685101981032101054249860279864869741101695057=41^{15}\cdot 73^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1814.89$
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:  $41, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{73} a^{8} + \frac{33}{73} a^{7} - \frac{30}{73} a^{6} + \frac{8}{73} a^{5} + \frac{23}{73} a^{4} - \frac{19}{73} a^{3} + \frac{11}{73} a^{2} + \frac{26}{73} a + \frac{2}{73}$, $\frac{1}{73} a^{9} - \frac{24}{73} a^{7} - \frac{24}{73} a^{6} - \frac{22}{73} a^{5} + \frac{25}{73} a^{4} - \frac{19}{73} a^{3} + \frac{28}{73} a^{2} + \frac{20}{73} a + \frac{7}{73}$, $\frac{1}{73} a^{10} - \frac{35}{73} a^{7} - \frac{12}{73} a^{6} - \frac{2}{73} a^{5} + \frac{22}{73} a^{4} + \frac{10}{73} a^{3} - \frac{8}{73} a^{2} - \frac{26}{73} a - \frac{25}{73}$, $\frac{1}{73} a^{11} - \frac{25}{73} a^{7} - \frac{30}{73} a^{6} + \frac{10}{73} a^{5} + \frac{12}{73} a^{4} - \frac{16}{73} a^{3} - \frac{6}{73} a^{2} + \frac{9}{73} a - \frac{3}{73}$, $\frac{1}{73} a^{12} - \frac{8}{73} a^{7} - \frac{10}{73} a^{6} - \frac{7}{73} a^{5} - \frac{25}{73} a^{4} + \frac{30}{73} a^{3} - \frac{8}{73} a^{2} - \frac{10}{73} a - \frac{23}{73}$, $\frac{1}{73} a^{13} + \frac{35}{73} a^{7} - \frac{28}{73} a^{6} - \frac{34}{73} a^{5} - \frac{5}{73} a^{4} - \frac{14}{73} a^{3} + \frac{5}{73} a^{2} - \frac{34}{73} a + \frac{16}{73}$, $\frac{1}{73} a^{14} - \frac{15}{73} a^{7} - \frac{6}{73} a^{6} + \frac{7}{73} a^{5} - \frac{16}{73} a^{4} + \frac{13}{73} a^{3} + \frac{19}{73} a^{2} - \frac{18}{73} a + \frac{3}{73}$, $\frac{1}{35261909491409121113235152852320203766336141229055058266505234174140406973833363660490988844465466622198413} a^{15} + \frac{123586321264654276182454794069788303797849106726206311334989307830352717366718149692176782367737232178193}{35261909491409121113235152852320203766336141229055058266505234174140406973833363660490988844465466622198413} a^{14} - \frac{182099079169855065889126388117006178626804864323372128780131494163121228901383571205648843688909706805588}{35261909491409121113235152852320203766336141229055058266505234174140406973833363660490988844465466622198413} a^{13} + \frac{217973939499495460760795580020293061325961464164179206541504677061929232630527564594172761487398808538177}{35261909491409121113235152852320203766336141229055058266505234174140406973833363660490988844465466622198413} a^{12} - \frac{72244642951589044035171852505050141192746603964143462347629973579920649125314559137245809329786237113634}{35261909491409121113235152852320203766336141229055058266505234174140406973833363660490988844465466622198413} a^{11} - \frac{52063758933871335420712672447060562935925184192125240373590831056577767222496228196856195168350947049241}{35261909491409121113235152852320203766336141229055058266505234174140406973833363660490988844465466622198413} a^{10} + \frac{116137508403817395643434648258200433223793330123888310809120938922059792556848301664236013102689559343657}{35261909491409121113235152852320203766336141229055058266505234174140406973833363660490988844465466622198413} a^{9} - \frac{172088385304545902846558236569166718319503672426828541490459426505885740827427920097109722880671219928595}{35261909491409121113235152852320203766336141229055058266505234174140406973833363660490988844465466622198413} a^{8} + \frac{14901254946842086741961608895440704461038538798471433599879282085120186080468037642126609358817467844367386}{35261909491409121113235152852320203766336141229055058266505234174140406973833363660490988844465466622198413} a^{7} + \frac{9713860521231807952967180753143729815988280364086133235415523517829257998517404323177111462808869217827101}{35261909491409121113235152852320203766336141229055058266505234174140406973833363660490988844465466622198413} a^{6} - \frac{1425895723524122942942682063947491531387760901541958461647403687272510394630077141034012144015658340625490}{35261909491409121113235152852320203766336141229055058266505234174140406973833363660490988844465466622198413} a^{5} - \frac{7987457917764735186170632986484945067874101081285673776471311542289260697378570037984872664618251605372569}{35261909491409121113235152852320203766336141229055058266505234174140406973833363660490988844465466622198413} a^{4} - \frac{747414549114428812794610258946639537794949802330468298355480627567951760150836980370347895350554520389513}{35261909491409121113235152852320203766336141229055058266505234174140406973833363660490988844465466622198413} a^{3} + \frac{10870501653571694006153754436904480025327237661060777928204495299667530907733207288865577741974606613036039}{35261909491409121113235152852320203766336141229055058266505234174140406973833363660490988844465466622198413} a^{2} - \frac{13764008602434114079836207682610273594139394557609737558791725790948440465026022195631303498045630383278437}{35261909491409121113235152852320203766336141229055058266505234174140406973833363660490988844465466622198413} a + \frac{1967894014435230806499490383005111002741522002807376111127550015293887109902886977235593989855534350272956}{35261909491409121113235152852320203766336141229055058266505234174140406973833363660490988844465466622198413}$

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

Class group and class number

$C_{48}$, which has order $48$ (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:  \( 2367139577690000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_8$ (as 16T104):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $C_2^3.C_8$
Character table for $C_2^3.C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{2993}) \), 4.4.26811440657.3, 8.8.2151528076860770946805457.4

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ R $16$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

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
41Data not computed
73Data not computed