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Properties

Label 16.8.20231328175...1609.1
Degree $16$
Signature $[8, 4]$
Discriminant $47^{12}\cdot 89^{15}$
Root discriminant $1206.78$
Ramified primes $47, 89$
Class number $1$ (GRH)
Class group Trivial (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![-3108481636851712, -2193741464668160, 178672951198304, 27704092920824, -53240095283656, 2645214889372, -252544335297, -61011866388, 5460971989, 195916584, 19436086, 1318432, -186046, -4036, -37, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 37*x^14 - 4036*x^13 - 186046*x^12 + 1318432*x^11 + 19436086*x^10 + 195916584*x^9 + 5460971989*x^8 - 61011866388*x^7 - 252544335297*x^6 + 2645214889372*x^5 - 53240095283656*x^4 + 27704092920824*x^3 + 178672951198304*x^2 - 2193741464668160*x - 3108481636851712)
 
gp: K = bnfinit(x^16 - 4*x^15 - 37*x^14 - 4036*x^13 - 186046*x^12 + 1318432*x^11 + 19436086*x^10 + 195916584*x^9 + 5460971989*x^8 - 61011866388*x^7 - 252544335297*x^6 + 2645214889372*x^5 - 53240095283656*x^4 + 27704092920824*x^3 + 178672951198304*x^2 - 2193741464668160*x - 3108481636851712, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 37 x^{14} - 4036 x^{13} - 186046 x^{12} + 1318432 x^{11} + 19436086 x^{10} + 195916584 x^{9} + 5460971989 x^{8} - 61011866388 x^{7} - 252544335297 x^{6} + 2645214889372 x^{5} - 53240095283656 x^{4} + 27704092920824 x^{3} + 178672951198304 x^{2} - 2193741464668160 x - 3108481636851712 \)

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:  \(20231328175647090122615673426308895930590180621609=47^{12}\cdot 89^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1206.78$
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:  $47, 89$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{4}$, $\frac{1}{32} a^{7} + \frac{1}{32} a^{6} - \frac{1}{32} a^{5} - \frac{1}{32} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{3008} a^{8} + \frac{45}{3008} a^{7} - \frac{185}{3008} a^{6} - \frac{85}{3008} a^{5} - \frac{13}{188} a^{4} + \frac{79}{376} a^{3} + \frac{51}{376} a^{2} - \frac{6}{47} a + \frac{7}{47}$, $\frac{1}{24064} a^{9} - \frac{1}{12032} a^{8} + \frac{9}{1504} a^{7} + \frac{451}{12032} a^{6} - \frac{537}{24064} a^{5} - \frac{359}{6016} a^{4} + \frac{145}{3008} a^{3} + \frac{751}{3008} a^{2} + \frac{155}{752} a$, $\frac{1}{48128} a^{10} - \frac{1}{48128} a^{9} - \frac{1}{24064} a^{8} - \frac{85}{24064} a^{7} - \frac{819}{48128} a^{6} - \frac{1013}{48128} a^{5} - \frac{665}{12032} a^{4} - \frac{91}{376} a^{3} + \frac{43}{6016} a^{2} + \frac{191}{1504} a + \frac{15}{94}$, $\frac{1}{385024} a^{11} + \frac{3}{385024} a^{10} - \frac{1}{192512} a^{9} - \frac{13}{192512} a^{8} - \frac{2747}{385024} a^{7} + \frac{2807}{385024} a^{6} - \frac{3359}{96256} a^{5} - \frac{347}{6016} a^{4} - \frac{7601}{48128} a^{3} + \frac{769}{12032} a^{2} + \frac{245}{1504} a + \frac{25}{94}$, $\frac{1}{1540096} a^{12} - \frac{3}{1540096} a^{10} + \frac{1}{385024} a^{9} - \frac{189}{1540096} a^{8} - \frac{357}{192512} a^{7} + \frac{20167}{1540096} a^{6} - \frac{11077}{385024} a^{5} - \frac{3549}{192512} a^{4} - \frac{37577}{192512} a^{3} + \frac{11315}{48128} a^{2} - \frac{183}{1504} a + \frac{63}{188}$, $\frac{1}{135528448} a^{13} + \frac{41}{135528448} a^{12} - \frac{163}{135528448} a^{11} - \frac{1399}{135528448} a^{10} - \frac{1593}{135528448} a^{9} - \frac{1255}{12320768} a^{8} - \frac{216705}{135528448} a^{7} + \frac{6365899}{135528448} a^{6} + \frac{178819}{3080192} a^{5} - \frac{352623}{8470528} a^{4} + \frac{2034363}{16941056} a^{3} - \frac{457293}{4235264} a^{2} - \frac{13293}{66176} a - \frac{1}{1504}$, $\frac{1}{4191253033910272} a^{14} + \frac{500969}{190511501541376} a^{13} - \frac{441155295}{2095626516955136} a^{12} + \frac{1205414377}{2095626516955136} a^{11} - \frac{4218562525}{1047813258477568} a^{10} + \frac{27491930095}{2095626516955136} a^{9} - \frac{346890084813}{2095626516955136} a^{8} - \frac{1839877126985}{2095626516955136} a^{7} + \frac{179574099887427}{4191253033910272} a^{6} - \frac{23860221125439}{1047813258477568} a^{5} - \frac{21865931555179}{523906629238784} a^{4} + \frac{90460444486587}{523906629238784} a^{3} - \frac{14794435171641}{130976657309696} a^{2} + \frac{968460519979}{2046510270464} a - \frac{12203877245}{46511597056}$, $\frac{1}{9389449309344093030381689638772443171903020874985600950705586176} a^{15} + \frac{262219078042267562628418812360632784316683454915}{9389449309344093030381689638772443171903020874985600950705586176} a^{14} - \frac{108701600824326479884375478485307856241633328738191247}{53349143803091437672623236583934336203994436789690914492645376} a^{13} - \frac{653038227149811578161406789925322156835185110510158920605}{2347362327336023257595422409693110792975755218746400237676396544} a^{12} + \frac{2731926968596200990944801209940210200725064635229765845707}{4694724654672046515190844819386221585951510437492800475352793088} a^{11} + \frac{46606015622640589201489988705183674578361278502200826117661}{4694724654672046515190844819386221585951510437492800475352793088} a^{10} + \frac{3686802808516724645092851957805632951552889815067894966547}{2347362327336023257595422409693110792975755218746400237676396544} a^{9} - \frac{16268900569766051817973243904785864615819591254034379298899}{213396575212365750690492946335737344815977747158763657970581504} a^{8} + \frac{34980666719150693960802491135778580169078269066480983617643641}{9389449309344093030381689638772443171903020874985600950705586176} a^{7} + \frac{265747646426556945046033864726419416585514545343055887133992827}{9389449309344093030381689638772443171903020874985600950705586176} a^{6} - \frac{98387891009881859363427187271941168230248487342891163120988761}{2347362327336023257595422409693110792975755218746400237676396544} a^{5} + \frac{11251152641169172804933564963747019315156627196938537023694255}{293420290917002907199427801211638849121969402343300029709549568} a^{4} + \frac{262043381891501640902600826587974069580984715874095667180523051}{1173681163668011628797711204846555396487877609373200118838198272} a^{3} + \frac{61171673193912037676822391898140302296121482719727268273917355}{293420290917002907199427801211638849121969402343300029709549568} a^{2} + \frac{694718337795234863259559823958460367124690313550507176881987}{4584692045578170424991059393931857017530771911614062964211712} a - \frac{18728548163717432288664616609346327454311725709335662451529}{104197546490412964204342258952996750398426634354865067368448}$

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:  $11$
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:  \( 4882461092630000000000 \) (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

\(\Q(\sqrt{89}) \), 4.4.704969.1, 8.8.215834804494349846249.1

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ $16$ $16$ R ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

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
$47$47.8.6.3$x^{8} - 47 x^{4} + 28717$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
47.8.6.3$x^{8} - 47 x^{4} + 28717$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
89Data not computed