↑

Properties

Label 16.0.14542935268...3777.2
Degree $16$
Signature $[0, 8]$
Discriminant $17^{13}\cdot 59^{8}$
Root discriminant $76.77$
Ramified primes $17, 59$
Class number $12$ (GRH)
Class group $[12]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T289)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1422185, -4906712, 8457612, -9199270, 7120382, -4264318, 2128095, -894616, 300789, -79564, 17177, -2562, -132, 194, -34, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 34*x^14 + 194*x^13 - 132*x^12 - 2562*x^11 + 17177*x^10 - 79564*x^9 + 300789*x^8 - 894616*x^7 + 2128095*x^6 - 4264318*x^5 + 7120382*x^4 - 9199270*x^3 + 8457612*x^2 - 4906712*x + 1422185)
 
gp: K = bnfinit(x^16 - 2*x^15 - 34*x^14 + 194*x^13 - 132*x^12 - 2562*x^11 + 17177*x^10 - 79564*x^9 + 300789*x^8 - 894616*x^7 + 2128095*x^6 - 4264318*x^5 + 7120382*x^4 - 9199270*x^3 + 8457612*x^2 - 4906712*x + 1422185, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 34 x^{14} + 194 x^{13} - 132 x^{12} - 2562 x^{11} + 17177 x^{10} - 79564 x^{9} + 300789 x^{8} - 894616 x^{7} + 2128095 x^{6} - 4264318 x^{5} + 7120382 x^{4} - 9199270 x^{3} + 8457612 x^{2} - 4906712 x + 1422185 \)

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:  \(1454293526857723611028217753777=17^{13}\cdot 59^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $76.77$
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:  $17, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{42} a^{12} - \frac{2}{21} a^{11} - \frac{5}{42} a^{10} + \frac{1}{21} a^{9} - \frac{5}{42} a^{8} - \frac{5}{21} a^{7} + \frac{3}{7} a^{6} - \frac{3}{7} a^{5} + \frac{4}{21} a^{4} - \frac{1}{7} a^{3} - \frac{5}{14} a^{2} - \frac{1}{2} a - \frac{4}{21}$, $\frac{1}{630} a^{13} + \frac{7}{30} a^{11} - \frac{8}{35} a^{10} + \frac{29}{210} a^{9} + \frac{3}{35} a^{8} + \frac{2}{63} a^{7} + \frac{16}{105} a^{6} - \frac{11}{315} a^{5} - \frac{92}{315} a^{4} + \frac{1}{210} a^{3} - \frac{23}{70} a^{2} + \frac{17}{315} a - \frac{20}{63}$, $\frac{1}{630} a^{14} - \frac{1}{210} a^{12} + \frac{47}{210} a^{11} - \frac{6}{35} a^{10} + \frac{23}{210} a^{9} + \frac{2}{9} a^{8} - \frac{7}{15} a^{7} + \frac{113}{630} a^{6} + \frac{311}{630} a^{5} + \frac{1}{10} a^{4} + \frac{1}{10} a^{3} + \frac{79}{630} a^{2} + \frac{23}{126} a + \frac{17}{42}$, $\frac{1}{13395907809729022032750281279147443890} a^{15} - \frac{511467892065974031492952514661187}{744217100540501224041682293285969105} a^{14} - \frac{82385222497319735103605095365769}{1339590780972902203275028127914744389} a^{13} - \frac{10359866949490979089030433445886589}{1488434201081002448083364586571938210} a^{12} - \frac{771996083962686538044177660436064681}{4465302603243007344250093759715814630} a^{11} - \frac{1804393254934205778166664833567132}{148843420108100244808336458657193821} a^{10} - \frac{143275089084508287598363215056683589}{6697953904864511016375140639573721945} a^{9} + \frac{11406921973555412623483068866816098}{2232651301621503672125046879857907315} a^{8} + \frac{728629699176747725032305411582391237}{2232651301621503672125046879857907315} a^{7} - \frac{5516944861804407044117518613223135079}{13395907809729022032750281279147443890} a^{6} + \frac{3685365798028883156236418961905801161}{13395907809729022032750281279147443890} a^{5} - \frac{663178846696792429634006466347416261}{6697953904864511016375140639573721945} a^{4} + \frac{125302954351247021094897081824033116}{1339590780972902203275028127914744389} a^{3} - \frac{825679809483145510329093061139886049}{2679181561945804406550056255829488778} a^{2} + \frac{92105393708003387110466366019136498}{956850557837787288053591519939103135} a + \frac{492658401631900772541456422242822313}{1339590780972902203275028127914744389}$

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

Class group and class number

$C_{12}$, which has order $12$ (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:  \( 295149176.043 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T289):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 44 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-59}) \), 4.0.59177.1, 8.0.17204919837377.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ R

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
$17$17.8.7.8$x^{8} + 4131$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.6.3$x^{8} - 17 x^{4} + 867$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$
59Data not computed