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Properties

Label 16.0.66787881998...0625.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 149^{6}$
Root discriminant $26.70$
Ramified primes $5, 149$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4\wr C_2$ (as 16T28)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![14851, -30977, 5334, 38408, -27252, -17167, 21941, 1389, -8872, 2103, 1689, -846, -52, 91, -4, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 4*x^14 + 91*x^13 - 52*x^12 - 846*x^11 + 1689*x^10 + 2103*x^9 - 8872*x^8 + 1389*x^7 + 21941*x^6 - 17167*x^5 - 27252*x^4 + 38408*x^3 + 5334*x^2 - 30977*x + 14851)
 
gp: K = bnfinit(x^16 - 6*x^15 - 4*x^14 + 91*x^13 - 52*x^12 - 846*x^11 + 1689*x^10 + 2103*x^9 - 8872*x^8 + 1389*x^7 + 21941*x^6 - 17167*x^5 - 27252*x^4 + 38408*x^3 + 5334*x^2 - 30977*x + 14851, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 4 x^{14} + 91 x^{13} - 52 x^{12} - 846 x^{11} + 1689 x^{10} + 2103 x^{9} - 8872 x^{8} + 1389 x^{7} + 21941 x^{6} - 17167 x^{5} - 27252 x^{4} + 38408 x^{3} + 5334 x^{2} - 30977 x + 14851 \)

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:  \(66787881998297119140625=5^{14}\cdot 149^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.70$
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:  $5, 149$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{22} a^{13} - \frac{5}{22} a^{12} - \frac{2}{11} a^{11} - \frac{1}{22} a^{10} - \frac{1}{2} a^{9} - \frac{5}{22} a^{8} - \frac{1}{11} a^{7} + \frac{4}{11} a^{6} + \frac{5}{11} a^{5} - \frac{4}{11} a^{4} + \frac{7}{22} a^{3} - \frac{5}{11} a^{2} + \frac{7}{22} a - \frac{2}{11}$, $\frac{1}{44} a^{14} - \frac{1}{44} a^{13} - \frac{1}{22} a^{12} - \frac{3}{22} a^{11} - \frac{1}{11} a^{10} - \frac{4}{11} a^{9} + \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{9}{44} a^{6} - \frac{3}{11} a^{5} + \frac{2}{11} a^{4} + \frac{7}{44} a^{3} - \frac{1}{4} a^{2} + \frac{1}{22} a + \frac{17}{44}$, $\frac{1}{14889891036872412555564454676} a^{15} - \frac{35624288612607474347287697}{3722472759218103138891113669} a^{14} - \frac{181519180753317397065411201}{14889891036872412555564454676} a^{13} - \frac{1168931044916208169123156}{8253819865228610064060119} a^{12} - \frac{43829429672042230401731605}{7444945518436206277782227338} a^{11} - \frac{778047098206635753076140390}{3722472759218103138891113669} a^{10} - \frac{5185275922676252883153956967}{14889891036872412555564454676} a^{9} - \frac{3832620569723273081025626085}{14889891036872412555564454676} a^{8} + \frac{82770901292759600235494915}{1353626457897492050505859516} a^{7} + \frac{5254413251272627455792984527}{14889891036872412555564454676} a^{6} + \frac{527882460871306669071238098}{3722472759218103138891113669} a^{5} - \frac{128136031526976850036686237}{1353626457897492050505859516} a^{4} + \frac{1055752391778592087279125675}{3722472759218103138891113669} a^{3} + \frac{1571076608707280345212178033}{14889891036872412555564454676} a^{2} - \frac{5649177397616126707989526895}{14889891036872412555564454676} a + \frac{3507710914919261059628980505}{14889891036872412555564454676}$

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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{2979643316639244454267103}{3722472759218103138891113669} a^{15} - \frac{53293492938184583056478881}{14889891036872412555564454676} a^{14} - \frac{140266633850503972557693473}{14889891036872412555564454676} a^{13} + \frac{1038139660685408901392183}{16507639730457220128120238} a^{12} + \frac{209577142130870256622944521}{3722472759218103138891113669} a^{11} - \frac{4840624120307847535488749691}{7444945518436206277782227338} a^{10} + \frac{1470573661307212391268043842}{3722472759218103138891113669} a^{9} + \frac{41795208952492839579247024377}{14889891036872412555564454676} a^{8} - \frac{1333667752774356104383291154}{338406614474373012626464879} a^{7} - \frac{82832561397149054314439830949}{14889891036872412555564454676} a^{6} + \frac{49043500037228665323436625486}{3722472759218103138891113669} a^{5} + \frac{1647878631292425221904611951}{338406614474373012626464879} a^{4} - \frac{329859041180291264949202847629}{14889891036872412555564454676} a^{3} + \frac{56136958434563993018857151137}{14889891036872412555564454676} a^{2} + \frac{59748529162666123522993548448}{3722472759218103138891113669} a - \frac{148693709149378333665689376371}{14889891036872412555564454676} \) (order $10$)
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:  \( 346477.938234 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\wr C_2$ (as 16T28):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_4\wr C_2$
Character table for $C_4\wr C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.18625.1, \(\Q(\zeta_{5})\), 4.0.3725.1, 8.0.346890625.1

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

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
5Data not computed
$149$149.4.2.2$x^{4} - 149 x^{2} + 66603$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
149.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
149.8.4.1$x^{8} + 88804 x^{4} - 3307949 x^{2} + 1971537604$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$