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Properties

Label 16.0.13735096585...809.11
Degree $16$
Signature $[0, 8]$
Discriminant $13^{8}\cdot 17^{14}$
Root discriminant $43.01$
Ramified primes $13, 17$
Class number $16$ (GRH)
Class group $[2, 2, 4]$ (GRH)
Galois group $C_2^2 : C_8$ (as 16T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![17407, -40729, 105638, -87451, 129016, -83044, 69763, -36402, 22378, -8465, 4498, -1176, 539, -86, 34, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 34*x^14 - 86*x^13 + 539*x^12 - 1176*x^11 + 4498*x^10 - 8465*x^9 + 22378*x^8 - 36402*x^7 + 69763*x^6 - 83044*x^5 + 129016*x^4 - 87451*x^3 + 105638*x^2 - 40729*x + 17407)
 
gp: K = bnfinit(x^16 - 2*x^15 + 34*x^14 - 86*x^13 + 539*x^12 - 1176*x^11 + 4498*x^10 - 8465*x^9 + 22378*x^8 - 36402*x^7 + 69763*x^6 - 83044*x^5 + 129016*x^4 - 87451*x^3 + 105638*x^2 - 40729*x + 17407, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 34 x^{14} - 86 x^{13} + 539 x^{12} - 1176 x^{11} + 4498 x^{10} - 8465 x^{9} + 22378 x^{8} - 36402 x^{7} + 69763 x^{6} - 83044 x^{5} + 129016 x^{4} - 87451 x^{3} + 105638 x^{2} - 40729 x + 17407 \)

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:  \(137350965859713069141239809=13^{8}\cdot 17^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.01$
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:  $13, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2678} a^{14} - \frac{131}{1339} a^{13} + \frac{186}{1339} a^{12} - \frac{671}{2678} a^{11} + \frac{367}{1339} a^{10} - \frac{315}{1339} a^{9} + \frac{63}{206} a^{8} - \frac{599}{1339} a^{7} + \frac{295}{1339} a^{6} - \frac{483}{2678} a^{5} + \frac{191}{1339} a^{4} - \frac{4}{103} a^{3} + \frac{251}{2678} a^{2} - \frac{22}{103} a$, $\frac{1}{1425856334649434686434406189846874} a^{15} - \frac{69148850231781591939059866703}{712928167324717343217203094923437} a^{14} - \frac{41532201861679903057094626339288}{712928167324717343217203094923437} a^{13} - \frac{2817921980658918017044464013525}{13843265384945967829460254270358} a^{12} + \frac{180902625518741595123294501084}{7058694725987300427893099949737} a^{11} - \frac{260929031940957143646302377849417}{712928167324717343217203094923437} a^{10} + \frac{62617153955435759207067805080497}{1425856334649434686434406189846874} a^{9} - \frac{308571857078233462112015872851649}{712928167324717343217203094923437} a^{8} + \frac{102600169618832069457286009459876}{712928167324717343217203094923437} a^{7} + \frac{9214563374636263467798215464021}{1425856334649434686434406189846874} a^{6} - \frac{101695975400882745185495937861899}{712928167324717343217203094923437} a^{5} + \frac{5675910445871712859648359380071}{15168684411164198791855384998371} a^{4} - \frac{186800023603427372274958394027581}{1425856334649434686434406189846874} a^{3} - \frac{60782113566283088638514941101642}{712928167324717343217203094923437} a^{2} + \frac{10928150292603627133791900329350}{54840628255747487939784853455649} a + \frac{137775273928735144846350430850}{532433284036383378056163625783}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}$, which has order $16$ (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:  \( 232376.238125 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_8$ (as 16T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2^2 : C_8$
Character table for $C_2^2 : C_8$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 4.0.3757.1, 4.0.63869.1, 8.8.11719682839553.1, 8.0.69347235737.1, 8.0.4079249161.1

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 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$13$13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$17$17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$