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Properties

Label 16.0.24209958949...0137.1
Degree $16$
Signature $[0, 8]$
Discriminant $23^{12}\cdot 73^{7}$
Root discriminant $68.63$
Ramified primes $23, 73$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T289)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![15512391, -11428128, -148671, 2236473, -197231, -44446, 141119, -26060, -3048, 2288, 1069, 72, -257, 76, 9, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 9*x^14 + 76*x^13 - 257*x^12 + 72*x^11 + 1069*x^10 + 2288*x^9 - 3048*x^8 - 26060*x^7 + 141119*x^6 - 44446*x^5 - 197231*x^4 + 2236473*x^3 - 148671*x^2 - 11428128*x + 15512391)
 
gp: K = bnfinit(x^16 - 6*x^15 + 9*x^14 + 76*x^13 - 257*x^12 + 72*x^11 + 1069*x^10 + 2288*x^9 - 3048*x^8 - 26060*x^7 + 141119*x^6 - 44446*x^5 - 197231*x^4 + 2236473*x^3 - 148671*x^2 - 11428128*x + 15512391, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 9 x^{14} + 76 x^{13} - 257 x^{12} + 72 x^{11} + 1069 x^{10} + 2288 x^{9} - 3048 x^{8} - 26060 x^{7} + 141119 x^{6} - 44446 x^{5} - 197231 x^{4} + 2236473 x^{3} - 148671 x^{2} - 11428128 x + 15512391 \)

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:  \(242099589496868228963210570137=23^{12}\cdot 73^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68.63$
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:  $23, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{12} + \frac{1}{9} a^{11} + \frac{1}{3} a^{10} - \frac{2}{9} a^{9} - \frac{1}{9} a^{8} - \frac{1}{9} a^{7} - \frac{1}{3} a^{5} - \frac{2}{9} a^{4} - \frac{2}{9} a^{3} + \frac{1}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{369} a^{14} - \frac{13}{369} a^{13} + \frac{49}{369} a^{12} + \frac{3}{41} a^{11} + \frac{7}{369} a^{10} + \frac{122}{369} a^{9} - \frac{142}{369} a^{8} + \frac{4}{123} a^{7} + \frac{17}{123} a^{6} - \frac{92}{369} a^{5} + \frac{49}{369} a^{4} + \frac{124}{369} a^{3} + \frac{46}{123} a^{2} + \frac{6}{41} a$, $\frac{1}{16445999195633399158247385809975630564107743579} a^{15} - \frac{1517191936540481053582032656644093988296574}{5481999731877799719415795269991876854702581193} a^{14} + \frac{35568848186601359323620052195212956471760437}{1827333243959266573138598423330625618234193731} a^{13} - \frac{1178074292952259646682145275123358577140258778}{16445999195633399158247385809975630564107743579} a^{12} - \frac{1708178421756648517497567529667370728339941814}{16445999195633399158247385809975630564107743579} a^{11} - \frac{115111074655728470678758151179012542670221698}{1827333243959266573138598423330625618234193731} a^{10} - \frac{5699322869998064485328696311363532102108718260}{16445999195633399158247385809975630564107743579} a^{9} + \frac{2513870439366815714669651970856138737771938192}{16445999195633399158247385809975630564107743579} a^{8} + \frac{1517178622381643575844516074937366615510836765}{5481999731877799719415795269991876854702581193} a^{7} - \frac{6442760005041882371060174428736478273814429532}{16445999195633399158247385809975630564107743579} a^{6} + \frac{2112085270591514852613505142113518286170586292}{16445999195633399158247385809975630564107743579} a^{5} - \frac{5421027200801097761149362919021294101417728164}{16445999195633399158247385809975630564107743579} a^{4} - \frac{7720247138475260962756950546812833229485475345}{16445999195633399158247385809975630564107743579} a^{3} + \frac{882539649111727364901704040012217820945443909}{1827333243959266573138598423330625618234193731} a^{2} + \frac{549070150311625974778567935054960925602696602}{1827333243959266573138598423330625618234193731} a + \frac{10902597023911526103297991306028042482857}{28624986198588069194019117805201146956063}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 704664552.281 \) (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

\(\Q(\sqrt{-23}) \), 4.0.38617.1, 8.0.108862906297.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{8}$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
23Data not computed
$73$73.8.0.1$x^{8} - x + 40$$1$$8$$0$$C_8$$[\ ]^{8}$
73.8.7.6$x^{8} + 9125$$8$$1$$7$$C_8$$[\ ]_{8}$