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Properties

Label 16.0.20160743165...3424.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{62}\cdot 137^{12}$
Root discriminant $587.53$
Ramified primes $2, 137$
Class number $1668388896$ (GRH)
Class group $[2, 6, 139032408]$ (GRH)
Galois group $C_8\times C_2$ (as 16T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![360729969664, 0, 721459939328, 0, 254419951232, 0, 33736151360, 0, 1983245154, 0, 49550160, 0, 388532, 0, 1096, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 1096*x^14 + 388532*x^12 + 49550160*x^10 + 1983245154*x^8 + 33736151360*x^6 + 254419951232*x^4 + 721459939328*x^2 + 360729969664)
 
gp: K = bnfinit(x^16 + 1096*x^14 + 388532*x^12 + 49550160*x^10 + 1983245154*x^8 + 33736151360*x^6 + 254419951232*x^4 + 721459939328*x^2 + 360729969664, 1)
 

Normalized defining polynomial

\( x^{16} + 1096 x^{14} + 388532 x^{12} + 49550160 x^{10} + 1983245154 x^{8} + 33736151360 x^{6} + 254419951232 x^{4} + 721459939328 x^{2} + 360729969664 \)

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:  \(201607431658701025979129418461200790453223424=2^{62}\cdot 137^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $587.53$
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:  $2, 137$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4384=2^{5}\cdot 137\)
Dirichlet character group:    $\lbrace$$\chi_{4384}(1,·)$, $\chi_{4384}(3051,·)$, $\chi_{4384}(1097,·)$, $\chi_{4384}(4147,·)$, $\chi_{4384}(273,·)$, $\chi_{4384}(1059,·)$, $\chi_{4384}(1369,·)$, $\chi_{4384}(3289,·)$, $\chi_{4384}(859,·)$, $\chi_{4384}(2465,·)$, $\chi_{4384}(1955,·)$, $\chi_{4384}(2193,·)$, $\chi_{4384}(3561,·)$, $\chi_{4384}(2155,·)$, $\chi_{4384}(3251,·)$, $\chi_{4384}(4347,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{137} a^{4}$, $\frac{1}{137} a^{5}$, $\frac{1}{137} a^{6}$, $\frac{1}{137} a^{7}$, $\frac{1}{412918} a^{8} + \frac{2}{1507} a^{6} - \frac{4}{1507} a^{4} + \frac{5}{11} a^{2} + \frac{5}{11}$, $\frac{1}{825836} a^{9} + \frac{1}{1507} a^{7} - \frac{2}{1507} a^{5} - \frac{3}{11} a^{3} + \frac{5}{22} a$, $\frac{1}{1651672} a^{10} - \frac{1}{274} a^{6} - \frac{4}{1507} a^{4} - \frac{7}{44} a^{2} - \frac{3}{11}$, $\frac{1}{3303344} a^{11} - \frac{1}{548} a^{7} - \frac{2}{1507} a^{5} - \frac{7}{88} a^{3} + \frac{4}{11} a$, $\frac{1}{15386976352} a^{12} + \frac{7}{28078424} a^{10} - \frac{27}{28078424} a^{8} + \frac{60}{25619} a^{6} + \frac{945}{409904} a^{4} - \frac{167}{748} a^{2} - \frac{52}{187}$, $\frac{1}{30773952704} a^{13} + \frac{7}{56156848} a^{11} - \frac{27}{56156848} a^{9} - \frac{127}{51238} a^{7} - \frac{2047}{819808} a^{5} + \frac{581}{1496} a^{3} - \frac{26}{187} a$, $\frac{1}{52345368022957803904} a^{14} + \frac{38300155}{13086342005739450976} a^{12} + \frac{27838578193}{95520744567441248} a^{10} + \frac{4033454019}{11940093070930156} a^{8} + \frac{1248491294049}{1394463424342208} a^{6} + \frac{50301010625}{20506815063856} a^{4} + \frac{158635408329}{636160321324} a^{2} + \frac{78297617606}{159040080331}$, $\frac{1}{104690736045915607808} a^{15} + \frac{38300155}{26172684011478901952} a^{13} + \frac{27838578193}{191041489134882496} a^{11} + \frac{4033454019}{23880186141860312} a^{9} - \frac{8930073847135}{2788926848684416} a^{7} + \frac{50301010625}{41013630127712} a^{5} - \frac{477524912995}{1272320642648} a^{3} - \frac{80742462725}{318080160662} a$

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

Class group and class number

$C_{2}\times C_{6}\times C_{139032408}$, which has order $1668388896$ (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:  \( 7631128.109331426 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_8$ (as 16T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_8\times C_2$
Character table for $C_8\times C_2$

Intermediate fields

\(\Q(\sqrt{137}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{274}) \), \(\Q(\sqrt{2}, \sqrt{137})\), 4.4.38438912.2, \(\Q(\zeta_{16})^+\), 8.8.1477549955743744.2, 8.0.14198853181109417541632.58, 8.0.14198853181109417541632.38

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.8.31.2$x^{8} + 24 x^{6} + 4 x^{4} + 16 x^{2} + 34$$8$$1$$31$$C_8$$[3, 4, 5]$
2.8.31.2$x^{8} + 24 x^{6} + 4 x^{4} + 16 x^{2} + 34$$8$$1$$31$$C_8$$[3, 4, 5]$
$137$137.8.6.2$x^{8} + 1507 x^{4} + 675684$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
137.8.6.2$x^{8} + 1507 x^{4} + 675684$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$