↑

Properties

Label 16.0.24857298334...3296.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 41^{14}$
Root discriminant $51.55$
Ramified primes $2, 41$
Class number $8$ (GRH)
Class group $[8]$ (GRH)
Galois group $C_8\times C_2$ (as 16T5)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3481, 0, 7546, 0, 11641, 0, -14529, 0, 4458, 0, -555, 0, 75, 0, -5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^14 + 75*x^12 - 555*x^10 + 4458*x^8 - 14529*x^6 + 11641*x^4 + 7546*x^2 + 3481)
 
gp: K = bnfinit(x^16 - 5*x^14 + 75*x^12 - 555*x^10 + 4458*x^8 - 14529*x^6 + 11641*x^4 + 7546*x^2 + 3481, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{14} + 75 x^{12} - 555 x^{10} + 4458 x^{8} - 14529 x^{6} + 11641 x^{4} + 7546 x^{2} + 3481 \)

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:  \(2485729833445986061658423296=2^{16}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.55$
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:  $2, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(164=2^{2}\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{164}(1,·)$, $\chi_{164}(3,·)$, $\chi_{164}(73,·)$, $\chi_{164}(55,·)$, $\chi_{164}(79,·)$, $\chi_{164}(81,·)$, $\chi_{164}(83,·)$, $\chi_{164}(85,·)$, $\chi_{164}(137,·)$, $\chi_{164}(27,·)$, $\chi_{164}(161,·)$, $\chi_{164}(163,·)$, $\chi_{164}(109,·)$, $\chi_{164}(91,·)$, $\chi_{164}(155,·)$, $\chi_{164}(9,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{37} a^{8} - \frac{15}{37} a^{6} - \frac{9}{37} a^{4} + \frac{2}{37} a^{2} - \frac{11}{37}$, $\frac{1}{37} a^{9} - \frac{15}{37} a^{7} - \frac{9}{37} a^{5} + \frac{2}{37} a^{3} - \frac{11}{37} a$, $\frac{1}{37} a^{10} - \frac{12}{37} a^{6} + \frac{15}{37} a^{4} - \frac{18}{37} a^{2} - \frac{17}{37}$, $\frac{1}{37} a^{11} - \frac{12}{37} a^{7} + \frac{15}{37} a^{5} - \frac{18}{37} a^{3} - \frac{17}{37} a$, $\frac{1}{148} a^{12} - \frac{1}{74} a^{8} - \frac{61}{148} a^{6} + \frac{3}{148} a^{4} + \frac{3}{148} a^{2} + \frac{1}{148}$, $\frac{1}{148} a^{13} - \frac{1}{74} a^{9} - \frac{61}{148} a^{7} + \frac{3}{148} a^{5} + \frac{3}{148} a^{3} + \frac{1}{148} a$, $\frac{1}{70947335276} a^{14} - \frac{26106966}{17736833819} a^{12} - \frac{13126457}{35473667638} a^{10} + \frac{176768667}{70947335276} a^{8} + \frac{17189079275}{70947335276} a^{6} + \frac{15007561027}{70947335276} a^{4} - \frac{23306913371}{70947335276} a^{2} + \frac{4432778936}{17736833819}$, $\frac{1}{4185892781284} a^{15} + \frac{2370762469}{1046473195321} a^{13} - \frac{1930622005}{2092946390642} a^{11} - \frac{22833177909}{4185892781284} a^{9} - \frac{464102303273}{4185892781284} a^{7} - \frac{1043449981469}{4185892781284} a^{5} + \frac{1982393429837}{4185892781284} a^{3} - \frac{143693752147}{1046473195321} a$

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

Class group and class number

$C_{8}$, which has order $8$ (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:  \( \frac{16057}{90705833} a^{15} - \frac{131483}{181411666} a^{13} + \frac{1162326}{90705833} a^{11} - \frac{7897189}{90705833} a^{9} + \frac{131174055}{181411666} a^{7} - \frac{359188081}{181411666} a^{5} + \frac{165678755}{181411666} a^{3} + \frac{275836633}{181411666} a \) (order $4$)
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:  \( 1737529.057604532 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{41}) \), \(\Q(\sqrt{-41}) \), \(\Q(i, \sqrt{41})\), 4.4.68921.1, 4.0.1102736.1, 8.0.1216026685696.1, 8.8.49857094113536.1, 8.0.194754273881.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{16}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$41$41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$