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Properties

Label 16.0.53112317075...5569.1
Degree $16$
Signature $[0, 8]$
Discriminant $7^{14}\cdot 23^{8}$
Root discriminant $26.32$
Ramified primes $7, 23$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $D_{8}$ (as 16T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![14641, 0, 24684, 0, 12222, 0, 462, 0, -1113, 0, -238, 0, 7, 0, 9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 9*x^14 + 7*x^12 - 238*x^10 - 1113*x^8 + 462*x^6 + 12222*x^4 + 24684*x^2 + 14641)
 
gp: K = bnfinit(x^16 + 9*x^14 + 7*x^12 - 238*x^10 - 1113*x^8 + 462*x^6 + 12222*x^4 + 24684*x^2 + 14641, 1)
 

Normalized defining polynomial

\( x^{16} + 9 x^{14} + 7 x^{12} - 238 x^{10} - 1113 x^{8} + 462 x^{6} + 12222 x^{4} + 24684 x^{2} + 14641 \)

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:  \(53112317075112629735569=7^{14}\cdot 23^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.32$
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:  $7, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{290} a^{10} + \frac{11}{58} a^{8} - \frac{57}{145} a^{6} - \frac{1}{2} a^{5} - \frac{111}{290} a^{4} - \frac{1}{58} a^{2} - \frac{1}{2} a - \frac{91}{290}$, $\frac{1}{3190} a^{11} - \frac{105}{638} a^{9} + \frac{8}{145} a^{7} - \frac{1}{2} a^{6} + \frac{1049}{3190} a^{5} + \frac{173}{638} a^{3} - \frac{1}{2} a^{2} + \frac{1069}{3190} a$, $\frac{1}{3190} a^{12} + \frac{3}{3190} a^{10} + \frac{23}{145} a^{8} - \frac{1}{2} a^{7} + \frac{1467}{3190} a^{6} - \frac{323}{3190} a^{4} - \frac{1}{2} a^{3} - \frac{1571}{3190} a^{2} - \frac{9}{145}$, $\frac{1}{3190} a^{13} + \frac{243}{1595} a^{9} - \frac{328}{1595} a^{7} - \frac{1}{2} a^{6} + \frac{263}{638} a^{5} - \frac{1}{2} a^{4} - \frac{488}{1595} a^{3} - \frac{1}{2} a^{2} + \frac{138}{319} a$, $\frac{1}{6772370} a^{14} + \frac{1043}{6772370} a^{12} - \frac{8507}{6772370} a^{10} - \frac{167099}{3386185} a^{8} + \frac{1584252}{3386185} a^{6} - \frac{1}{2} a^{5} + \frac{289967}{615670} a^{4} + \frac{1288057}{6772370} a^{2} + \frac{4364}{27985}$, $\frac{1}{74496070} a^{15} - \frac{108}{7449607} a^{13} + \frac{8477}{74496070} a^{11} + \frac{8403452}{37248035} a^{9} + \frac{7161373}{37248035} a^{7} + \frac{1443584}{3386185} a^{5} - \frac{1}{2} a^{4} - \frac{3162117}{14899214} a^{3} - \frac{1}{2} a^{2} - \frac{72643}{307835} a$

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:  \( 24713.1006295 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_8$ (as 16T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $D_{8}$
Character table for $D_{8}$

Intermediate fields

\(\Q(\sqrt{-23}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{161}) \), \(\Q(\sqrt{-7}, \sqrt{-23})\), 4.0.7889.1 x2, 4.2.181447.1 x2, 8.0.32923013809.1, 8.0.10020047681.1 x4, 8.2.230461096663.1 x4

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

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/29.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
7Data not computed
$23$23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$