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Properties

Label 16.0.27340995738...2241.2
Degree $16$
Signature $[0, 8]$
Discriminant $41^{12}\cdot 59^{4}$
Root discriminant $44.91$
Ramified primes $41, 59$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\wr C_4$ (as 16T158)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![256, -2304, 5120, 13280, 34784, 36664, 26536, 4824, 587, -924, 172, 163, 42, -37, 2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 2*x^14 - 37*x^13 + 42*x^12 + 163*x^11 + 172*x^10 - 924*x^9 + 587*x^8 + 4824*x^7 + 26536*x^6 + 36664*x^5 + 34784*x^4 + 13280*x^3 + 5120*x^2 - 2304*x + 256)
 
gp: K = bnfinit(x^16 - 2*x^15 + 2*x^14 - 37*x^13 + 42*x^12 + 163*x^11 + 172*x^10 - 924*x^9 + 587*x^8 + 4824*x^7 + 26536*x^6 + 36664*x^5 + 34784*x^4 + 13280*x^3 + 5120*x^2 - 2304*x + 256, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 2 x^{14} - 37 x^{13} + 42 x^{12} + 163 x^{11} + 172 x^{10} - 924 x^{9} + 587 x^{8} + 4824 x^{7} + 26536 x^{6} + 36664 x^{5} + 34784 x^{4} + 13280 x^{3} + 5120 x^{2} - 2304 x + 256 \)

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:  \(273409957389535508936652241=41^{12}\cdot 59^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.91$
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:  $41, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{4} a^{10} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{11} - \frac{1}{4} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{3}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{16} a^{12} - \frac{1}{8} a^{10} - \frac{1}{16} a^{9} + \frac{7}{16} a^{7} + \frac{1}{8} a^{6} - \frac{1}{4} a^{5} + \frac{3}{16} a^{4} - \frac{1}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{992} a^{13} + \frac{11}{496} a^{12} - \frac{21}{496} a^{11} - \frac{29}{992} a^{10} + \frac{85}{496} a^{9} + \frac{191}{992} a^{8} + \frac{47}{248} a^{7} + \frac{4}{31} a^{6} - \frac{165}{992} a^{5} + \frac{15}{31} a^{4} + \frac{85}{248} a^{3} + \frac{39}{124} a^{2} + \frac{13}{62} a - \frac{6}{31}$, $\frac{1}{819392} a^{14} - \frac{15}{204848} a^{13} + \frac{5277}{409696} a^{12} + \frac{40615}{819392} a^{11} - \frac{23791}{204848} a^{10} + \frac{54203}{819392} a^{9} + \frac{82783}{409696} a^{8} - \frac{2753}{14632} a^{7} - \frac{39925}{819392} a^{6} - \frac{155931}{409696} a^{5} + \frac{57577}{204848} a^{4} + \frac{849}{3658} a^{3} - \frac{23627}{51212} a^{2} - \frac{10769}{25606} a - \frac{2389}{12803}$, $\frac{1}{1002783673831710202928384} a^{15} - \frac{20698583377732511}{35813702636846792961728} a^{14} + \frac{64955782676920268957}{501391836915855101464192} a^{13} - \frac{128710790072151942111}{143254810547387171846912} a^{12} + \frac{973381139002653319767}{250695918457927550732096} a^{11} - \frac{73401275939503686703621}{1002783673831710202928384} a^{10} + \frac{751896835371066213749}{16173930223092100047232} a^{9} + \frac{10512607020040434566097}{125347959228963775366048} a^{8} - \frac{316440936760747028273957}{1002783673831710202928384} a^{7} - \frac{28691161561031770528233}{71627405273693585923456} a^{6} - \frac{120041330034114396634611}{250695918457927550732096} a^{5} + \frac{18412399272092819953843}{62673979614481887683024} a^{4} + \frac{855630479371012897544}{3917123725905117980189} a^{3} + \frac{844540633726716063901}{4476712829605849120216} a^{2} + \frac{2879523553914499719569}{15668494903620471920756} a - \frac{842264651240244335037}{7834247451810235960378}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 12530682.7114 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T158):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $C_2\wr C_4$
Character table for $C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 4.2.99179.1, 4.2.4066339.1, 8.2.280256150219.1 x2, 8.0.403295435681.1 x2, 8.4.16535112862921.1

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
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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ R

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
$41$41.8.6.1$x^{8} - 9881 x^{4} + 34857216$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
41.8.6.1$x^{8} - 9881 x^{4} + 34857216$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$59$$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$