↑

Properties

Label 16.0.27340995738...2241.1
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 16T157)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, -160, 760, -1606, 6167, -10210, 11303, -1530, 2263, -1008, 331, -84, 88, -22, 6, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 6*x^14 - 22*x^13 + 88*x^12 - 84*x^11 + 331*x^10 - 1008*x^9 + 2263*x^8 - 1530*x^7 + 11303*x^6 - 10210*x^5 + 6167*x^4 - 1606*x^3 + 760*x^2 - 160*x + 16)
 
gp: K = bnfinit(x^16 - 2*x^15 + 6*x^14 - 22*x^13 + 88*x^12 - 84*x^11 + 331*x^10 - 1008*x^9 + 2263*x^8 - 1530*x^7 + 11303*x^6 - 10210*x^5 + 6167*x^4 - 1606*x^3 + 760*x^2 - 160*x + 16, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 6 x^{14} - 22 x^{13} + 88 x^{12} - 84 x^{11} + 331 x^{10} - 1008 x^{9} + 2263 x^{8} - 1530 x^{7} + 11303 x^{6} - 10210 x^{5} + 6167 x^{4} - 1606 x^{3} + 760 x^{2} - 160 x + 16 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{6} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{12} - \frac{1}{4} a^{11} - \frac{1}{8} a^{10} + \frac{1}{8} a^{9} - \frac{1}{2} a^{6} + \frac{3}{8} a^{5} + \frac{1}{8} a^{4} - \frac{1}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{248} a^{14} - \frac{5}{124} a^{13} + \frac{17}{248} a^{12} - \frac{57}{248} a^{11} + \frac{11}{124} a^{10} - \frac{47}{248} a^{9} + \frac{13}{124} a^{8} + \frac{3}{62} a^{7} - \frac{61}{248} a^{6} + \frac{57}{124} a^{5} - \frac{4}{31} a^{4} + \frac{5}{248} a^{3} + \frac{15}{31} a^{2} - \frac{7}{62} a - \frac{13}{31}$, $\frac{1}{247850471442840146449144} a^{15} - \frac{106552464744795175731}{247850471442840146449144} a^{14} - \frac{4307645621141779257457}{123925235721420073224572} a^{13} + \frac{4292124631379138001643}{247850471442840146449144} a^{12} + \frac{1698498713007421243495}{7995176498156133756424} a^{11} - \frac{12626806033582494211323}{61962617860710036612286} a^{10} + \frac{12014921179206774355577}{61962617860710036612286} a^{9} + \frac{13463660976654117803331}{123925235721420073224572} a^{8} - \frac{8803377322772196304089}{247850471442840146449144} a^{7} + \frac{4247323420758689970599}{247850471442840146449144} a^{6} + \frac{18693549580744583026047}{247850471442840146449144} a^{5} + \frac{27551334079087524489385}{61962617860710036612286} a^{4} - \frac{7445474631610393968228}{30981308930355018306143} a^{3} - \frac{862681859431479083097}{3349330695173515492556} a^{2} - \frac{540914324668146878431}{30981308930355018306143} a - \frac{11287782804512840083674}{30981308930355018306143}$

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:  \( 4737067.86092 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T157):

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, 8.2.280256150219.1 x2, 8.4.16535112862921.2

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