↑

Properties

Label 16.0.94438653106...6913.3
Degree $16$
Signature $[0, 8]$
Discriminant $43^{8}\cdot 97^{7}$
Root discriminant $48.52$
Ramified primes $43, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4.D_4:C_4$ (as 16T289)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![114563, 275623, 228396, 42851, -28551, 483, 12621, 1226, -2060, -298, -203, 90, 72, -32, 12, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 12*x^14 - 32*x^13 + 72*x^12 + 90*x^11 - 203*x^10 - 298*x^9 - 2060*x^8 + 1226*x^7 + 12621*x^6 + 483*x^5 - 28551*x^4 + 42851*x^3 + 228396*x^2 + 275623*x + 114563)
 
gp: K = bnfinit(x^16 - 5*x^15 + 12*x^14 - 32*x^13 + 72*x^12 + 90*x^11 - 203*x^10 - 298*x^9 - 2060*x^8 + 1226*x^7 + 12621*x^6 + 483*x^5 - 28551*x^4 + 42851*x^3 + 228396*x^2 + 275623*x + 114563, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 12 x^{14} - 32 x^{13} + 72 x^{12} + 90 x^{11} - 203 x^{10} - 298 x^{9} - 2060 x^{8} + 1226 x^{7} + 12621 x^{6} + 483 x^{5} - 28551 x^{4} + 42851 x^{3} + 228396 x^{2} + 275623 x + 114563 \)

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:  \(944386531066764936008646913=43^{8}\cdot 97^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.52$
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:  $43, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{17} a^{13} + \frac{6}{17} a^{12} + \frac{1}{17} a^{11} + \frac{6}{17} a^{10} + \frac{3}{17} a^{9} - \frac{3}{17} a^{8} + \frac{2}{17} a^{7} - \frac{6}{17} a^{6} - \frac{7}{17} a^{5} + \frac{5}{17} a^{4} - \frac{4}{17} a^{3} - \frac{3}{17} a^{2} - \frac{8}{17} a$, $\frac{1}{17} a^{14} - \frac{1}{17} a^{12} + \frac{1}{17} a^{10} - \frac{4}{17} a^{9} + \frac{3}{17} a^{8} - \frac{1}{17} a^{7} - \frac{5}{17} a^{6} - \frac{4}{17} a^{5} + \frac{4}{17} a^{3} - \frac{7}{17} a^{2} - \frac{3}{17} a$, $\frac{1}{346068351841230873382240481139575479} a^{15} - \frac{7794215512863052321712521462615844}{346068351841230873382240481139575479} a^{14} - \frac{3215657481613786547583383445469323}{346068351841230873382240481139575479} a^{13} - \frac{3079676285362890520359440899177099}{15046450080053516234010455701720673} a^{12} - \frac{4465998830353389273232331022704517}{31460759258293715762021861921779589} a^{11} - \frac{20659750428006881265220493457093179}{346068351841230873382240481139575479} a^{10} - \frac{121818898373362870653693772714002112}{346068351841230873382240481139575479} a^{9} + \frac{20069465012377546103909061189042604}{346068351841230873382240481139575479} a^{8} + \frac{856712413552259568784418111304188}{346068351841230873382240481139575479} a^{7} - \frac{77025003844166129967696539700700765}{346068351841230873382240481139575479} a^{6} - \frac{48974087931636630973749912833396657}{346068351841230873382240481139575479} a^{5} + \frac{150646903254907034710411869921141953}{346068351841230873382240481139575479} a^{4} + \frac{39383085118580958581133145216054359}{346068351841230873382240481139575479} a^{3} - \frac{100638879546075135748075852445669063}{346068351841230873382240481139575479} a^{2} + \frac{39609702740793792500557240122198}{96317381531096819755702889267903} a + \frac{374379990317207705349220047827612}{885085298826677425530026805983569}$

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:  \( 9377944.14506 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T289):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 44 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-43}) \), 4.0.179353.1, 8.0.3120247365073.1

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

Sibling fields

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$43$43.8.4.1$x^{8} + 73960 x^{4} - 79507 x^{2} + 1367520400$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
43.8.4.1$x^{8} + 73960 x^{4} - 79507 x^{2} + 1367520400$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$97$$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
97.8.7.1$x^{8} - 97$$8$$1$$7$$C_8$$[\ ]_{8}$