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Properties

Label 16.0.40092926956...969.18
Degree $16$
Signature $[0, 8]$
Discriminant $17^{14}\cdot 47^{8}$
Root discriminant $81.79$
Ramified primes $17, 47$
Class number $1280$ (GRH)
Class group $[4, 4, 80]$ (GRH)
Galois group $C_8: C_2$ (as 16T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![47343743, -2357794, 37249515, -16343158, 14771709, -6549280, 4667230, -1352592, 508573, -143816, 35104, -8130, 1684, -268, 50, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 50*x^14 - 268*x^13 + 1684*x^12 - 8130*x^11 + 35104*x^10 - 143816*x^9 + 508573*x^8 - 1352592*x^7 + 4667230*x^6 - 6549280*x^5 + 14771709*x^4 - 16343158*x^3 + 37249515*x^2 - 2357794*x + 47343743)
 
gp: K = bnfinit(x^16 - 4*x^15 + 50*x^14 - 268*x^13 + 1684*x^12 - 8130*x^11 + 35104*x^10 - 143816*x^9 + 508573*x^8 - 1352592*x^7 + 4667230*x^6 - 6549280*x^5 + 14771709*x^4 - 16343158*x^3 + 37249515*x^2 - 2357794*x + 47343743, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 50 x^{14} - 268 x^{13} + 1684 x^{12} - 8130 x^{11} + 35104 x^{10} - 143816 x^{9} + 508573 x^{8} - 1352592 x^{7} + 4667230 x^{6} - 6549280 x^{5} + 14771709 x^{4} - 16343158 x^{3} + 37249515 x^{2} - 2357794 x + 47343743 \)

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:  \(4009292695690170390860412175969=17^{14}\cdot 47^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81.79$
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:  $17, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{34} a^{8} - \frac{1}{17} a^{7} + \frac{3}{17} a^{6} + \frac{7}{17} a^{5} - \frac{15}{34} a^{4} - \frac{7}{17} a^{3} - \frac{11}{34} a^{2} - \frac{15}{34} a + \frac{1}{34}$, $\frac{1}{34} a^{9} + \frac{1}{17} a^{7} - \frac{4}{17} a^{6} + \frac{13}{34} a^{5} - \frac{5}{17} a^{4} - \frac{5}{34} a^{3} - \frac{3}{34} a^{2} + \frac{5}{34} a + \frac{1}{17}$, $\frac{1}{34} a^{10} - \frac{2}{17} a^{7} + \frac{1}{34} a^{6} - \frac{2}{17} a^{5} - \frac{9}{34} a^{4} - \frac{9}{34} a^{3} - \frac{7}{34} a^{2} - \frac{1}{17} a - \frac{1}{17}$, $\frac{1}{34} a^{11} - \frac{7}{34} a^{7} - \frac{7}{17} a^{6} + \frac{13}{34} a^{5} - \frac{1}{34} a^{4} + \frac{5}{34} a^{3} - \frac{6}{17} a^{2} + \frac{3}{17} a + \frac{2}{17}$, $\frac{1}{2278} a^{12} + \frac{3}{2278} a^{11} - \frac{19}{2278} a^{10} + \frac{8}{1139} a^{9} + \frac{15}{1139} a^{8} + \frac{305}{2278} a^{7} + \frac{193}{1139} a^{6} + \frac{250}{1139} a^{5} - \frac{50}{1139} a^{4} - \frac{229}{1139} a^{3} - \frac{142}{1139} a^{2} - \frac{279}{2278} a - \frac{45}{134}$, $\frac{1}{2278} a^{13} - \frac{14}{1139} a^{11} + \frac{3}{1139} a^{10} - \frac{9}{1139} a^{9} + \frac{7}{1139} a^{8} + \frac{141}{2278} a^{7} + \frac{347}{2278} a^{6} + \frac{205}{1139} a^{5} - \frac{548}{1139} a^{4} - \frac{49}{2278} a^{3} + \frac{975}{2278} a^{2} + \frac{943}{2278} a - \frac{25}{1139}$, $\frac{1}{2278} a^{14} + \frac{23}{2278} a^{11} - \frac{7}{1139} a^{10} - \frac{7}{2278} a^{9} - \frac{12}{1139} a^{8} - \frac{414}{1139} a^{7} - \frac{488}{1139} a^{6} + \frac{556}{1139} a^{5} + \frac{769}{2278} a^{4} - \frac{593}{2278} a^{3} + \frac{227}{2278} a^{2} + \frac{558}{1139} a + \frac{355}{2278}$, $\frac{1}{152650193675367820480024131650735083327334} a^{15} - \frac{14744353719158186798291758481354385354}{76325096837683910240012065825367541663667} a^{14} + \frac{14562524341868136460892966904688683657}{76325096837683910240012065825367541663667} a^{13} + \frac{2140913943682241663039405730574055396}{76325096837683910240012065825367541663667} a^{12} + \frac{800596448057995191701558572593411336505}{76325096837683910240012065825367541663667} a^{11} - \frac{177492472520825714857514265823455153218}{76325096837683910240012065825367541663667} a^{10} + \frac{190468373454988051550707537107633860627}{152650193675367820480024131650735083327334} a^{9} + \frac{821099741036371262846855312812965556064}{76325096837683910240012065825367541663667} a^{8} - \frac{14840928800880598109238305751307676442856}{76325096837683910240012065825367541663667} a^{7} - \frac{12082406329107628614046231239857482554038}{76325096837683910240012065825367541663667} a^{6} + \frac{74044802069276707426677266269229261368917}{152650193675367820480024131650735083327334} a^{5} - \frac{14679778472900310497490117550617088687984}{76325096837683910240012065825367541663667} a^{4} - \frac{60138060323098979457238041339597563932219}{152650193675367820480024131650735083327334} a^{3} - \frac{40182479542672553895773601412270197855789}{152650193675367820480024131650735083327334} a^{2} + \frac{10381153867233970865855931649469872731193}{152650193675367820480024131650735083327334} a + \frac{18752141503014973193472354018816517951331}{152650193675367820480024131650735083327334}$

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

Class group and class number

$C_{4}\times C_{4}\times C_{80}$, which has order $1280$ (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:  \( 282136.204408 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}$ (as 16T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_8: C_2$
Character table for $C_8: C_2$

Intermediate fields

\(\Q(\sqrt{-799}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-47}) \), \(\Q(\sqrt{17}, \sqrt{-47})\), 4.4.4913.1, 4.0.10852817.2, 8.0.117783636835489.4, 8.4.906438128657.1 x2

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

Sibling fields

Degree 8 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$17$17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
$47$47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$