↑

Properties

Label 16.4.18149808491...0641.7
Degree $16$
Signature $[4, 6]$
Discriminant $17^{14}\cdot 47^{6}$
Root discriminant $50.55$
Ramified primes $17, 47$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group $D_4:C_4$ (as 16T26)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![256, -3008, 11280, -9104, -13540, 20672, -6237, 1354, -2178, 328, 909, -646, 229, -84, 32, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 32*x^14 - 84*x^13 + 229*x^12 - 646*x^11 + 909*x^10 + 328*x^9 - 2178*x^8 + 1354*x^7 - 6237*x^6 + 20672*x^5 - 13540*x^4 - 9104*x^3 + 11280*x^2 - 3008*x + 256)
 
gp: K = bnfinit(x^16 - 8*x^15 + 32*x^14 - 84*x^13 + 229*x^12 - 646*x^11 + 909*x^10 + 328*x^9 - 2178*x^8 + 1354*x^7 - 6237*x^6 + 20672*x^5 - 13540*x^4 - 9104*x^3 + 11280*x^2 - 3008*x + 256, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 32 x^{14} - 84 x^{13} + 229 x^{12} - 646 x^{11} + 909 x^{10} + 328 x^{9} - 2178 x^{8} + 1354 x^{7} - 6237 x^{6} + 20672 x^{5} - 13540 x^{4} - 9104 x^{3} + 11280 x^{2} - 3008 x + 256 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1814980849112797822933640641=17^{14}\cdot 47^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.55$
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 not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{1}{8} a^{5} - \frac{3}{16} a^{4} - \frac{3}{16} a^{3} - \frac{3}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{9} - \frac{1}{8} a^{8} + \frac{1}{8} a^{7} + \frac{1}{8} a^{6} + \frac{3}{16} a^{5} + \frac{1}{8} a^{4} - \frac{1}{16} a^{3} + \frac{3}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{1888} a^{12} - \frac{3}{944} a^{11} - \frac{19}{1888} a^{10} + \frac{75}{944} a^{9} - \frac{117}{944} a^{8} - \frac{15}{944} a^{7} + \frac{271}{1888} a^{6} - \frac{3}{472} a^{5} - \frac{415}{1888} a^{4} + \frac{241}{944} a^{3} + \frac{181}{472} a^{2} - \frac{57}{118} a + \frac{5}{59}$, $\frac{1}{103840} a^{13} + \frac{21}{103840} a^{12} + \frac{1589}{103840} a^{11} - \frac{33}{9440} a^{10} + \frac{3383}{51920} a^{9} - \frac{309}{3245} a^{8} + \frac{4889}{103840} a^{7} - \frac{14643}{103840} a^{6} - \frac{9589}{103840} a^{5} - \frac{16623}{103840} a^{4} + \frac{34}{295} a^{3} + \frac{3539}{12980} a^{2} + \frac{3727}{12980} a + \frac{1256}{3245}$, $\frac{1}{148491200} a^{14} - \frac{7}{148491200} a^{13} - \frac{1029}{13499200} a^{12} + \frac{13601}{29698240} a^{11} + \frac{215907}{14849120} a^{10} - \frac{713681}{9280700} a^{9} - \frac{15915967}{148491200} a^{8} - \frac{239471}{2284480} a^{7} - \frac{4580309}{29698240} a^{6} + \frac{230969}{148491200} a^{5} + \frac{836689}{18561400} a^{4} + \frac{33959}{314600} a^{3} - \frac{1781493}{3712280} a^{2} - \frac{571849}{2320175} a + \frac{665516}{2320175}$, $\frac{1}{8167016000} a^{15} + \frac{1}{408350800} a^{14} - \frac{2877}{2041754000} a^{13} - \frac{29701}{1020877000} a^{12} + \frac{41634121}{1633403200} a^{11} - \frac{2509623}{371228000} a^{10} - \frac{55085859}{8167016000} a^{9} - \frac{185569781}{2041754000} a^{8} - \frac{11462059}{163340320} a^{7} + \frac{22526057}{371228000} a^{6} - \frac{962539}{5939648} a^{5} + \frac{394894443}{2041754000} a^{4} + \frac{36043613}{157058000} a^{3} + \frac{101723207}{255219250} a^{2} - \frac{56777453}{510438500} a + \frac{52771557}{127609625}$

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

Class group and class number

$C_{6}$, which has order $6$ (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:  $9$
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:  \( 32034366.963 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4:C_4$ (as 16T26):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $D_4:C_4$
Character table for $D_4:C_4$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.2.230911.1, 4.4.4913.1, 4.2.13583.1, 8.2.42602592046879.2, 8.2.42602592046879.1, 8.4.53319889921.1

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

Sibling fields

Galois closure: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$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$$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$