↑

Properties

Label 16.0.17983880206...3984.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{30}\cdot 7^{4}\cdot 17^{8}$
Root discriminant $24.60$
Ramified primes $2, 7, 17$
Class number $6$
Class group $[6]$
Galois group $C_2^2.C_2^5.C_2$ (as 16T511)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![676, -728, 1712, -1060, 1187, 464, -768, 1196, -262, -348, 740, -600, 346, -136, 40, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 40*x^14 - 136*x^13 + 346*x^12 - 600*x^11 + 740*x^10 - 348*x^9 - 262*x^8 + 1196*x^7 - 768*x^6 + 464*x^5 + 1187*x^4 - 1060*x^3 + 1712*x^2 - 728*x + 676)
 
gp: K = bnfinit(x^16 - 8*x^15 + 40*x^14 - 136*x^13 + 346*x^12 - 600*x^11 + 740*x^10 - 348*x^9 - 262*x^8 + 1196*x^7 - 768*x^6 + 464*x^5 + 1187*x^4 - 1060*x^3 + 1712*x^2 - 728*x + 676, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 40 x^{14} - 136 x^{13} + 346 x^{12} - 600 x^{11} + 740 x^{10} - 348 x^{9} - 262 x^{8} + 1196 x^{7} - 768 x^{6} + 464 x^{5} + 1187 x^{4} - 1060 x^{3} + 1712 x^{2} - 728 x + 676 \)

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:  \(17983880206872670633984=2^{30}\cdot 7^{4}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.60$
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:  $2, 7, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{68} a^{14} + \frac{1}{17} a^{13} - \frac{4}{17} a^{12} + \frac{1}{17} a^{11} - \frac{4}{17} a^{10} + \frac{4}{17} a^{9} - \frac{11}{34} a^{8} - \frac{8}{17} a^{7} - \frac{6}{17} a^{6} + \frac{5}{17} a^{5} + \frac{15}{34} a^{4} - \frac{8}{17} a^{3} + \frac{29}{68} a^{2} + \frac{8}{17} a - \frac{1}{34}$, $\frac{1}{13691478642827302674916} a^{15} - \frac{42503742656841123441}{6845739321413651337458} a^{14} - \frac{1269660511483876413141}{6845739321413651337458} a^{13} - \frac{479251038002709171649}{3422869660706825668729} a^{12} - \frac{3400936260739695735703}{6845739321413651337458} a^{11} - \frac{160799519013102705678}{3422869660706825668729} a^{10} + \frac{786194519704941405516}{3422869660706825668729} a^{9} - \frac{241537447125097415795}{3422869660706825668729} a^{8} + \frac{1084706155155753286909}{6845739321413651337458} a^{7} - \frac{42130707762393568637}{263297666208217359133} a^{6} + \frac{1555658951694379170772}{3422869660706825668729} a^{5} - \frac{4646525576306778694}{201345274159225039337} a^{4} + \frac{4419054846382859807271}{13691478642827302674916} a^{3} - \frac{1581221058761825381963}{6845739321413651337458} a^{2} - \frac{2555834331178047352617}{6845739321413651337458} a - \frac{16213944628269071820}{263297666208217359133}$

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

Class group and class number

$C_{6}$, which has order $6$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 8309.39203867 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.C_2^5.C_2$ (as 16T511):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 46 conjugacy class representatives for $C_2^2.C_2^5.C_2$
Character table for $C_2^2.C_2^5.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.9248.1, 4.2.1156.1, 4.2.2312.1, 8.4.134103990272.4, 8.0.134103990272.4, 8.4.85525504.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$2$2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.8.18.1$x^{8} + 14 x^{6} + 10 x^{4} + 12 x^{2} + 16 x + 4$$4$$2$$18$$D_4\times C_2$$[2, 3, 7/2]^{2}$
$7$7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$