↑

Properties

Label 16.8.21943166735...8209.1
Degree $16$
Signature $[8, 4]$
Discriminant $17^{14}\cdot 19^{4}$
Root discriminant $24.91$
Ramified primes $17, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2 : C_8$ (as 16T24)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, 16, -64, -258, 153, 669, -227, -551, 357, 348, -165, -191, 0, 38, -2, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 2*x^14 + 38*x^13 - 191*x^11 - 165*x^10 + 348*x^9 + 357*x^8 - 551*x^7 - 227*x^6 + 669*x^5 + 153*x^4 - 258*x^3 - 64*x^2 + 16*x + 16)
 
gp: K = bnfinit(x^16 - 3*x^15 - 2*x^14 + 38*x^13 - 191*x^11 - 165*x^10 + 348*x^9 + 357*x^8 - 551*x^7 - 227*x^6 + 669*x^5 + 153*x^4 - 258*x^3 - 64*x^2 + 16*x + 16, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 2 x^{14} + 38 x^{13} - 191 x^{11} - 165 x^{10} + 348 x^{9} + 357 x^{8} - 551 x^{7} - 227 x^{6} + 669 x^{5} + 153 x^{4} - 258 x^{3} - 64 x^{2} + 16 x + 16 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(21943166735047688468209=17^{14}\cdot 19^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.91$
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:  $17, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{5252} a^{14} - \frac{69}{404} a^{13} + \frac{215}{2626} a^{12} - \frac{597}{2626} a^{11} + \frac{583}{2626} a^{10} - \frac{587}{5252} a^{9} - \frac{261}{5252} a^{8} - \frac{955}{2626} a^{7} - \frac{943}{5252} a^{6} + \frac{601}{5252} a^{5} + \frac{2543}{5252} a^{4} + \frac{53}{404} a^{3} + \frac{177}{5252} a^{2} - \frac{12}{101} a + \frac{614}{1313}$, $\frac{1}{4468235035876664} a^{15} - \frac{274194540085}{4468235035876664} a^{14} - \frac{29104051423260}{558529379484583} a^{13} - \frac{112251596492475}{2234117517938332} a^{12} - \frac{7552588510707}{85927596843782} a^{11} + \frac{501264788856705}{4468235035876664} a^{10} + \frac{846009115587813}{4468235035876664} a^{9} - \frac{928816205538761}{2234117517938332} a^{8} - \frac{660482838119655}{4468235035876664} a^{7} - \frac{6210642672903}{17801733210664} a^{6} - \frac{345747396554273}{4468235035876664} a^{5} + \frac{535284388232259}{4468235035876664} a^{4} + \frac{1287693270835343}{4468235035876664} a^{3} + \frac{519275445443707}{1117058758969166} a^{2} - \frac{525334922656411}{1117058758969166} a - \frac{173893739232704}{558529379484583}$

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:  $11$
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:  \( 153986.024054 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_8$ (as 16T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2^2 : C_8$
Character table for $C_2^2 : C_8$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.2.5491.1, 4.4.4913.1, 4.2.93347.1, \(\Q(\zeta_{17})^+\), 8.4.148132260953.1, 8.4.8713662409.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R R ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
17Data not computed
$19$19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.8.4.1$x^{8} + 7220 x^{4} - 27436 x^{2} + 13032100$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$