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Properties

Label 16.0.24219616215...6961.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 157^{8}$
Root discriminant $21.70$
Ramified primes $3, 157$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $D_{8}$ (as 16T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![729, -3159, 5616, -6354, 6762, -6093, 4132, -2843, 1884, -934, 518, -261, 107, -37, 15, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 15*x^14 - 37*x^13 + 107*x^12 - 261*x^11 + 518*x^10 - 934*x^9 + 1884*x^8 - 2843*x^7 + 4132*x^6 - 6093*x^5 + 6762*x^4 - 6354*x^3 + 5616*x^2 - 3159*x + 729)
 
gp: K = bnfinit(x^16 - 2*x^15 + 15*x^14 - 37*x^13 + 107*x^12 - 261*x^11 + 518*x^10 - 934*x^9 + 1884*x^8 - 2843*x^7 + 4132*x^6 - 6093*x^5 + 6762*x^4 - 6354*x^3 + 5616*x^2 - 3159*x + 729, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 15 x^{14} - 37 x^{13} + 107 x^{12} - 261 x^{11} + 518 x^{10} - 934 x^{9} + 1884 x^{8} - 2843 x^{7} + 4132 x^{6} - 6093 x^{5} + 6762 x^{4} - 6354 x^{3} + 5616 x^{2} - 3159 x + 729 \)

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:  \(2421961621595988176961=3^{8}\cdot 157^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.70$
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:  $3, 157$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{9} - \frac{1}{3} a^{5} - \frac{4}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{9} a^{10} - \frac{1}{9} a^{4}$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{5}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{6}$, $\frac{1}{27} a^{13} - \frac{1}{27} a^{12} - \frac{1}{27} a^{11} + \frac{1}{27} a^{10} + \frac{1}{9} a^{8} - \frac{1}{27} a^{7} + \frac{1}{27} a^{6} + \frac{1}{27} a^{5} - \frac{10}{27} a^{4} + \frac{1}{3} a^{3} + \frac{2}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{351} a^{14} - \frac{4}{351} a^{13} + \frac{8}{351} a^{12} + \frac{7}{351} a^{11} + \frac{4}{117} a^{10} + \frac{2}{27} a^{8} - \frac{50}{351} a^{7} - \frac{44}{351} a^{6} + \frac{101}{351} a^{5} - \frac{4}{117} a^{4} + \frac{11}{39} a^{3} + \frac{4}{39} a^{2} - \frac{8}{39} a - \frac{6}{13}$, $\frac{1}{17943112340995023} a^{15} + \frac{17312325028204}{17943112340995023} a^{14} - \frac{4412473152461}{460079803615257} a^{13} - \frac{678908150958547}{17943112340995023} a^{12} - \frac{960803781991}{197177058692253} a^{11} - \frac{257071312244687}{5981037446998341} a^{10} - \frac{34824158365798}{1380239410845771} a^{9} - \frac{997984452368551}{17943112340995023} a^{8} + \frac{257681107614311}{5981037446998341} a^{7} + \frac{90916320824719}{1380239410845771} a^{6} - \frac{1704739930573178}{17943112340995023} a^{5} + \frac{1219410520904315}{5981037446998341} a^{4} - \frac{65250255350606}{460079803615257} a^{3} - \frac{20647074966983}{153359934538419} a^{2} - \frac{479402638045}{13030582673199} a + \frac{6920295832518}{73839968481461}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( -\frac{2610536285}{431459647029} a^{15} + \frac{3947185810}{431459647029} a^{14} - \frac{12397139234}{143819882343} a^{13} + \frac{79193318672}{431459647029} a^{12} - \frac{240431173762}{431459647029} a^{11} + \frac{190849287130}{143819882343} a^{10} - \frac{1079406520147}{431459647029} a^{9} + \frac{1945497824936}{431459647029} a^{8} - \frac{1342710093058}{143819882343} a^{7} + \frac{5544483366049}{431459647029} a^{6} - \frac{8185049950280}{431459647029} a^{5} + \frac{4112805605471}{143819882343} a^{4} - \frac{3952710667364}{143819882343} a^{3} + \frac{1239716946181}{47939960781} a^{2} - \frac{2381491259}{104444359} a + \frac{15298246709}{1775554103} \) (order $6$)
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:  \( 219939.617486 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_8$ (as 16T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $D_{8}$
Character table for $D_{8}$

Intermediate fields

\(\Q(\sqrt{-471}) \), \(\Q(\sqrt{157}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{157})\), 4.2.73947.1 x2, 4.0.1413.1 x2, 8.0.49213429281.1, 8.2.16404476427.1 x4, 8.0.313461333.1 x4

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

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$157$157.2.1.1$x^{2} - 157$$2$$1$$1$$C_2$$[\ ]_{2}$
157.2.1.1$x^{2} - 157$$2$$1$$1$$C_2$$[\ ]_{2}$
157.2.1.1$x^{2} - 157$$2$$1$$1$$C_2$$[\ ]_{2}$
157.2.1.1$x^{2} - 157$$2$$1$$1$$C_2$$[\ ]_{2}$
157.2.1.1$x^{2} - 157$$2$$1$$1$$C_2$$[\ ]_{2}$
157.2.1.1$x^{2} - 157$$2$$1$$1$$C_2$$[\ ]_{2}$
157.2.1.1$x^{2} - 157$$2$$1$$1$$C_2$$[\ ]_{2}$
157.2.1.1$x^{2} - 157$$2$$1$$1$$C_2$$[\ ]_{2}$