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Properties

Label 16.0.17382115550...0625.2
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{8}\cdot 7^{14}$
Root discriminant $21.26$
Ramified primes $3, 5, 7$
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![81, 108, 177, -21, -35, 7, 119, 192, -38, 47, 112, -28, -14, 21, 2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 2*x^14 + 21*x^13 - 14*x^12 - 28*x^11 + 112*x^10 + 47*x^9 - 38*x^8 + 192*x^7 + 119*x^6 + 7*x^5 - 35*x^4 - 21*x^3 + 177*x^2 + 108*x + 81)
 
gp: K = bnfinit(x^16 - 2*x^15 + 2*x^14 + 21*x^13 - 14*x^12 - 28*x^11 + 112*x^10 + 47*x^9 - 38*x^8 + 192*x^7 + 119*x^6 + 7*x^5 - 35*x^4 - 21*x^3 + 177*x^2 + 108*x + 81, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 2 x^{14} + 21 x^{13} - 14 x^{12} - 28 x^{11} + 112 x^{10} + 47 x^{9} - 38 x^{8} + 192 x^{7} + 119 x^{6} + 7 x^{5} - 35 x^{4} - 21 x^{3} + 177 x^{2} + 108 x + 81 \)

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:  \(1738211555063394140625=3^{8}\cdot 5^{8}\cdot 7^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.26$
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:  $3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{9} - \frac{1}{3} a^{5} - \frac{1}{9} a^{3} + \frac{1}{3} a$, $\frac{1}{9} a^{10} + \frac{2}{9} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{27} a^{11} + \frac{1}{27} a^{10} - \frac{1}{9} a^{6} - \frac{7}{27} a^{5} - \frac{1}{27} a^{4} - \frac{1}{9} a^{3} - \frac{2}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{27} a^{12} - \frac{1}{27} a^{10} - \frac{1}{9} a^{7} - \frac{4}{27} a^{6} + \frac{2}{9} a^{5} - \frac{2}{27} a^{4} - \frac{1}{9} a^{3} - \frac{1}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{27} a^{13} + \frac{1}{27} a^{10} - \frac{1}{9} a^{8} - \frac{4}{27} a^{7} + \frac{1}{9} a^{6} - \frac{1}{3} a^{5} - \frac{4}{27} a^{4} - \frac{2}{9} a^{3} + \frac{1}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{1215} a^{14} + \frac{11}{1215} a^{13} - \frac{2}{135} a^{12} + \frac{19}{1215} a^{11} + \frac{62}{1215} a^{10} - \frac{8}{405} a^{9} - \frac{46}{1215} a^{8} + \frac{121}{1215} a^{7} + \frac{41}{405} a^{6} - \frac{517}{1215} a^{5} - \frac{56}{1215} a^{4} - \frac{37}{81} a^{3} + \frac{41}{405} a^{2} - \frac{11}{45} a + \frac{7}{15}$, $\frac{1}{11906852985} a^{15} + \frac{233230}{2381370597} a^{14} - \frac{1250461}{1322983665} a^{13} - \frac{187606133}{11906852985} a^{12} - \frac{44872552}{11906852985} a^{11} + \frac{160610953}{3968950995} a^{10} - \frac{586492642}{11906852985} a^{9} - \frac{1935293873}{11906852985} a^{8} - \frac{134958436}{3968950995} a^{7} + \frac{14798575}{2381370597} a^{6} + \frac{650951791}{11906852985} a^{5} - \frac{1635925463}{3968950995} a^{4} + \frac{477187016}{3968950995} a^{3} - \frac{39468395}{88198911} a^{2} - \frac{3576237}{48999395} a - \frac{511659}{48999395}$

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:  \( -1 \) (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:  \( 63368.9368668 \) (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

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{-15}, \sqrt{21})\), 4.2.15435.1 x2, 4.0.25725.1 x2, 8.0.5955980625.1, 8.2.8338372875.2 x4, 8.0.13897288125.2 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 R R ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7Data not computed