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Properties

Label 16.0.13872985106...9121.7
Degree $16$
Signature $[0, 8]$
Discriminant $17^{12}\cdot 47^{8}$
Root discriminant $57.40$
Ramified primes $17, 47$
Class number $16$ (GRH)
Class group $[2, 2, 4]$ (GRH)
Galois group $D_4:C_4$ (as 16T26)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![38399, 81780, 11928, -292762, 444127, -459372, 341320, -199490, 102318, -42616, 16738, -5214, 1506, -322, 66, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 66*x^14 - 322*x^13 + 1506*x^12 - 5214*x^11 + 16738*x^10 - 42616*x^9 + 102318*x^8 - 199490*x^7 + 341320*x^6 - 459372*x^5 + 444127*x^4 - 292762*x^3 + 11928*x^2 + 81780*x + 38399)
 
gp: K = bnfinit(x^16 - 8*x^15 + 66*x^14 - 322*x^13 + 1506*x^12 - 5214*x^11 + 16738*x^10 - 42616*x^9 + 102318*x^8 - 199490*x^7 + 341320*x^6 - 459372*x^5 + 444127*x^4 - 292762*x^3 + 11928*x^2 + 81780*x + 38399, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 66 x^{14} - 322 x^{13} + 1506 x^{12} - 5214 x^{11} + 16738 x^{10} - 42616 x^{9} + 102318 x^{8} - 199490 x^{7} + 341320 x^{6} - 459372 x^{5} + 444127 x^{4} - 292762 x^{3} + 11928 x^{2} + 81780 x + 38399 \)

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:  \(13872985106194361214049869121=17^{12}\cdot 47^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.40$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{16} a^{12} - \frac{1}{8} a^{11} - \frac{1}{16} a^{9} + \frac{1}{16} a^{8} - \frac{1}{16} a^{6} + \frac{3}{16} a^{4} - \frac{5}{16} a^{3} - \frac{7}{16} a^{2} + \frac{7}{16} a - \frac{1}{16}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{8} a^{8} - \frac{1}{16} a^{7} - \frac{1}{8} a^{6} - \frac{1}{16} a^{5} + \frac{1}{16} a^{4} - \frac{5}{16} a^{3} - \frac{3}{16} a^{2} - \frac{3}{16} a - \frac{3}{8}$, $\frac{1}{579658511861552} a^{14} - \frac{7}{579658511861552} a^{13} - \frac{2004705483217}{289829255930776} a^{12} + \frac{24056465798695}{579658511861552} a^{11} + \frac{10036744339021}{289829255930776} a^{10} - \frac{31055790614305}{579658511861552} a^{9} - \frac{7085472792741}{579658511861552} a^{8} - \frac{100360753218827}{579658511861552} a^{7} + \frac{11820280847279}{579658511861552} a^{6} - \frac{34304875457031}{144914627965388} a^{5} - \frac{29355794408219}{289829255930776} a^{4} - \frac{126405659396221}{289829255930776} a^{3} - \frac{62965905497229}{144914627965388} a^{2} - \frac{82320544098111}{579658511861552} a - \frac{1016128402511}{144914627965388}$, $\frac{1}{37300445579779009648} a^{15} + \frac{32167}{37300445579779009648} a^{14} + \frac{33437045697417455}{18650222789889504824} a^{13} - \frac{281841193603710737}{18650222789889504824} a^{12} - \frac{368904526076509669}{18650222789889504824} a^{11} - \frac{245339135736444503}{37300445579779009648} a^{10} + \frac{286285471490848943}{9325111394944752412} a^{9} - \frac{520356683679886159}{18650222789889504824} a^{8} - \frac{1588489007899517765}{37300445579779009648} a^{7} - \frac{1352611938760609241}{37300445579779009648} a^{6} - \frac{499986535793791479}{2331277848736188103} a^{5} - \frac{978269889851448339}{37300445579779009648} a^{4} - \frac{8479643478862511469}{37300445579779009648} a^{3} + \frac{1447648312436255485}{18650222789889504824} a^{2} - \frac{4659094570061035123}{37300445579779009648} a + \frac{11638288706034474029}{37300445579779009648}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}$, which has order $16$ (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:  \( 2123124.68871 \) (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

\(\Q(\sqrt{17}) \), 4.0.10852817.2, 4.2.13583.1, 4.2.230911.1, 8.2.8671400783.1, 8.2.2506034826287.2, 8.0.117783636835489.3

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$17$17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
$47$47.2.1.2$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}$
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}$