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Properties

Label 16.8.66688975910...2433.1
Degree $16$
Signature $[8, 4]$
Discriminant $13^{12}\cdot 17^{15}$
Root discriminant $97.50$
Ramified primes $13, 17$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group $(C_2\times C_8).D_4$ (as 16T306)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![31417, 167061, -75136, -401668, 755326, -679458, 207532, 104338, -143598, 89830, -29406, 6162, -936, -70, 30, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 30*x^14 - 70*x^13 - 936*x^12 + 6162*x^11 - 29406*x^10 + 89830*x^9 - 143598*x^8 + 104338*x^7 + 207532*x^6 - 679458*x^5 + 755326*x^4 - 401668*x^3 - 75136*x^2 + 167061*x + 31417)
 
gp: K = bnfinit(x^16 - 8*x^15 + 30*x^14 - 70*x^13 - 936*x^12 + 6162*x^11 - 29406*x^10 + 89830*x^9 - 143598*x^8 + 104338*x^7 + 207532*x^6 - 679458*x^5 + 755326*x^4 - 401668*x^3 - 75136*x^2 + 167061*x + 31417, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 30 x^{14} - 70 x^{13} - 936 x^{12} + 6162 x^{11} - 29406 x^{10} + 89830 x^{9} - 143598 x^{8} + 104338 x^{7} + 207532 x^{6} - 679458 x^{5} + 755326 x^{4} - 401668 x^{3} - 75136 x^{2} + 167061 x + 31417 \)

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:  \(66688975910627504451630153142433=13^{12}\cdot 17^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $97.50$
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:  $13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{13} a^{4} - \frac{2}{13} a^{3} - \frac{5}{13} a^{2} + \frac{6}{13} a - \frac{4}{13}$, $\frac{1}{13} a^{5} + \frac{4}{13} a^{3} - \frac{4}{13} a^{2} - \frac{5}{13} a + \frac{5}{13}$, $\frac{1}{13} a^{6} + \frac{4}{13} a^{3} + \frac{2}{13} a^{2} - \frac{6}{13} a + \frac{3}{13}$, $\frac{1}{13} a^{7} - \frac{3}{13} a^{3} + \frac{1}{13} a^{2} + \frac{5}{13} a + \frac{3}{13}$, $\frac{1}{169} a^{8} - \frac{4}{169} a^{7} - \frac{6}{169} a^{6} + \frac{6}{169} a^{5} + \frac{6}{169} a^{4} - \frac{5}{169} a^{3} - \frac{54}{169} a^{2} - \frac{9}{169} a + \frac{3}{169}$, $\frac{1}{169} a^{9} + \frac{4}{169} a^{7} - \frac{5}{169} a^{6} + \frac{4}{169} a^{5} + \frac{6}{169} a^{4} - \frac{9}{169} a^{3} - \frac{4}{169} a^{2} + \frac{71}{169} a + \frac{51}{169}$, $\frac{1}{169} a^{10} - \frac{2}{169} a^{7} + \frac{2}{169} a^{6} - \frac{5}{169} a^{5} + \frac{6}{169} a^{4} - \frac{75}{169} a^{3} - \frac{25}{169} a^{2} + \frac{9}{169} a - \frac{51}{169}$, $\frac{1}{169} a^{11} - \frac{6}{169} a^{7} - \frac{4}{169} a^{6} + \frac{5}{169} a^{5} + \frac{2}{169} a^{4} + \frac{4}{169} a^{3} - \frac{8}{169} a^{2} - \frac{30}{169} a + \frac{58}{169}$, $\frac{1}{103259} a^{12} - \frac{6}{103259} a^{11} + \frac{114}{103259} a^{10} + \frac{96}{103259} a^{9} - \frac{277}{103259} a^{8} - \frac{1367}{103259} a^{7} - \frac{2289}{103259} a^{6} - \frac{527}{103259} a^{5} + \frac{3838}{103259} a^{4} + \frac{15517}{103259} a^{3} - \frac{49084}{103259} a^{2} - \frac{19784}{103259} a + \frac{21880}{103259}$, $\frac{1}{103259} a^{13} + \frac{6}{7943} a^{11} + \frac{1}{611} a^{10} + \frac{23}{7943} a^{9} + \frac{2}{7943} a^{8} + \frac{180}{7943} a^{7} - \frac{157}{7943} a^{6} - \frac{136}{7943} a^{5} - \frac{184}{7943} a^{4} + \frac{3292}{7943} a^{3} - \frac{253}{7943} a^{2} - \frac{210}{7943} a + \frac{28632}{103259}$, $\frac{1}{23533035877} a^{14} - \frac{7}{23533035877} a^{13} + \frac{7032}{1810233529} a^{12} - \frac{3245}{139248733} a^{11} + \frac{4890601}{1810233529} a^{10} - \frac{2643440}{1810233529} a^{9} + \frac{3633099}{1810233529} a^{8} - \frac{63404415}{1810233529} a^{7} + \frac{41204211}{1810233529} a^{6} + \frac{61956925}{1810233529} a^{5} - \frac{19961623}{1810233529} a^{4} - \frac{539289997}{1810233529} a^{3} - \frac{568649034}{1810233529} a^{2} - \frac{8627658735}{23533035877} a + \frac{9864466206}{23533035877}$, $\frac{1}{2376836623577} a^{15} + \frac{43}{2376836623577} a^{14} - \frac{5606509}{2376836623577} a^{13} + \frac{502256}{182833586429} a^{12} - \frac{471730226}{182833586429} a^{11} - \frac{120829780}{182833586429} a^{10} + \frac{348058865}{182833586429} a^{9} + \frac{32015546}{182833586429} a^{8} - \frac{2618671598}{182833586429} a^{7} - \frac{27666241}{14064122033} a^{6} + \frac{1298207631}{182833586429} a^{5} - \frac{6604215829}{182833586429} a^{4} - \frac{821084987}{1810233529} a^{3} + \frac{358084095089}{2376836623577} a^{2} + \frac{169526309171}{2376836623577} a - \frac{11220608445}{2376836623577}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  \( 1231555092.81 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_8).D_4$ (as 16T306):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $(C_2\times C_8).D_4$
Character table for $(C_2\times C_8).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 8.8.11719682839553.1

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

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ R R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$13$13.4.3.3$x^{4} + 26$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.3$x^{4} + 26$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.3$x^{4} + 26$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.4$x^{4} + 104$$4$$1$$3$$C_4$$[\ ]_{4}$
17Data not computed