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Properties

Label 16.8.95581759383...8929.1
Degree $16$
Signature $[8, 4]$
Discriminant $11^{4}\cdot 97^{14}$
Root discriminant $99.72$
Ramified primes $11, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2 : C_8$ (as 16T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-364019, -2647585, 6802389, 2459585, -12369897, 9020278, -1648945, -931559, 507772, -54968, -19527, 6168, -55, -104, -22, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 22*x^14 - 104*x^13 - 55*x^12 + 6168*x^11 - 19527*x^10 - 54968*x^9 + 507772*x^8 - 931559*x^7 - 1648945*x^6 + 9020278*x^5 - 12369897*x^4 + 2459585*x^3 + 6802389*x^2 - 2647585*x - 364019)
 
gp: K = bnfinit(x^16 - x^15 - 22*x^14 - 104*x^13 - 55*x^12 + 6168*x^11 - 19527*x^10 - 54968*x^9 + 507772*x^8 - 931559*x^7 - 1648945*x^6 + 9020278*x^5 - 12369897*x^4 + 2459585*x^3 + 6802389*x^2 - 2647585*x - 364019, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 22 x^{14} - 104 x^{13} - 55 x^{12} + 6168 x^{11} - 19527 x^{10} - 54968 x^{9} + 507772 x^{8} - 931559 x^{7} - 1648945 x^{6} + 9020278 x^{5} - 12369897 x^{4} + 2459585 x^{3} + 6802389 x^{2} - 2647585 x - 364019 \)

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:  \(95581759383007560835666012898929=11^{4}\cdot 97^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $99.72$
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:  $11, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{83959722606159773224152439421984758113407805514189306} a^{15} - \frac{19304239164763285180021802451437001524021321998375779}{83959722606159773224152439421984758113407805514189306} a^{14} + \frac{8621617381510110183825734858188693215733549978318352}{41979861303079886612076219710992379056703902757094653} a^{13} - \frac{8584530609271924010768479808427477573216376244872402}{41979861303079886612076219710992379056703902757094653} a^{12} - \frac{8699276358543051959087498258655884349317618174590186}{41979861303079886612076219710992379056703902757094653} a^{11} + \frac{940396764199393424091167795700224076175247924974070}{41979861303079886612076219710992379056703902757094653} a^{10} - \frac{40024283033069279824883426298288068641437574050343103}{83959722606159773224152439421984758113407805514189306} a^{9} - \frac{10997699419564359101757632573524681211235789549819755}{41979861303079886612076219710992379056703902757094653} a^{8} - \frac{3020309536008667248759266319045486044476033197505887}{83959722606159773224152439421984758113407805514189306} a^{7} - \frac{11776614025825587937710762240874133072558417006823648}{41979861303079886612076219710992379056703902757094653} a^{6} - \frac{18235332575375319130671213987709401549753450422673311}{41979861303079886612076219710992379056703902757094653} a^{5} + \frac{16446914049688605873548938745110244309876756738576609}{83959722606159773224152439421984758113407805514189306} a^{4} - \frac{33082954069277887212716545333220697756576396623253515}{83959722606159773224152439421984758113407805514189306} a^{3} + \frac{4541145899855864887524784744474530361575878695226064}{41979861303079886612076219710992379056703902757094653} a^{2} - \frac{31884616389435168351430930107658172183458392217449331}{83959722606159773224152439421984758113407805514189306} a - \frac{12093807084915248322850251760422307340996492445630795}{41979861303079886612076219710992379056703902757094653}$

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:  \( 8828543184.49 \) (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

\(\Q(\sqrt{97}) \), 4.2.10039403.1, 4.4.912673.1, 4.2.103499.1, 8.4.9776592421851673.1, 8.8.80798284478113.1, 8.4.100789612596409.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$11$11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
97Data not computed