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Properties

Label 16.0.17690457232...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 11^{12}\cdot 31^{4}$
Root discriminant $58.28$
Ramified primes $5, 11, 31$
Class number $16$ (GRH)
Class group $[2, 2, 4]$ (GRH)
Galois group 16T1161

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7051, 22638, 27016, 28017, 32641, 8509, 4304, 991, 1407, 121, 209, -36, 36, 2, 6, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 6*x^14 + 2*x^13 + 36*x^12 - 36*x^11 + 209*x^10 + 121*x^9 + 1407*x^8 + 991*x^7 + 4304*x^6 + 8509*x^5 + 32641*x^4 + 28017*x^3 + 27016*x^2 + 22638*x + 7051)
 
gp: K = bnfinit(x^16 - 2*x^15 + 6*x^14 + 2*x^13 + 36*x^12 - 36*x^11 + 209*x^10 + 121*x^9 + 1407*x^8 + 991*x^7 + 4304*x^6 + 8509*x^5 + 32641*x^4 + 28017*x^3 + 27016*x^2 + 22638*x + 7051, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 6 x^{14} + 2 x^{13} + 36 x^{12} - 36 x^{11} + 209 x^{10} + 121 x^{9} + 1407 x^{8} + 991 x^{7} + 4304 x^{6} + 8509 x^{5} + 32641 x^{4} + 28017 x^{3} + 27016 x^{2} + 22638 x + 7051 \)

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:  \(17690457232041959478759765625=5^{14}\cdot 11^{12}\cdot 31^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.28$
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:  $5, 11, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{209} a^{14} + \frac{103}{209} a^{13} + \frac{43}{209} a^{12} - \frac{7}{209} a^{11} + \frac{36}{209} a^{10} - \frac{21}{209} a^{9} - \frac{10}{209} a^{8} - \frac{102}{209} a^{7} + \frac{2}{11} a^{6} - \frac{1}{11} a^{5} + \frac{86}{209} a^{4} - \frac{5}{19} a^{3} - \frac{8}{19} a^{2} + \frac{3}{19} a - \frac{1}{19}$, $\frac{1}{5403446793947377578446108425481} a^{15} - \frac{879965805788244452518159045}{491222435813397961676918947771} a^{14} - \frac{123616641447524257261728827741}{5403446793947377578446108425481} a^{13} - \frac{1947914846826340856271775763865}{5403446793947377578446108425481} a^{12} - \frac{670362475703155842746911823724}{5403446793947377578446108425481} a^{11} + \frac{99782842719925980954185617238}{491222435813397961676918947771} a^{10} - \frac{1641547393736749828222680529284}{5403446793947377578446108425481} a^{9} - \frac{1664970315644323690940676916987}{5403446793947377578446108425481} a^{8} + \frac{644998248029628940805303201427}{5403446793947377578446108425481} a^{7} - \frac{132439373729687252789846350514}{284391936523546188339268864499} a^{6} - \frac{465755147476257522774835584929}{5403446793947377578446108425481} a^{5} + \frac{2195979967743765753267997135667}{5403446793947377578446108425481} a^{4} + \frac{38591919274792072392458312788}{491222435813397961676918947771} a^{3} + \frac{153135002091621297410198609220}{491222435813397961676918947771} a^{2} - \frac{196101910504443837265240885902}{491222435813397961676918947771} a - \frac{102638848304065656403256807072}{491222435813397961676918947771}$

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:  \( 2789844.44275 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1161:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 43 conjugacy class representatives for t16n1161
Character table for t16n1161 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.15125.1, 8.4.78009078125.1

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

Sibling fields

Degree 16 siblings: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
5Data not computed
$11$11.4.3.1$x^{4} + 33$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.4.3.1$x^{4} + 33$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.8.6.1$x^{8} + 143 x^{4} + 5929$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
31Data not computed