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Properties

Label 16.16.3345657390...2593.1
Degree $16$
Signature $[16, 0]$
Discriminant $17^{15}\cdot 43^{8}$
Root discriminant $93.39$
Ramified primes $17, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{16}$ (as 16T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![766028671, 2878072306, -2878072306, -1097310578, 1097310578, 167583976, -167583976, -13115246, 13115246, 574089, -574089, -14213, 14213, 186, -186, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 186*x^14 + 186*x^13 + 14213*x^12 - 14213*x^11 - 574089*x^10 + 574089*x^9 + 13115246*x^8 - 13115246*x^7 - 167583976*x^6 + 167583976*x^5 + 1097310578*x^4 - 1097310578*x^3 - 2878072306*x^2 + 2878072306*x + 766028671)
 
gp: K = bnfinit(x^16 - x^15 - 186*x^14 + 186*x^13 + 14213*x^12 - 14213*x^11 - 574089*x^10 + 574089*x^9 + 13115246*x^8 - 13115246*x^7 - 167583976*x^6 + 167583976*x^5 + 1097310578*x^4 - 1097310578*x^3 - 2878072306*x^2 + 2878072306*x + 766028671, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 186 x^{14} + 186 x^{13} + 14213 x^{12} - 14213 x^{11} - 574089 x^{10} + 574089 x^{9} + 13115246 x^{8} - 13115246 x^{7} - 167583976 x^{6} + 167583976 x^{5} + 1097310578 x^{4} - 1097310578 x^{3} - 2878072306 x^{2} + 2878072306 x + 766028671 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(33456573905268530473918973952593=17^{15}\cdot 43^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $93.39$
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:  $17, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(731=17\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{731}(1,·)$, $\chi_{731}(386,·)$, $\chi_{731}(515,·)$, $\chi_{731}(87,·)$, $\chi_{731}(259,·)$, $\chi_{731}(302,·)$, $\chi_{731}(214,·)$, $\chi_{731}(343,·)$, $\chi_{731}(601,·)$, $\chi_{731}(474,·)$, $\chi_{731}(603,·)$, $\chi_{731}(300,·)$, $\chi_{731}(558,·)$, $\chi_{731}(687,·)$, $\chi_{731}(560,·)$, $\chi_{731}(689,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{182679551} a^{9} - \frac{65395694}{182679551} a^{8} - \frac{99}{182679551} a^{7} - \frac{90924560}{182679551} a^{6} + \frac{3267}{182679551} a^{5} - \frac{57088314}{182679551} a^{4} - \frac{39930}{182679551} a^{3} - \frac{82197400}{182679551} a^{2} + \frac{131769}{182679551} a - \frac{69658126}{182679551}$, $\frac{1}{182679551} a^{10} - \frac{110}{182679551} a^{8} + \frac{11365570}{182679551} a^{7} + \frac{4235}{182679551} a^{6} + \frac{38248865}{182679551} a^{5} - \frac{66550}{182679551} a^{4} + \frac{71922725}{182679551} a^{3} + \frac{366025}{182679551} a^{2} + \frac{61123890}{182679551} a - \frac{322102}{182679551}$, $\frac{1}{182679551} a^{11} - \frac{57658281}{182679551} a^{8} - \frac{6655}{182679551} a^{7} + \frac{83922570}{182679551} a^{6} + \frac{292820}{182679551} a^{5} + \frac{3312919}{182679551} a^{4} - \frac{4026275}{182679551} a^{3} - \frac{29292111}{182679551} a^{2} + \frac{14172488}{182679551} a + \frac{10147282}{182679551}$, $\frac{1}{182679551} a^{12} - \frac{7986}{182679551} a^{8} + \frac{38818832}{182679551} a^{7} + \frac{409948}{182679551} a^{6} + \frac{30299865}{182679551} a^{5} - \frac{7247295}{182679551} a^{4} - \frac{14071188}{182679551} a^{3} + \frac{42517464}{182679551} a^{2} - \frac{58349719}{182679551} a - \frac{38974342}{182679551}$, $\frac{1}{182679551} a^{13} + \frac{69642857}{182679551} a^{8} - \frac{380666}{182679551} a^{7} + \frac{57978930}{182679551} a^{6} + \frac{18842967}{182679551} a^{5} + \frac{46812504}{182679551} a^{4} + \frac{88995586}{182679551} a^{3} + \frac{63520175}{182679551} a^{2} - \frac{82744414}{182679551} a - \frac{30561441}{182679551}$, $\frac{1}{182679551} a^{14} - \frac{484484}{182679551} a^{8} + \frac{10798835}{182679551} a^{7} + \frac{27978951}{182679551} a^{6} - \frac{40360320}{182679551} a^{5} + \frac{20435577}{182679551} a^{4} - \frac{28004688}{182679551} a^{3} - \frac{63990898}{182679551} a^{2} - \frac{75620540}{182679551} a + \frac{65553691}{182679551}$, $\frac{1}{182679551} a^{15} + \frac{53994175}{182679551} a^{8} - \frac{19984965}{182679551} a^{7} - \frac{5279669}{182679551} a^{6} - \frac{40871154}{182679551} a^{5} + \frac{12014940}{182679551} a^{4} - \frac{45404612}{182679551} a^{3} + \frac{10637656}{182679551} a^{2} - \frac{32316963}{182679551} a - \frac{27265244}{182679551}$

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:  $15$
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:  \( 23684006739.7 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{16}$ (as 16T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 16
The 16 conjugacy class representatives for $C_{16}$
Character table for $C_{16}$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ R ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
17Data not computed
$43$43.8.4.2$x^{8} - 79507 x^{2} + 68376020$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
43.8.4.2$x^{8} - 79507 x^{2} + 68376020$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$