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Properties

Label 16.16.2241591694...2833.1
Degree $16$
Signature $[16, 0]$
Discriminant $17^{15}\cdot 23^{8}$
Root discriminant $68.30$
Ramified primes $17, 23$
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![-2845289, 31398761, -31398761, -25708183, 25708183, 7604201, -7604201, -1120471, 1120471, 91289, -91289, -4183, 4183, 101, -101, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 101*x^14 + 101*x^13 + 4183*x^12 - 4183*x^11 - 91289*x^10 + 91289*x^9 + 1120471*x^8 - 1120471*x^7 - 7604201*x^6 + 7604201*x^5 + 25708183*x^4 - 25708183*x^3 - 31398761*x^2 + 31398761*x - 2845289)
 
gp: K = bnfinit(x^16 - x^15 - 101*x^14 + 101*x^13 + 4183*x^12 - 4183*x^11 - 91289*x^10 + 91289*x^9 + 1120471*x^8 - 1120471*x^7 - 7604201*x^6 + 7604201*x^5 + 25708183*x^4 - 25708183*x^3 - 31398761*x^2 + 31398761*x - 2845289, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 101 x^{14} + 101 x^{13} + 4183 x^{12} - 4183 x^{11} - 91289 x^{10} + 91289 x^{9} + 1120471 x^{8} - 1120471 x^{7} - 7604201 x^{6} + 7604201 x^{5} + 25708183 x^{4} - 25708183 x^{3} - 31398761 x^{2} + 31398761 x - 2845289 \)

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:  \(224159169454780289392644342833=17^{15}\cdot 23^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68.30$
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:  $17, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(391=17\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{391}(1,·)$, $\chi_{391}(70,·)$, $\chi_{391}(208,·)$, $\chi_{391}(275,·)$, $\chi_{391}(22,·)$, $\chi_{391}(367,·)$, $\chi_{391}(93,·)$, $\chi_{391}(160,·)$, $\chi_{391}(162,·)$, $\chi_{391}(91,·)$, $\chi_{391}(45,·)$, $\chi_{391}(47,·)$, $\chi_{391}(114,·)$, $\chi_{391}(185,·)$, $\chi_{391}(252,·)$, $\chi_{391}(254,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{1609901} a^{9} - \frac{379328}{1609901} a^{8} - \frac{54}{1609901} a^{7} + \frac{498833}{1609901} a^{6} + \frac{972}{1609901} a^{5} + \frac{567010}{1609901} a^{4} - \frac{6480}{1609901} a^{3} + \frac{498154}{1609901} a^{2} + \frac{11664}{1609901} a + \frac{431335}{1609901}$, $\frac{1}{1609901} a^{10} - \frac{60}{1609901} a^{8} - \frac{666067}{1609901} a^{7} + \frac{1260}{1609901} a^{6} + \frac{606497}{1609901} a^{5} - \frac{10800}{1609901} a^{4} + \frac{771541}{1609901} a^{3} + \frac{32400}{1609901} a^{2} - \frac{704722}{1609901} a - \frac{15552}{1609901}$, $\frac{1}{1609901} a^{11} + \frac{722768}{1609901} a^{8} - \frac{1980}{1609901} a^{7} - \frac{51642}{1609901} a^{6} + \frac{47520}{1609901} a^{5} - \frac{625681}{1609901} a^{4} - \frac{356400}{1609901} a^{3} + \frac{206300}{1609901} a^{2} + \frac{684288}{1609901} a + \frac{121684}{1609901}$, $\frac{1}{1609901} a^{12} - \frac{2376}{1609901} a^{8} + \frac{340206}{1609901} a^{7} + \frac{66528}{1609901} a^{6} + \frac{370560}{1609901} a^{5} - \frac{641520}{1609901} a^{4} + \frac{540931}{1609901} a^{3} + \frac{442963}{1609901} a^{2} - \frac{802632}{1609901} a + \frac{583469}{1609901}$, $\frac{1}{1609901} a^{13} + \frac{601438}{1609901} a^{8} - \frac{61776}{1609901} a^{7} + \frac{710632}{1609901} a^{6} + \frac{58051}{1609901} a^{5} + \frac{269554}{1609901} a^{4} - \frac{464408}{1609901} a^{3} - \frac{465963}{1609901} a^{2} - \frac{681085}{1609901} a - \frac{654977}{1609901}$, $\frac{1}{1609901} a^{14} - \frac{78624}{1609901} a^{8} - \frac{619637}{1609901} a^{7} - \frac{743146}{1609901} a^{6} + \frac{65881}{1609901} a^{5} + \frac{284240}{1609901} a^{4} - \frac{718044}{1609901} a^{3} - \frac{410833}{1609901} a^{2} + \frac{120749}{1609901} a - \frac{202689}{1609901}$, $\frac{1}{1609901} a^{15} + \frac{121617}{1609901} a^{8} - \frac{159139}{1609901} a^{7} - \frac{96489}{1609901} a^{6} - \frac{568480}{1609901} a^{5} + \frac{107605}{1609901} a^{4} + \frac{444264}{1609901} a^{3} - \frac{300584}{1609901} a^{2} - \frac{775923}{1609901} a + \frac{718475}{1609901}$

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:  \( 2165743017.62 \) (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 ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ R $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
23Data not computed