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Properties

Label 16.16.6668897591...2433.1
Degree $16$
Signature $[16, 0]$
Discriminant $13^{12}\cdot 17^{15}$
Root discriminant $97.50$
Ramified primes $13, 17$
Class number $4$ (GRH)
Class group $[4]$ (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![-3484829, -9372458, -5013027, 5988419, 5808017, -1155899, -2183632, 13038, 417368, 20127, -45100, -2245, 2738, 84, -84, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 84*x^14 + 84*x^13 + 2738*x^12 - 2245*x^11 - 45100*x^10 + 20127*x^9 + 417368*x^8 + 13038*x^7 - 2183632*x^6 - 1155899*x^5 + 5808017*x^4 + 5988419*x^3 - 5013027*x^2 - 9372458*x - 3484829)
 
gp: K = bnfinit(x^16 - x^15 - 84*x^14 + 84*x^13 + 2738*x^12 - 2245*x^11 - 45100*x^10 + 20127*x^9 + 417368*x^8 + 13038*x^7 - 2183632*x^6 - 1155899*x^5 + 5808017*x^4 + 5988419*x^3 - 5013027*x^2 - 9372458*x - 3484829, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 84 x^{14} + 84 x^{13} + 2738 x^{12} - 2245 x^{11} - 45100 x^{10} + 20127 x^{9} + 417368 x^{8} + 13038 x^{7} - 2183632 x^{6} - 1155899 x^{5} + 5808017 x^{4} + 5988419 x^{3} - 5013027 x^{2} - 9372458 x - 3484829 \)

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:  \(66688975910627504451630153142433=13^{12}\cdot 17^{15}\)
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 Galois and abelian over $\Q$.
Conductor:  \(221=13\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{221}(1,·)$, $\chi_{221}(73,·)$, $\chi_{221}(77,·)$, $\chi_{221}(216,·)$, $\chi_{221}(25,·)$, $\chi_{221}(155,·)$, $\chi_{221}(157,·)$, $\chi_{221}(96,·)$, $\chi_{221}(99,·)$, $\chi_{221}(168,·)$, $\chi_{221}(44,·)$, $\chi_{221}(109,·)$, $\chi_{221}(118,·)$, $\chi_{221}(183,·)$, $\chi_{221}(57,·)$, $\chi_{221}(190,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{101} a^{13} + \frac{42}{101} a^{12} + \frac{22}{101} a^{11} - \frac{32}{101} a^{10} - \frac{35}{101} a^{9} - \frac{37}{101} a^{8} - \frac{49}{101} a^{7} - \frac{37}{101} a^{6} - \frac{3}{101} a^{5} + \frac{21}{101} a^{4} - \frac{7}{101} a^{3} - \frac{24}{101} a^{2} + \frac{46}{101} a - \frac{9}{101}$, $\frac{1}{3650443} a^{14} - \frac{14292}{3650443} a^{13} - \frac{1480706}{3650443} a^{12} - \frac{244478}{3650443} a^{11} + \frac{95154}{3650443} a^{10} + \frac{1793847}{3650443} a^{9} - \frac{130029}{3650443} a^{8} + \frac{76735}{3650443} a^{7} - \frac{819813}{3650443} a^{6} - \frac{719123}{3650443} a^{5} - \frac{510495}{3650443} a^{4} + \frac{838422}{3650443} a^{3} + \frac{69342}{3650443} a^{2} + \frac{1679989}{3650443} a - \frac{600517}{3650443}$, $\frac{1}{828523875747811593230268173} a^{15} + \frac{30586985559967174169}{828523875747811593230268173} a^{14} + \frac{263963367238170478576238}{828523875747811593230268173} a^{13} + \frac{711150911345017172139421}{828523875747811593230268173} a^{12} - \frac{324004693860507896070470030}{828523875747811593230268173} a^{11} + \frac{297739655484982208346296298}{828523875747811593230268173} a^{10} + \frac{389321905557731704890979572}{828523875747811593230268173} a^{9} - \frac{292873151295975224594681778}{828523875747811593230268173} a^{8} + \frac{225437116891952383731920622}{828523875747811593230268173} a^{7} - \frac{319349384654434030398598411}{828523875747811593230268173} a^{6} + \frac{33532940027754257190311647}{828523875747811593230268173} a^{5} + \frac{122421019139463875762701878}{828523875747811593230268173} a^{4} - \frac{297327932560836832238832500}{828523875747811593230268173} a^{3} + \frac{262658438955068060018522422}{828523875747811593230268173} a^{2} + \frac{58709840744352613768135452}{828523875747811593230268173} a + \frac{65386209100600121924618985}{828523875747811593230268173}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 14623244314.8 \) (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, 8.8.11719682839553.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ R R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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