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Properties

Label 16.16.4861414234...6513.1
Degree $16$
Signature $[16, 0]$
Discriminant $17^{15}\cdot 19^{8}$
Root discriminant $62.08$
Ramified primes $17, 19$
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![-1114859, 7755484, -7755484, -8182016, 8182016, 2974234, -2974234, -532016, 532016, 52359, -52359, -2891, 2891, 84, -84, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 84*x^14 + 84*x^13 + 2891*x^12 - 2891*x^11 - 52359*x^10 + 52359*x^9 + 532016*x^8 - 532016*x^7 - 2974234*x^6 + 2974234*x^5 + 8182016*x^4 - 8182016*x^3 - 7755484*x^2 + 7755484*x - 1114859)
 
gp: K = bnfinit(x^16 - x^15 - 84*x^14 + 84*x^13 + 2891*x^12 - 2891*x^11 - 52359*x^10 + 52359*x^9 + 532016*x^8 - 532016*x^7 - 2974234*x^6 + 2974234*x^5 + 8182016*x^4 - 8182016*x^3 - 7755484*x^2 + 7755484*x - 1114859, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 84 x^{14} + 84 x^{13} + 2891 x^{12} - 2891 x^{11} - 52359 x^{10} + 52359 x^{9} + 532016 x^{8} - 532016 x^{7} - 2974234 x^{6} + 2974234 x^{5} + 8182016 x^{4} - 8182016 x^{3} - 7755484 x^{2} + 7755484 x - 1114859 \)

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:  \(48614142345328546750712906513=17^{15}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.08$
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:  $17, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(323=17\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{323}(1,·)$, $\chi_{323}(134,·)$, $\chi_{323}(265,·)$, $\chi_{323}(75,·)$, $\chi_{323}(77,·)$, $\chi_{323}(284,·)$, $\chi_{323}(37,·)$, $\chi_{323}(227,·)$, $\chi_{323}(229,·)$, $\chi_{323}(113,·)$, $\chi_{323}(172,·)$, $\chi_{323}(303,·)$, $\chi_{323}(305,·)$, $\chi_{323}(115,·)$, $\chi_{323}(56,·)$, $\chi_{323}(191,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{308549} a^{9} - \frac{61932}{308549} a^{8} - \frac{45}{308549} a^{7} + \frac{8888}{308549} a^{6} + \frac{675}{308549} a^{5} - \frac{111100}{308549} a^{4} - \frac{3750}{308549} a^{3} + \frac{135851}{308549} a^{2} + \frac{5625}{308549} a + \frac{30799}{308549}$, $\frac{1}{308549} a^{10} - \frac{50}{308549} a^{8} - \frac{1111}{308549} a^{7} + \frac{875}{308549} a^{6} + \frac{38885}{308549} a^{5} - \frac{6250}{308549} a^{4} - \frac{80301}{308549} a^{3} + \frac{15625}{308549} a^{2} + \frac{46478}{308549} a - \frac{6250}{308549}$, $\frac{1}{308549} a^{11} - \frac{12221}{308549} a^{8} - \frac{1375}{308549} a^{7} - \frac{133813}{308549} a^{6} + \frac{27500}{308549} a^{5} - \frac{81419}{308549} a^{4} + \frac{136674}{308549} a^{3} + \frac{50950}{308549} a^{2} - \frac{33549}{308549} a - \frac{2795}{308549}$, $\frac{1}{308549} a^{12} - \frac{1650}{308549} a^{8} - \frac{66660}{308549} a^{7} + \frac{38500}{308549} a^{6} + \frac{145482}{308549} a^{5} - \frac{826}{308549} a^{4} - \frac{112548}{308549} a^{3} - \frac{100647}{308549} a^{2} - \frac{66097}{308549} a - \frac{35201}{308549}$, $\frac{1}{308549} a^{13} - \frac{124741}{308549} a^{8} - \frac{35750}{308549} a^{7} + \frac{330}{308549} a^{6} - \frac{121272}{308549} a^{5} - \frac{149442}{308549} a^{4} - \frac{117167}{308549} a^{3} + \frac{81479}{308549} a^{2} - \frac{10421}{308549} a - \frac{92235}{308549}$, $\frac{1}{308549} a^{14} - \frac{45500}{308549} a^{8} - \frac{59133}{308549} a^{7} - \frac{39821}{308549} a^{6} + \frac{125405}{308549} a^{5} - \frac{55383}{308549} a^{4} + \frac{63013}{308549} a^{3} + \frac{50992}{308549} a^{2} - \frac{64536}{308549} a - \frac{154089}{308549}$, $\frac{1}{308549} a^{15} + \frac{12884}{308549} a^{8} + \frac{72522}{308549} a^{7} + \frac{21666}{308549} a^{6} + \frac{110766}{308549} a^{5} - \frac{28720}{308549} a^{4} + \frac{53589}{308549} a^{3} - \frac{6153}{308549} a^{2} - \frac{3710}{308549} a - \frac{75058}{308549}$

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:  \( 1110998666.2 \) (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 R $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$19$19.8.4.2$x^{8} - 13718 x^{2} + 1303210$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
19.8.4.2$x^{8} - 13718 x^{2} + 1303210$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$