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Properties

Label 16.0.76098811128...3472.4
Degree $16$
Signature $[0, 8]$
Discriminant $2^{62}\cdot 7^{8}\cdot 17^{15}$
Root discriminant $552.82$
Ramified primes $2, 7, 17$
Class number $1224405536$ (GRH)
Class group $[2, 2, 306101384]$ (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![1568025872, 0, 2592170770112, 0, 2124099047056, 0, 152944237824, 0, 4085046994, 0, 51499392, 0, 323204, 0, 952, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 952*x^14 + 323204*x^12 + 51499392*x^10 + 4085046994*x^8 + 152944237824*x^6 + 2124099047056*x^4 + 2592170770112*x^2 + 1568025872)
 
gp: K = bnfinit(x^16 + 952*x^14 + 323204*x^12 + 51499392*x^10 + 4085046994*x^8 + 152944237824*x^6 + 2124099047056*x^4 + 2592170770112*x^2 + 1568025872, 1)
 

Normalized defining polynomial

\( x^{16} + 952 x^{14} + 323204 x^{12} + 51499392 x^{10} + 4085046994 x^{8} + 152944237824 x^{6} + 2124099047056 x^{4} + 2592170770112 x^{2} + 1568025872 \)

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:  \(76098811128269790757290873194547302704873472=2^{62}\cdot 7^{8}\cdot 17^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $552.82$
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:  $2, 7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3808=2^{5}\cdot 7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{3808}(1,·)$, $\chi_{3808}(3779,·)$, $\chi_{3808}(2185,·)$, $\chi_{3808}(1931,·)$, $\chi_{3808}(2129,·)$, $\chi_{3808}(729,·)$, $\chi_{3808}(1371,·)$, $\chi_{3808}(2995,·)$, $\chi_{3808}(1121,·)$, $\chi_{3808}(1763,·)$, $\chi_{3808}(1707,·)$, $\chi_{3808}(2801,·)$, $\chi_{3808}(2547,·)$, $\chi_{3808}(2267,·)$, $\chi_{3808}(841,·)$, $\chi_{3808}(2297,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{7} a^{2}$, $\frac{1}{7} a^{3}$, $\frac{1}{49} a^{4}$, $\frac{1}{49} a^{5}$, $\frac{1}{343} a^{6}$, $\frac{1}{343} a^{7}$, $\frac{1}{4802} a^{8}$, $\frac{1}{4802} a^{9}$, $\frac{1}{638666} a^{10} + \frac{1}{91238} a^{8} - \frac{9}{6517} a^{6} + \frac{4}{931} a^{4} + \frac{4}{133} a^{2} + \frac{4}{19}$, $\frac{1}{1277332} a^{11} - \frac{9}{91238} a^{9} + \frac{5}{6517} a^{7} + \frac{2}{931} a^{5} - \frac{15}{266} a^{3} + \frac{2}{19} a$, $\frac{1}{8941324} a^{12} - \frac{3}{6517} a^{6} - \frac{1}{98} a^{4} - \frac{2}{19}$, $\frac{1}{8941324} a^{13} - \frac{3}{6517} a^{7} - \frac{1}{98} a^{5} - \frac{2}{19} a$, $\frac{1}{201701349261341964600386296} a^{14} + \frac{108536148243301195}{14407239232952997471456164} a^{12} + \frac{625639879184124041}{1029088516639499819389726} a^{10} - \frac{3803800113127727222}{73506322617107129956409} a^{8} - \frac{6679909987014406403}{42003612924061217117948} a^{6} - \frac{28421739694464063657}{3000258066004372651282} a^{4} + \frac{1496127638097600686}{30614878224534414809} a^{2} + \frac{14152487020840781060}{30614878224534414809}$, $\frac{1}{201701349261341964600386296} a^{15} + \frac{108536148243301195}{14407239232952997471456164} a^{13} - \frac{360029621870405329}{2058177033278999638779452} a^{11} + \frac{6894184195892426255}{147012645234214259912818} a^{9} - \frac{38906097591787474623}{42003612924061217117948} a^{7} + \frac{26362779233650152317}{3000258066004372651282} a^{5} - \frac{16114328812122618849}{428608295143481807326} a^{3} + \frac{10929868260363474238}{30614878224534414809} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{306101384}$, which has order $1224405536$ (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:  \( 11854712.241874253 \) (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

\(\Q(\sqrt{17}) \), 4.4.314432.1, 8.8.1721085137518592.1

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 R $16$ $16$ R $16$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/19.1.0.1}{1} }^{16}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.31.85$x^{8} + 24 x^{6} + 12 x^{4} + 10$$8$$1$$31$$C_8$$[3, 4, 5]$
2.8.31.85$x^{8} + 24 x^{6} + 12 x^{4} + 10$$8$$1$$31$$C_8$$[3, 4, 5]$
7Data not computed
17Data not computed