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Properties

Label 16.0.94779071185...3856.5
Degree $16$
Signature $[0, 8]$
Discriminant $2^{64}\cdot 3^{8}\cdot 23^{8}$
Root discriminant $132.91$
Ramified primes $2, 3, 23$
Class number $1026560$ (GRH)
Class group $[2, 16, 32080]$ (GRH)
Galois group $C_8\times C_2$ (as 16T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![63285955489, 0, 26261675072, 0, 8110223184, 0, 954143904, 0, 55123860, 0, 1729376, 0, 30056, 0, 272, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 272*x^14 + 30056*x^12 + 1729376*x^10 + 55123860*x^8 + 954143904*x^6 + 8110223184*x^4 + 26261675072*x^2 + 63285955489)
 
gp: K = bnfinit(x^16 + 272*x^14 + 30056*x^12 + 1729376*x^10 + 55123860*x^8 + 954143904*x^6 + 8110223184*x^4 + 26261675072*x^2 + 63285955489, 1)
 

Normalized defining polynomial

\( x^{16} + 272 x^{14} + 30056 x^{12} + 1729376 x^{10} + 55123860 x^{8} + 954143904 x^{6} + 8110223184 x^{4} + 26261675072 x^{2} + 63285955489 \)

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:  \(9477907118573134595068808298233856=2^{64}\cdot 3^{8}\cdot 23^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $132.91$
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:  $2, 3, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2208=2^{5}\cdot 3\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{2208}(1,·)$, $\chi_{2208}(1931,·)$, $\chi_{2208}(1933,·)$, $\chi_{2208}(1103,·)$, $\chi_{2208}(1105,·)$, $\chi_{2208}(275,·)$, $\chi_{2208}(277,·)$, $\chi_{2208}(2207,·)$, $\chi_{2208}(1379,·)$, $\chi_{2208}(1381,·)$, $\chi_{2208}(551,·)$, $\chi_{2208}(553,·)$, $\chi_{2208}(1655,·)$, $\chi_{2208}(1657,·)$, $\chi_{2208}(827,·)$, $\chi_{2208}(829,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{22645} a^{8} + \frac{136}{22645} a^{6} + \frac{1156}{4529} a^{4} + \frac{10673}{22645} a^{2} + \frac{8527}{22645}$, $\frac{1}{5696734715} a^{9} - \frac{2410514824}{5696734715} a^{7} - \frac{402907742}{1139346943} a^{5} - \frac{133617472}{5696734715} a^{3} - \frac{1136249638}{5696734715} a$, $\frac{1}{5696734715} a^{10} + \frac{34}{1139346943} a^{8} + \frac{1101621719}{5696734715} a^{6} - \frac{224295006}{813819245} a^{4} - \frac{163691616}{5696734715} a^{2} + \frac{2554}{22645}$, $\frac{1}{5696734715} a^{11} + \frac{724242319}{5696734715} a^{7} - \frac{902567242}{5696734715} a^{5} - \frac{33665748}{813819245} a^{3} + \frac{115960268}{5696734715} a$, $\frac{1}{5696734715} a^{12} - \frac{19074}{5696734715} a^{8} - \frac{511525307}{1139346943} a^{6} + \frac{633503749}{5696734715} a^{4} + \frac{543121034}{5696734715} a^{2} - \frac{2053}{22645}$, $\frac{1}{5696734715} a^{13} - \frac{2371494746}{5696734715} a^{7} - \frac{202198116}{5696734715} a^{5} - \frac{1636122289}{5696734715} a^{3} + \frac{2733528312}{5696734715} a$, $\frac{1}{5696734715} a^{14} + \frac{3909}{813819245} a^{8} - \frac{341295721}{813819245} a^{6} - \frac{83150937}{813819245} a^{4} - \frac{2300075791}{5696734715} a^{2} - \frac{5721}{22645}$, $\frac{1}{5696734715} a^{15} - \frac{7613463}{162763849} a^{7} + \frac{233651833}{813819245} a^{5} + \frac{453571646}{1139346943} a^{3} + \frac{2678290032}{5696734715} a$

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

Class group and class number

$C_{2}\times C_{16}\times C_{32080}$, which has order $1026560$ (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:  \( 15753.94986242651 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_8$ (as 16T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_8\times C_2$
Character table for $C_8\times C_2$

Intermediate fields

\(\Q(\sqrt{-138}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-69}) \), \(\Q(\sqrt{2}, \sqrt{-69})\), \(\Q(\zeta_{16})^+\), 4.0.9750528.5, 8.0.380291185115136.75, \(\Q(\zeta_{32})^+\), 8.0.48677271694737408.34

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 R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
3Data not computed
$23$23.8.4.1$x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
23.8.4.1$x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$