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Properties

Label 16.0.72399383432...5584.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 3^{8}\cdot 17^{14}$
Root discriminant $41.33$
Ramified primes $2, 3, 17$
Class number $20$ (GRH)
Class group $[2, 10]$ (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![289, 0, -1445, 0, 6069, 0, -5202, 0, 3162, 0, -986, 0, 221, 0, -17, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 17*x^14 + 221*x^12 - 986*x^10 + 3162*x^8 - 5202*x^6 + 6069*x^4 - 1445*x^2 + 289)
 
gp: K = bnfinit(x^16 - 17*x^14 + 221*x^12 - 986*x^10 + 3162*x^8 - 5202*x^6 + 6069*x^4 - 1445*x^2 + 289, 1)
 

Normalized defining polynomial

\( x^{16} - 17 x^{14} + 221 x^{12} - 986 x^{10} + 3162 x^{8} - 5202 x^{6} + 6069 x^{4} - 1445 x^{2} + 289 \)

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:  \(72399383432805056195395584=2^{16}\cdot 3^{8}\cdot 17^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.33$
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:  $2, 3, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(204=2^{2}\cdot 3\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{204}(19,·)$, $\chi_{204}(1,·)$, $\chi_{204}(137,·)$, $\chi_{204}(13,·)$, $\chi_{204}(83,·)$, $\chi_{204}(149,·)$, $\chi_{204}(151,·)$, $\chi_{204}(89,·)$, $\chi_{204}(155,·)$, $\chi_{204}(157,·)$, $\chi_{204}(101,·)$, $\chi_{204}(169,·)$, $\chi_{204}(43,·)$, $\chi_{204}(179,·)$, $\chi_{204}(59,·)$, $\chi_{204}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{4} a^{6} + \frac{1}{4}$, $\frac{1}{4} a^{7} + \frac{1}{4} a$, $\frac{1}{68} a^{8} + \frac{1}{4} a^{2}$, $\frac{1}{68} a^{9} + \frac{1}{4} a^{3}$, $\frac{1}{884} a^{10} - \frac{5}{884} a^{8} + \frac{3}{52} a^{6} - \frac{15}{52} a^{4} + \frac{23}{52} a^{2} + \frac{15}{52}$, $\frac{1}{884} a^{11} - \frac{5}{884} a^{9} + \frac{3}{52} a^{7} - \frac{15}{52} a^{5} + \frac{23}{52} a^{3} + \frac{15}{52} a$, $\frac{1}{7072} a^{12} - \frac{1}{16} a^{6} + \frac{101}{416}$, $\frac{1}{7072} a^{13} - \frac{1}{16} a^{7} + \frac{101}{416} a$, $\frac{1}{91936} a^{14} + \frac{1}{91936} a^{12} - \frac{1}{11492} a^{10} + \frac{163}{45968} a^{8} + \frac{131}{2704} a^{6} - \frac{297}{676} a^{4} + \frac{645}{5408} a^{2} - \frac{955}{5408}$, $\frac{1}{91936} a^{15} + \frac{1}{91936} a^{13} - \frac{1}{11492} a^{11} + \frac{163}{45968} a^{9} + \frac{131}{2704} a^{7} - \frac{297}{676} a^{5} + \frac{645}{5408} a^{3} - \frac{955}{5408} a$

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

Class group and class number

$C_{2}\times C_{10}$, which has order $20$ (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:  \( \frac{83}{91936} a^{14} - \frac{693}{45968} a^{12} + \frac{33}{169} a^{10} - \frac{38419}{45968} a^{8} + \frac{3597}{1352} a^{6} - \frac{693}{169} a^{4} + \frac{27431}{5408} a^{2} - \frac{561}{2704} \) (order $6$)
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:  \( 351247.794557 \) (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

\(\Q(\sqrt{-51}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{17})\), 4.4.4913.1, 4.0.44217.1, 8.0.1955143089.1, 8.8.8508782723328.1, 8.0.105046700288.1

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 R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
3Data not computed
17Data not computed