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Properties

Label 16.0.47490403326...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{12}\cdot 5^{14}\cdot 11^{4}$
Root discriminant $16.97$
Ramified primes $3, 5, 11$
Class number $2$
Class group $[2]$
Galois group $C_2 \times (C_8:C_2)$ (as 16T15)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3721, 0, -4526, 0, 5747, 0, -5153, 0, 2725, 0, -862, 0, 167, 0, -19, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 19*x^14 + 167*x^12 - 862*x^10 + 2725*x^8 - 5153*x^6 + 5747*x^4 - 4526*x^2 + 3721)
 
gp: K = bnfinit(x^16 - 19*x^14 + 167*x^12 - 862*x^10 + 2725*x^8 - 5153*x^6 + 5747*x^4 - 4526*x^2 + 3721, 1)
 

Normalized defining polynomial

\( x^{16} - 19 x^{14} + 167 x^{12} - 862 x^{10} + 2725 x^{8} - 5153 x^{6} + 5747 x^{4} - 4526 x^{2} + 3721 \)

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:  \(47490403326416015625=3^{12}\cdot 5^{14}\cdot 11^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.97$
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:  $3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{62} a^{10} - \frac{1}{62} a^{8} - \frac{6}{31} a^{6} + \frac{23}{62} a^{4} + \frac{4}{31} a^{2} - \frac{1}{2} a + \frac{3}{62}$, $\frac{1}{62} a^{11} - \frac{1}{62} a^{9} - \frac{6}{31} a^{7} + \frac{23}{62} a^{5} + \frac{4}{31} a^{3} - \frac{1}{2} a^{2} + \frac{3}{62} a$, $\frac{1}{62} a^{12} - \frac{13}{62} a^{8} + \frac{11}{62} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{11}{62} a^{2} - \frac{1}{2} a + \frac{3}{62}$, $\frac{1}{62} a^{13} - \frac{13}{62} a^{9} + \frac{11}{62} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{11}{62} a^{3} - \frac{1}{2} a^{2} + \frac{3}{62} a$, $\frac{1}{44660522} a^{14} - \frac{37131}{22330261} a^{12} + \frac{20359}{44660522} a^{10} + \frac{230335}{22330261} a^{8} - \frac{1}{2} a^{7} + \frac{4580049}{22330261} a^{6} - \frac{1}{2} a^{5} - \frac{7573423}{22330261} a^{4} - \frac{9888793}{22330261} a^{2} + \frac{15238169}{44660522}$, $\frac{1}{2724291842} a^{15} + \frac{7166179}{1362145921} a^{13} + \frac{6132993}{1362145921} a^{11} + \frac{381515769}{2724291842} a^{9} + \frac{43015655}{2724291842} a^{7} - \frac{1073313085}{2724291842} a^{5} - \frac{1}{2} a^{4} - \frac{366256797}{2724291842} a^{3} + \frac{538863197}{1362145921} a - \frac{1}{2}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( \frac{946118}{1362145921} a^{15} + \frac{5865}{22330261} a^{14} - \frac{16817120}{1362145921} a^{13} - \frac{213081}{44660522} a^{12} + \frac{272295631}{2724291842} a^{11} + \frac{1822171}{44660522} a^{10} - \frac{637110721}{1362145921} a^{9} - \frac{4454017}{22330261} a^{8} + \frac{1754838629}{1362145921} a^{7} + \frac{12493755}{22330261} a^{6} - \frac{2714975788}{1362145921} a^{5} - \frac{18719159}{22330261} a^{4} + \frac{4870610295}{2724291842} a^{3} + \frac{22489003}{44660522} a^{2} - \frac{1408847861}{1362145921} a - \frac{28745541}{44660522} \) (order $30$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 10344.1260394 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times OD_{16}$ (as 16T15):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2 \times (C_8:C_2)$
Character table for $C_2 \times (C_8:C_2)$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{5})\), \(\Q(\sqrt{-3}, \sqrt{5})\), 8.0.6891328125.1, 8.8.6891328125.1, \(\Q(\zeta_{15})\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 16 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
3Data not computed
5Data not computed
$11$11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$