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Properties

Label 16.4.19131876000...0000.2
Degree $16$
Signature $[4, 6]$
Discriminant $2^{16}\cdot 3^{14}\cdot 5^{14}$
Root discriminant $21.39$
Ramified primes $2, 3, 5$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_8:C_2^2$ (as 16T45)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 18, -181, 438, 73, -792, 386, 264, -362, 48, 74, 24, -47, 6, 11, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 11*x^14 + 6*x^13 - 47*x^12 + 24*x^11 + 74*x^10 + 48*x^9 - 362*x^8 + 264*x^7 + 386*x^6 - 792*x^5 + 73*x^4 + 438*x^3 - 181*x^2 + 18*x + 1)
 
gp: K = bnfinit(x^16 - 6*x^15 + 11*x^14 + 6*x^13 - 47*x^12 + 24*x^11 + 74*x^10 + 48*x^9 - 362*x^8 + 264*x^7 + 386*x^6 - 792*x^5 + 73*x^4 + 438*x^3 - 181*x^2 + 18*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 11 x^{14} + 6 x^{13} - 47 x^{12} + 24 x^{11} + 74 x^{10} + 48 x^{9} - 362 x^{8} + 264 x^{7} + 386 x^{6} - 792 x^{5} + 73 x^{4} + 438 x^{3} - 181 x^{2} + 18 x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1913187600000000000000=2^{16}\cdot 3^{14}\cdot 5^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.39$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{4} a$, $\frac{1}{4} a^{12} - \frac{1}{4}$, $\frac{1}{76} a^{13} + \frac{3}{38} a^{12} - \frac{5}{76} a^{11} - \frac{1}{19} a^{10} + \frac{13}{76} a^{9} - \frac{1}{38} a^{8} - \frac{5}{76} a^{7} + \frac{1}{38} a^{6} - \frac{9}{76} a^{5} + \frac{4}{19} a^{4} + \frac{37}{76} a^{3} + \frac{17}{38} a^{2} - \frac{3}{38} a - \frac{3}{19}$, $\frac{1}{152} a^{14} + \frac{2}{19} a^{12} - \frac{3}{38} a^{11} - \frac{1}{152} a^{10} - \frac{1}{38} a^{9} + \frac{7}{152} a^{8} + \frac{4}{19} a^{7} - \frac{21}{152} a^{6} + \frac{4}{19} a^{5} + \frac{55}{152} a^{4} + \frac{5}{19} a^{3} - \frac{29}{76} a^{2} - \frac{13}{38} a - \frac{61}{152}$, $\frac{1}{105765317768} a^{15} + \frac{14311059}{52882658884} a^{14} + \frac{2387219}{695824459} a^{13} + \frac{6182297677}{52882658884} a^{12} - \frac{2101739}{506054152} a^{11} + \frac{2873640749}{52882658884} a^{10} - \frac{7846752119}{105765317768} a^{9} + \frac{783062259}{52882658884} a^{8} + \frac{5960797069}{105765317768} a^{7} - \frac{9746872097}{52882658884} a^{6} - \frac{20603449275}{105765317768} a^{5} + \frac{13179470499}{52882658884} a^{4} + \frac{230799299}{1201878611} a^{3} - \frac{4562763629}{26441329442} a^{2} + \frac{1817685903}{9615028888} a + \frac{3504545775}{13220664721}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $9$
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:  \( 31121.0187165 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8:C_2^2$ (as 16T45):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_8:C_2^2$
Character table for $C_8:C_2^2$

Intermediate fields

\(\Q(\sqrt{15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{3}) \), 4.2.54000.2, 4.2.54000.1, \(\Q(\sqrt{3}, \sqrt{5})\), 8.4.2916000000.1

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 8 siblings: data not computed
Degree 16 siblings: 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 R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
3Data not computed
5Data not computed