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Properties

Label 16.0.48003408671...0689.6
Degree $16$
Signature $[0, 8]$
Discriminant $17^{10}\cdot 47^{8}$
Root discriminant $40.28$
Ramified primes $17, 47$
Class number $20$ (GRH)
Class group $[20]$ (GRH)
Galois group $C_2^2.D_4$ (as 16T33)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5644, 18530, 20095, 934, -16, 10450, 2107, -954, 1977, -506, -181, 214, -11, 0, 4, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 4*x^14 - 11*x^12 + 214*x^11 - 181*x^10 - 506*x^9 + 1977*x^8 - 954*x^7 + 2107*x^6 + 10450*x^5 - 16*x^4 + 934*x^3 + 20095*x^2 + 18530*x + 5644)
 
gp: K = bnfinit(x^16 - 4*x^15 + 4*x^14 - 11*x^12 + 214*x^11 - 181*x^10 - 506*x^9 + 1977*x^8 - 954*x^7 + 2107*x^6 + 10450*x^5 - 16*x^4 + 934*x^3 + 20095*x^2 + 18530*x + 5644, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 4 x^{14} - 11 x^{12} + 214 x^{11} - 181 x^{10} - 506 x^{9} + 1977 x^{8} - 954 x^{7} + 2107 x^{6} + 10450 x^{5} - 16 x^{4} + 934 x^{3} + 20095 x^{2} + 18530 x + 5644 \)

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:  \(48003408671952806969030689=17^{10}\cdot 47^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.28$
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:  $17, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{16} a^{10} + \frac{1}{16} a^{9} + \frac{1}{16} a^{8} + \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{16} a^{4} + \frac{3}{16} a^{3} + \frac{7}{16} a^{2} + \frac{1}{8} a + \frac{1}{4}$, $\frac{1}{16} a^{11} + \frac{1}{16} a^{8} + \frac{1}{8} a^{7} + \frac{3}{16} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{5}{16} a^{2} + \frac{1}{8} a - \frac{1}{4}$, $\frac{1}{32} a^{12} - \frac{1}{32} a^{10} + \frac{1}{32} a^{8} + \frac{3}{16} a^{7} + \frac{7}{32} a^{6} - \frac{1}{4} a^{5} + \frac{5}{32} a^{4} + \frac{1}{4} a^{3} + \frac{3}{32} a^{2} - \frac{7}{16} a + \frac{3}{8}$, $\frac{1}{96} a^{13} + \frac{1}{96} a^{12} - \frac{1}{32} a^{11} - \frac{1}{96} a^{10} + \frac{1}{96} a^{9} - \frac{1}{32} a^{8} + \frac{3}{32} a^{7} + \frac{5}{32} a^{6} - \frac{3}{32} a^{5} + \frac{7}{32} a^{4} + \frac{19}{96} a^{3} + \frac{13}{32} a^{2} + \frac{13}{48} a - \frac{11}{24}$, $\frac{1}{576} a^{14} - \frac{1}{192} a^{13} + \frac{1}{288} a^{12} - \frac{13}{576} a^{11} + \frac{1}{288} a^{10} + \frac{23}{576} a^{9} - \frac{1}{16} a^{8} - \frac{11}{64} a^{7} + \frac{19}{96} a^{6} - \frac{37}{192} a^{5} - \frac{61}{288} a^{4} + \frac{173}{576} a^{3} + \frac{107}{576} a^{2} - \frac{11}{288} a + \frac{5}{144}$, $\frac{1}{464374761191109874542528} a^{15} - \frac{297621717696320455}{345516935410051989987} a^{14} + \frac{777012393883182275627}{464374761191109874542528} a^{13} - \frac{240163504345720591837}{464374761191109874542528} a^{12} + \frac{5841863841143362845593}{464374761191109874542528} a^{11} + \frac{14459991135495930850895}{464374761191109874542528} a^{10} - \frac{10537960320666452948779}{154791587063703291514176} a^{9} + \frac{20041201986601056637}{7371027955414442453056} a^{8} + \frac{28464902423694414764969}{154791587063703291514176} a^{7} - \frac{10760354881288296980401}{154791587063703291514176} a^{6} - \frac{1623732499272402067901}{464374761191109874542528} a^{5} - \frac{2379231936308475059719}{35721135476239221118656} a^{4} + \frac{22491786516220968343055}{116093690297777468635632} a^{3} - \frac{52311299916099272288431}{464374761191109874542528} a^{2} - \frac{3882470363793605091589}{13658081211503231604192} a + \frac{2765466849746722769}{27425865886552673904}$

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

Class group and class number

$C_{20}$, 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:  \( -1 \) (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:  \( 958441.505034 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.D_4$ (as 16T33):

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 $C_2^2.D_4$
Character table for $C_2^2.D_4$

Intermediate fields

\(\Q(\sqrt{-799}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-47}) \), 4.0.37553.1 x2, \(\Q(\sqrt{17}, \sqrt{-47})\), 4.2.13583.1 x2, 8.0.407555836801.1 x2, 8.0.407555836801.2, 8.0.6928449225617.1 x2

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$17$17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
$47$47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$