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Properties

Label 16.4.18149808491...0641.5
Degree $16$
Signature $[4, 6]$
Discriminant $17^{14}\cdot 47^{6}$
Root discriminant $50.55$
Ramified primes $17, 47$
Class number $10$ (GRH)
Class group $[10]$ (GRH)
Galois group $C_4\wr C_2$ (as 16T28)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-832, 17184, -51824, 88560, -114924, 115850, -93825, 58350, -26334, 7670, -315, -772, 497, -148, 36, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 36*x^14 - 148*x^13 + 497*x^12 - 772*x^11 - 315*x^10 + 7670*x^9 - 26334*x^8 + 58350*x^7 - 93825*x^6 + 115850*x^5 - 114924*x^4 + 88560*x^3 - 51824*x^2 + 17184*x - 832)
 
gp: K = bnfinit(x^16 - 6*x^15 + 36*x^14 - 148*x^13 + 497*x^12 - 772*x^11 - 315*x^10 + 7670*x^9 - 26334*x^8 + 58350*x^7 - 93825*x^6 + 115850*x^5 - 114924*x^4 + 88560*x^3 - 51824*x^2 + 17184*x - 832, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 36 x^{14} - 148 x^{13} + 497 x^{12} - 772 x^{11} - 315 x^{10} + 7670 x^{9} - 26334 x^{8} + 58350 x^{7} - 93825 x^{6} + 115850 x^{5} - 114924 x^{4} + 88560 x^{3} - 51824 x^{2} + 17184 x - 832 \)

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:  \(1814980849112797822933640641=17^{14}\cdot 47^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.55$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{68} a^{8} + \frac{7}{34} a^{7} - \frac{3}{17} a^{6} - \frac{2}{17} a^{5} + \frac{2}{17} a^{4} - \frac{4}{17} a^{3} + \frac{3}{68} a^{2} + \frac{5}{34} a + \frac{4}{17}$, $\frac{1}{68} a^{9} - \frac{1}{17} a^{7} - \frac{5}{34} a^{6} - \frac{4}{17} a^{5} + \frac{2}{17} a^{4} - \frac{11}{68} a^{3} - \frac{8}{17} a^{2} + \frac{3}{17} a - \frac{5}{17}$, $\frac{1}{272} a^{10} - \frac{1}{272} a^{9} - \frac{1}{136} a^{8} + \frac{11}{136} a^{7} + \frac{1}{68} a^{6} + \frac{21}{136} a^{5} - \frac{3}{272} a^{4} - \frac{19}{272} a^{3} - \frac{13}{68} a^{2} + \frac{7}{34} a + \frac{15}{34}$, $\frac{1}{544} a^{11} + \frac{1}{544} a^{9} - \frac{67}{272} a^{7} + \frac{55}{272} a^{6} + \frac{135}{544} a^{5} + \frac{61}{272} a^{4} - \frac{203}{544} a^{3} + \frac{11}{68} a^{2} - \frac{7}{34} a + \frac{33}{68}$, $\frac{1}{2176} a^{12} - \frac{1}{1088} a^{11} + \frac{1}{2176} a^{10} - \frac{5}{1088} a^{9} - \frac{3}{1088} a^{8} - \frac{123}{1088} a^{7} - \frac{453}{2176} a^{6} + \frac{79}{544} a^{5} + \frac{241}{2176} a^{4} + \frac{355}{1088} a^{3} - \frac{63}{136} a^{2} - \frac{131}{272} a - \frac{65}{136}$, $\frac{1}{4352} a^{13} - \frac{1}{4352} a^{12} + \frac{3}{4352} a^{11} + \frac{7}{4352} a^{10} + \frac{1}{1088} a^{9} - \frac{7}{1088} a^{8} - \frac{691}{4352} a^{7} - \frac{849}{4352} a^{6} + \frac{297}{4352} a^{5} + \frac{431}{4352} a^{4} - \frac{755}{2176} a^{3} - \frac{213}{544} a^{2} - \frac{109}{544} a - \frac{71}{272}$, $\frac{1}{4352} a^{14} - \frac{1}{2176} a^{11} - \frac{7}{4352} a^{10} - \frac{1}{1088} a^{9} + \frac{29}{4352} a^{8} - \frac{35}{544} a^{7} + \frac{481}{2176} a^{6} + \frac{21}{272} a^{5} - \frac{9}{4352} a^{4} - \frac{733}{2176} a^{3} + \frac{61}{272} a^{2} - \frac{5}{32} a + \frac{107}{272}$, $\frac{1}{69578014797824} a^{15} + \frac{24049365}{1087156481216} a^{14} - \frac{38321835}{17394503699456} a^{13} + \frac{3827393613}{17394503699456} a^{12} + \frac{53242793177}{69578014797824} a^{11} - \frac{37938160655}{34789007398912} a^{10} + \frac{290965364465}{69578014797824} a^{9} - \frac{19451603721}{17394503699456} a^{8} - \frac{122492301507}{34789007398912} a^{7} + \frac{112844743713}{2676077492224} a^{6} + \frac{3543043549483}{69578014797824} a^{5} + \frac{2531048999623}{17394503699456} a^{4} - \frac{1097561562289}{17394503699456} a^{3} - \frac{277290823665}{669019373056} a^{2} - \frac{40732667691}{2174312962432} a + \frac{3429710099}{167254843264}$

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

Class group and class number

$C_{10}$, which has order $10$ (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:  \( 18901226.1171 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\wr C_2$ (as 16T28):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_4\wr C_2$
Character table for $C_4\wr C_2$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 4.2.230911.1, 4.2.13583.1, 8.4.53319889921.1

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 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
47Data not computed