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Properties

Label 16.0.13735096585...809.10
Degree $16$
Signature $[0, 8]$
Discriminant $13^{8}\cdot 17^{14}$
Root discriminant $43.01$
Ramified primes $13, 17$
Class number $16$ (GRH)
Class group $[2, 2, 4]$ (GRH)
Galois group $C_2\wr C_4$ (as 16T158)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10183, -22763, 61982, -114223, 100861, -55794, 17731, 34, 951, 1501, -709, 559, -76, 4, 20, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 20*x^14 + 4*x^13 - 76*x^12 + 559*x^11 - 709*x^10 + 1501*x^9 + 951*x^8 + 34*x^7 + 17731*x^6 - 55794*x^5 + 100861*x^4 - 114223*x^3 + 61982*x^2 - 22763*x + 10183)
 
gp: K = bnfinit(x^16 - 6*x^15 + 20*x^14 + 4*x^13 - 76*x^12 + 559*x^11 - 709*x^10 + 1501*x^9 + 951*x^8 + 34*x^7 + 17731*x^6 - 55794*x^5 + 100861*x^4 - 114223*x^3 + 61982*x^2 - 22763*x + 10183, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 20 x^{14} + 4 x^{13} - 76 x^{12} + 559 x^{11} - 709 x^{10} + 1501 x^{9} + 951 x^{8} + 34 x^{7} + 17731 x^{6} - 55794 x^{5} + 100861 x^{4} - 114223 x^{3} + 61982 x^{2} - 22763 x + 10183 \)

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:  \(137350965859713069141239809=13^{8}\cdot 17^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.01$
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:  $13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{12} - \frac{1}{4} a^{11} + \frac{1}{8} a^{10} - \frac{5}{16} a^{9} + \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{8} a^{6} - \frac{7}{16} a^{5} - \frac{3}{16} a^{4} + \frac{7}{16} a^{3} + \frac{1}{16} a^{2} + \frac{7}{16}$, $\frac{1}{32} a^{14} + \frac{3}{32} a^{12} - \frac{1}{16} a^{11} + \frac{13}{32} a^{10} - \frac{1}{32} a^{9} + \frac{1}{4} a^{8} - \frac{5}{16} a^{7} - \frac{13}{32} a^{6} - \frac{1}{16} a^{5} + \frac{1}{8} a^{4} - \frac{1}{2} a^{3} + \frac{1}{32} a^{2} - \frac{1}{32} a + \frac{15}{32}$, $\frac{1}{98416827918936126484119268205889344} a^{15} - \frac{659299707180507507390079132349833}{98416827918936126484119268205889344} a^{14} + \frac{1590286367858334776398019670958971}{98416827918936126484119268205889344} a^{13} + \frac{12302037545427240067731184479839083}{98416827918936126484119268205889344} a^{12} - \frac{14814261391868059424216703392830497}{98416827918936126484119268205889344} a^{11} + \frac{17967330664104158068792711642160429}{49208413959468063242059634102944672} a^{10} - \frac{15143837410971317934823688453028615}{98416827918936126484119268205889344} a^{9} - \frac{18101139790631453438498500139889065}{49208413959468063242059634102944672} a^{8} - \frac{40323621156308182765999599655250483}{98416827918936126484119268205889344} a^{7} + \frac{5687236466887327122807505718990083}{98416827918936126484119268205889344} a^{6} + \frac{16531567114443964701680487719976407}{49208413959468063242059634102944672} a^{5} - \frac{10739135531947859392256510599372119}{24604206979734031621029817051472336} a^{4} - \frac{26485636777032704055060244795309191}{98416827918936126484119268205889344} a^{3} + \frac{12668739955272766893902442650368791}{49208413959468063242059634102944672} a^{2} + \frac{2647889080307118185360496586811871}{12302103489867015810514908525736168} a - \frac{49625354609250726059003897515993}{164301883003232264581167392664256}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}$, which has order $16$ (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:  \( 374046.195998 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T158):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $C_2\wr C_4$
Character table for $C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.0.63869.1, 4.4.4913.1, 4.0.3757.1, 8.0.11719682839553.1 x2, 8.4.901514064581.1 x2, 8.0.4079249161.1

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

Sibling fields

Degree 8 siblings: data not computed
Degree 16 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$17$17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$