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Properties

Label 16.0.44664132968...161.22
Degree $16$
Signature $[0, 8]$
Discriminant $13^{12}\cdot 61^{8}$
Root discriminant $53.47$
Ramified primes $13, 61$
Class number $160$ (GRH)
Class group $[2, 2, 40]$ (GRH)
Galois group $C_2\wr C_4$ (as 16T172)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![508743, -222270, 33908, -2739, 77250, -152713, 89768, -15861, 1306, 590, -1537, 471, 4, -7, 9, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 9*x^14 - 7*x^13 + 4*x^12 + 471*x^11 - 1537*x^10 + 590*x^9 + 1306*x^8 - 15861*x^7 + 89768*x^6 - 152713*x^5 + 77250*x^4 - 2739*x^3 + 33908*x^2 - 222270*x + 508743)
 
gp: K = bnfinit(x^16 - 6*x^15 + 9*x^14 - 7*x^13 + 4*x^12 + 471*x^11 - 1537*x^10 + 590*x^9 + 1306*x^8 - 15861*x^7 + 89768*x^6 - 152713*x^5 + 77250*x^4 - 2739*x^3 + 33908*x^2 - 222270*x + 508743, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 9 x^{14} - 7 x^{13} + 4 x^{12} + 471 x^{11} - 1537 x^{10} + 590 x^{9} + 1306 x^{8} - 15861 x^{7} + 89768 x^{6} - 152713 x^{5} + 77250 x^{4} - 2739 x^{3} + 33908 x^{2} - 222270 x + 508743 \)

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:  \(4466413296812760910104974161=13^{12}\cdot 61^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.47$
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:  $13, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{27} a^{14} + \frac{1}{27} a^{13} - \frac{4}{27} a^{12} - \frac{1}{27} a^{11} - \frac{4}{27} a^{10} + \frac{4}{27} a^{9} + \frac{2}{27} a^{8} + \frac{2}{27} a^{7} + \frac{2}{27} a^{6} - \frac{2}{27} a^{5} + \frac{2}{27} a^{4} - \frac{1}{27} a^{3} - \frac{8}{27} a^{2} - \frac{1}{9} a - \frac{1}{3}$, $\frac{1}{462500868628723909341194906517106841511} a^{15} + \frac{8283471009856557232659203835418248287}{462500868628723909341194906517106841511} a^{14} - \frac{8425389013531448807515205785714046540}{154166956209574636447064968839035613837} a^{13} + \frac{9584250447096366803125030007241718444}{462500868628723909341194906517106841511} a^{12} + \frac{43525838008261727068285206081495073258}{462500868628723909341194906517106841511} a^{11} - \frac{16651910744568735049531931534537828081}{154166956209574636447064968839035613837} a^{10} - \frac{17463775726216328340653985033237406280}{154166956209574636447064968839035613837} a^{9} - \frac{70671437496449492336415890244211497482}{462500868628723909341194906517106841511} a^{8} + \frac{72154923156843850200334732648006070761}{462500868628723909341194906517106841511} a^{7} - \frac{33604206518410484544449559247236797831}{154166956209574636447064968839035613837} a^{6} - \frac{19506294633413700303788353277574331600}{154166956209574636447064968839035613837} a^{5} - \frac{7016972894064016639241911905943685939}{462500868628723909341194906517106841511} a^{4} - \frac{19482565351818780213474915393189311606}{154166956209574636447064968839035613837} a^{3} - \frac{168654519956657758933660814055767032826}{462500868628723909341194906517106841511} a^{2} - \frac{33341323019564376071000013206493796846}{154166956209574636447064968839035613837} a + \frac{19524037500846854779812862667340999341}{51388985403191545482354989613011871279}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{40}$, which has order $160$ (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:  \( 206165.188146 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T172):

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

\(\Q(\sqrt{13}) \), 4.0.8175037.2, 4.4.10309.1, 4.0.134017.1, 8.4.1381581253.1, 8.4.5140863842413.3, 8.0.66831229951369.4

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
61Data not computed