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Properties

Label 16.4.19230048432...0625.1
Degree $16$
Signature $[4, 6]$
Discriminant $5^{12}\cdot 31^{12}$
Root discriminant $43.93$
Ramified primes $5, 31$
Class number $1$ (GRH)
Class group Trivial (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![74161, -40280, 6366, -24201, 1249, 10178, -2252, 4516, -2798, 492, -586, 118, -37, -20, -8, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 8*x^14 - 20*x^13 - 37*x^12 + 118*x^11 - 586*x^10 + 492*x^9 - 2798*x^8 + 4516*x^7 - 2252*x^6 + 10178*x^5 + 1249*x^4 - 24201*x^3 + 6366*x^2 - 40280*x + 74161)
 
gp: K = bnfinit(x^16 - 3*x^15 - 8*x^14 - 20*x^13 - 37*x^12 + 118*x^11 - 586*x^10 + 492*x^9 - 2798*x^8 + 4516*x^7 - 2252*x^6 + 10178*x^5 + 1249*x^4 - 24201*x^3 + 6366*x^2 - 40280*x + 74161, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 8 x^{14} - 20 x^{13} - 37 x^{12} + 118 x^{11} - 586 x^{10} + 492 x^{9} - 2798 x^{8} + 4516 x^{7} - 2252 x^{6} + 10178 x^{5} + 1249 x^{4} - 24201 x^{3} + 6366 x^{2} - 40280 x + 74161 \)

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:  \(192300484323376406494140625=5^{12}\cdot 31^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.93$
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:  $5, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{328} a^{13} + \frac{9}{328} a^{12} - \frac{1}{41} a^{11} - \frac{7}{328} a^{10} - \frac{3}{41} a^{9} - \frac{13}{164} a^{8} + \frac{31}{328} a^{7} - \frac{33}{82} a^{6} - \frac{21}{82} a^{5} - \frac{57}{328} a^{4} + \frac{7}{41} a^{3} - \frac{81}{164} a^{2} - \frac{17}{41} a - \frac{117}{328}$, $\frac{1}{656} a^{14} - \frac{7}{656} a^{12} - \frac{17}{656} a^{11} - \frac{43}{656} a^{10} + \frac{27}{164} a^{9} - \frac{145}{656} a^{8} + \frac{81}{656} a^{7} + \frac{15}{82} a^{6} + \frac{289}{656} a^{5} + \frac{159}{656} a^{4} + \frac{9}{82} a^{3} - \frac{9}{82} a^{2} + \frac{3}{16} a - \frac{13}{656}$, $\frac{1}{62620590735501044739424983536} a^{15} - \frac{6792857027997753785401701}{62620590735501044739424983536} a^{14} + \frac{77511494940085370896936049}{62620590735501044739424983536} a^{13} - \frac{3591379916591649868830885893}{31310295367750522369712491768} a^{12} + \frac{708947799715016666582869497}{31310295367750522369712491768} a^{11} + \frac{9877154906939165348074007}{703602143095517356622752624} a^{10} + \frac{11365510160099227310641998539}{62620590735501044739424983536} a^{9} - \frac{5778676266330623254910953411}{31310295367750522369712491768} a^{8} + \frac{8379741440110955740742281383}{62620590735501044739424983536} a^{7} - \frac{423998097907066294334707291}{1527331481353684018034755696} a^{6} - \frac{9925644767269688408323537595}{31310295367750522369712491768} a^{5} - \frac{14931731192719033313164744431}{62620590735501044739424983536} a^{4} - \frac{639726811417203857738329428}{3913786920968815296214061471} a^{3} - \frac{17447785730149819670317894537}{62620590735501044739424983536} a^{2} - \frac{1082902347972011873707061599}{3913786920968815296214061471} a - \frac{261560138568543556140727959}{1527331481353684018034755696}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 8464867.53427 \) (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

\(\Q(\sqrt{5}) \), 4.2.775.1, 4.4.120125.1, 4.2.3875.1, 8.4.14430015625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
5Data not computed
$31$31.8.6.2$x^{8} + 713 x^{4} + 138384$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
31.8.6.2$x^{8} + 713 x^{4} + 138384$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$