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Properties

Label 16.4.20836161997...0000.2
Degree $16$
Signature $[4, 6]$
Discriminant $2^{8}\cdot 5^{10}\cdot 11^{6}\cdot 19^{6}$
Root discriminant $28.67$
Ramified primes $2, 5, 11, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T790

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-199, 6557, -16176, -2366, -329, -9418, -1971, 119, -4131, 1279, -1000, 73, 16, -41, 19, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 19*x^14 - 41*x^13 + 16*x^12 + 73*x^11 - 1000*x^10 + 1279*x^9 - 4131*x^8 + 119*x^7 - 1971*x^6 - 9418*x^5 - 329*x^4 - 2366*x^3 - 16176*x^2 + 6557*x - 199)
 
gp: K = bnfinit(x^16 - 4*x^15 + 19*x^14 - 41*x^13 + 16*x^12 + 73*x^11 - 1000*x^10 + 1279*x^9 - 4131*x^8 + 119*x^7 - 1971*x^6 - 9418*x^5 - 329*x^4 - 2366*x^3 - 16176*x^2 + 6557*x - 199, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 19 x^{14} - 41 x^{13} + 16 x^{12} + 73 x^{11} - 1000 x^{10} + 1279 x^{9} - 4131 x^{8} + 119 x^{7} - 1971 x^{6} - 9418 x^{5} - 329 x^{4} - 2366 x^{3} - 16176 x^{2} + 6557 x - 199 \)

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:  \(208361619975602500000000=2^{8}\cdot 5^{10}\cdot 11^{6}\cdot 19^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.67$
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:  $2, 5, 11, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{14} - \frac{1}{6} a^{12} + \frac{1}{6} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{6} a^{4} + \frac{1}{6} a^{3} + \frac{1}{6} a^{2} - \frac{1}{2} a + \frac{1}{6}$, $\frac{1}{601490275931339845517600423502} a^{15} - \frac{24503106498240624801157982819}{300745137965669922758800211751} a^{14} + \frac{57297102463951028787571271827}{300745137965669922758800211751} a^{13} - \frac{55609044741805254165482059015}{300745137965669922758800211751} a^{12} - \frac{2887610918444764910596435701}{100248379321889974252933403917} a^{11} - \frac{5236653404063216835815079079}{300745137965669922758800211751} a^{10} + \frac{102087390818968362986931619624}{300745137965669922758800211751} a^{9} + \frac{71557494171442478764459152655}{300745137965669922758800211751} a^{8} - \frac{46990722250490357964116321953}{200496758643779948505866807834} a^{7} - \frac{143255021313505899389853604213}{300745137965669922758800211751} a^{6} - \frac{136785845809222377266769583892}{300745137965669922758800211751} a^{5} - \frac{106526407806225855986015583155}{300745137965669922758800211751} a^{4} + \frac{52797420699001966464831210515}{200496758643779948505866807834} a^{3} + \frac{135013111132642810432039450096}{300745137965669922758800211751} a^{2} + \frac{14784363571945838244501317785}{601490275931339845517600423502} a - \frac{149184958998683422970504698193}{300745137965669922758800211751}$

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:  \( 185563.883886 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T790:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 62 conjugacy class representatives for t16n790 are not computed
Character table for t16n790 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.5225.1, 4.2.275.1, 4.2.475.1, 8.4.27300625.1

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

Sibling fields

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$2$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$11$11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$19$19.4.3.1$x^{4} + 76$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.3.1$x^{4} + 76$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$