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Properties

Label 16.8.44118971794...8896.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{12}\cdot 41^{14}\cdot 73^{4}$
Root discriminant $126.70$
Ramified primes $2, 41, 73$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4.C_2^2:D_4$ (as 16T209)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![431493817, 689350461, -469772490, -455135461, 322248046, -134418294, 63839077, -11819846, 2019758, 213107, -101440, 22238, -3229, 86, -4, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 - 4*x^14 + 86*x^13 - 3229*x^12 + 22238*x^11 - 101440*x^10 + 213107*x^9 + 2019758*x^8 - 11819846*x^7 + 63839077*x^6 - 134418294*x^5 + 322248046*x^4 - 455135461*x^3 - 469772490*x^2 + 689350461*x + 431493817)
 
gp: K = bnfinit(x^16 - 8*x^15 - 4*x^14 + 86*x^13 - 3229*x^12 + 22238*x^11 - 101440*x^10 + 213107*x^9 + 2019758*x^8 - 11819846*x^7 + 63839077*x^6 - 134418294*x^5 + 322248046*x^4 - 455135461*x^3 - 469772490*x^2 + 689350461*x + 431493817, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} - 4 x^{14} + 86 x^{13} - 3229 x^{12} + 22238 x^{11} - 101440 x^{10} + 213107 x^{9} + 2019758 x^{8} - 11819846 x^{7} + 63839077 x^{6} - 134418294 x^{5} + 322248046 x^{4} - 455135461 x^{3} - 469772490 x^{2} + 689350461 x + 431493817 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(4411897179443060791351047777488896=2^{12}\cdot 41^{14}\cdot 73^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $126.70$
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:  $2, 41, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{2}{5} a^{11} - \frac{2}{5} a^{10} - \frac{2}{5} a^{8} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{13} - \frac{1}{5} a^{11} - \frac{1}{5} a^{10} - \frac{2}{5} a^{9} - \frac{1}{5} a^{8} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{1}{5}$, $\frac{1}{5} a^{14} + \frac{1}{5} a^{11} + \frac{1}{5} a^{10} - \frac{1}{5} a^{9} + \frac{2}{5} a^{8} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{1067771103851925522521090214184548157069047393357777030358182233305705} a^{15} + \frac{52618223562355534639567951066372861086915300563579435107104579936524}{1067771103851925522521090214184548157069047393357777030358182233305705} a^{14} - \frac{2871793813131015395274280301250790239887918633333948889566794636812}{213554220770385104504218042836909631413809478671555406071636446661141} a^{13} + \frac{22765874755292626180671791749075748585794870408576983387917289973289}{1067771103851925522521090214184548157069047393357777030358182233305705} a^{12} - \frac{266400573753264560441042047796933983433041997105717721310184159649049}{1067771103851925522521090214184548157069047393357777030358182233305705} a^{11} + \frac{250886580918312903369257781147336835608128416007397151181083474989942}{1067771103851925522521090214184548157069047393357777030358182233305705} a^{10} - \frac{296595830131973587272229419186018434422433668086493904735091045790577}{1067771103851925522521090214184548157069047393357777030358182233305705} a^{9} - \frac{130511066635428017137996106555166477859822371182605137805044926502134}{1067771103851925522521090214184548157069047393357777030358182233305705} a^{8} + \frac{25468036469394593665369023845591126875637665872239084656990130547329}{213554220770385104504218042836909631413809478671555406071636446661141} a^{7} + \frac{103363857951367669870604910230644590950662937435049385269591182251581}{213554220770385104504218042836909631413809478671555406071636446661141} a^{6} + \frac{192158160083839449944983909248696872833333306844271659234478491630751}{1067771103851925522521090214184548157069047393357777030358182233305705} a^{5} - \frac{150536041023690065330875391203704277195224611552269285093200338775586}{1067771103851925522521090214184548157069047393357777030358182233305705} a^{4} + \frac{239647141492227099718161038783440864894969761294762380129617677863319}{1067771103851925522521090214184548157069047393357777030358182233305705} a^{3} + \frac{169685916424931261494274690783568012677336499172169766424698624677496}{1067771103851925522521090214184548157069047393357777030358182233305705} a^{2} - \frac{90793989983768941578406180072547305325700059943621677985588321904953}{1067771103851925522521090214184548157069047393357777030358182233305705} a + \frac{231193440972258102250072950118854582848165175772848311959298398668346}{1067771103851925522521090214184548157069047393357777030358182233305705}$

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:  $11$
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:  \( 65042665241.1 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.C_2^2:D_4$ (as 16T209):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_4.C_2^2:D_4$
Character table for $C_4.C_2^2:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.13448.1, 4.4.68921.1, 4.4.551368.1, 8.8.304006671424.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
41Data not computed
$73$73.4.0.1$x^{4} - x + 13$$1$$4$$0$$C_4$$[\ ]^{4}$
73.4.2.2$x^{4} - 73 x^{2} + 58619$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
73.4.0.1$x^{4} - x + 13$$1$$4$$0$$C_4$$[\ ]^{4}$
73.4.2.2$x^{4} - 73 x^{2} + 58619$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$