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Properties

Label 16.0.34153198773...9969.1
Degree $16$
Signature $[0, 8]$
Discriminant $23^{4}\cdot 73^{14}$
Root discriminant $93.51$
Ramified primes $23, 73$
Class number $89$ (GRH)
Class group $[89]$ (GRH)
Galois group $C_2^2 : C_8$ (as 16T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![137861, -10639, 294148, -397603, 39313, -425376, 776806, -306130, -98968, 83350, -4322, -6688, 1029, 209, -44, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 44*x^14 + 209*x^13 + 1029*x^12 - 6688*x^11 - 4322*x^10 + 83350*x^9 - 98968*x^8 - 306130*x^7 + 776806*x^6 - 425376*x^5 + 39313*x^4 - 397603*x^3 + 294148*x^2 - 10639*x + 137861)
 
gp: K = bnfinit(x^16 - 3*x^15 - 44*x^14 + 209*x^13 + 1029*x^12 - 6688*x^11 - 4322*x^10 + 83350*x^9 - 98968*x^8 - 306130*x^7 + 776806*x^6 - 425376*x^5 + 39313*x^4 - 397603*x^3 + 294148*x^2 - 10639*x + 137861, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 44 x^{14} + 209 x^{13} + 1029 x^{12} - 6688 x^{11} - 4322 x^{10} + 83350 x^{9} - 98968 x^{8} - 306130 x^{7} + 776806 x^{6} - 425376 x^{5} + 39313 x^{4} - 397603 x^{3} + 294148 x^{2} - 10639 x + 137861 \)

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:  \(34153198773896725121035596949969=23^{4}\cdot 73^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $93.51$
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:  $23, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4}$, $\frac{1}{4} a^{7} - \frac{1}{4} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{16} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{1}{16} a^{6} + \frac{3}{16} a^{3} - \frac{3}{8} a^{2} - \frac{3}{8} a - \frac{3}{16}$, $\frac{1}{16} a^{10} - \frac{1}{8} a^{8} - \frac{1}{16} a^{7} - \frac{1}{8} a^{6} + \frac{3}{16} a^{4} - \frac{3}{8} a^{2} - \frac{3}{16} a - \frac{3}{8}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{8} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{3}{16} a^{5} - \frac{3}{16} a^{2} - \frac{3}{8} a - \frac{3}{8}$, $\frac{1}{32} a^{12} + \frac{1}{16} a^{6} - \frac{3}{32}$, $\frac{1}{192} a^{13} - \frac{1}{64} a^{12} - \frac{1}{96} a^{11} - \frac{1}{32} a^{10} + \frac{1}{32} a^{8} + \frac{1}{48} a^{7} + \frac{5}{96} a^{6} + \frac{13}{96} a^{5} - \frac{3}{32} a^{4} + \frac{1}{6} a^{3} + \frac{25}{96} a^{2} + \frac{67}{192} a + \frac{25}{192}$, $\frac{1}{576} a^{14} - \frac{1}{576} a^{13} - \frac{1}{288} a^{12} - \frac{5}{288} a^{11} + \frac{1}{48} a^{10} + \frac{1}{96} a^{9} + \frac{1}{36} a^{8} - \frac{3}{32} a^{7} - \frac{19}{288} a^{6} - \frac{31}{288} a^{5} - \frac{31}{144} a^{4} - \frac{13}{96} a^{3} - \frac{121}{576} a^{2} - \frac{19}{192} a - \frac{1}{9}$, $\frac{1}{688299834589821100275381589226112} a^{15} + \frac{40199758602778416362983989535}{172074958647455275068845397306528} a^{14} + \frac{703305226874824500542973156775}{344149917294910550137690794613056} a^{13} + \frac{7619598370678479241457084516629}{688299834589821100275381589226112} a^{12} + \frac{1002648811706837891659761563693}{86037479323727637534422698653264} a^{11} - \frac{392836192699298408347799544383}{57358319549151758356281799102176} a^{10} - \frac{6260120596010178185911929329611}{344149917294910550137690794613056} a^{9} + \frac{10105320617199281044728284445901}{86037479323727637534422698653264} a^{8} - \frac{1363885502046364275917065457413}{21509369830931909383605674663316} a^{7} - \frac{1044779830400833024208446563881}{12746293233144835190284844244928} a^{6} + \frac{6851821169236362536104345783199}{86037479323727637534422698653264} a^{5} - \frac{38037809415030579234346518133847}{172074958647455275068845397306528} a^{4} + \frac{122773128013476850485898123090805}{688299834589821100275381589226112} a^{3} + \frac{50598522762973172473627511820487}{172074958647455275068845397306528} a^{2} - \frac{74497619297810949328546265574287}{344149917294910550137690794613056} a + \frac{41173515117641469536753967696145}{688299834589821100275381589226112}$

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

Class group and class number

$C_{89}$, which has order $89$ (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:  \( 93249594.3322 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_8$ (as 16T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2^2 : C_8$
Character table for $C_2^2 : C_8$

Intermediate fields

\(\Q(\sqrt{73}) \), 4.2.122567.1, 4.4.389017.1, 4.2.8947391.1, 8.4.5844073816602313.1, 8.0.11047398519097.1, 8.4.80055805706881.1

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 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$23$23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.8.4.1$x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
73Data not computed