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Properties

Label 16.0.74256122186...2833.2
Degree $16$
Signature $[0, 8]$
Discriminant $13^{15}\cdot 29^{9}$
Root discriminant $73.61$
Ramified primes $13, 29$
Class number $16$ (GRH)
Class group $[4, 4]$ (GRH)
Galois group 16T1192

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10037, 23943, -21135, 4289, 39416, -150618, 220597, -196755, 139386, -69069, 27495, -7826, 1885, -300, 68, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 68*x^14 - 300*x^13 + 1885*x^12 - 7826*x^11 + 27495*x^10 - 69069*x^9 + 139386*x^8 - 196755*x^7 + 220597*x^6 - 150618*x^5 + 39416*x^4 + 4289*x^3 - 21135*x^2 + 23943*x + 10037)
 
gp: K = bnfinit(x^16 - 3*x^15 + 68*x^14 - 300*x^13 + 1885*x^12 - 7826*x^11 + 27495*x^10 - 69069*x^9 + 139386*x^8 - 196755*x^7 + 220597*x^6 - 150618*x^5 + 39416*x^4 + 4289*x^3 - 21135*x^2 + 23943*x + 10037, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 68 x^{14} - 300 x^{13} + 1885 x^{12} - 7826 x^{11} + 27495 x^{10} - 69069 x^{9} + 139386 x^{8} - 196755 x^{7} + 220597 x^{6} - 150618 x^{5} + 39416 x^{4} + 4289 x^{3} - 21135 x^{2} + 23943 x + 10037 \)

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:  \(742561221860627884726497942833=13^{15}\cdot 29^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.61$
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:  $13, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{6} a^{12} - \frac{1}{2} a^{10} + \frac{1}{3} a^{9} - \frac{1}{6} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{6} a^{4} - \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{6} a^{13} - \frac{1}{2} a^{11} + \frac{1}{3} a^{10} - \frac{1}{6} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{6} a^{5} - \frac{1}{3} a^{2} + \frac{1}{6} a$, $\frac{1}{522} a^{14} - \frac{17}{261} a^{13} - \frac{20}{261} a^{12} - \frac{47}{261} a^{11} - \frac{23}{87} a^{10} + \frac{118}{261} a^{9} + \frac{5}{261} a^{8} + \frac{71}{174} a^{7} - \frac{101}{261} a^{6} - \frac{41}{522} a^{5} + \frac{103}{522} a^{4} - \frac{2}{9} a^{3} - \frac{67}{174} a^{2} + \frac{98}{261} a - \frac{187}{522}$, $\frac{1}{35827430930717509993135474216693624518} a^{15} + \frac{12204926686415954647592617908798374}{17913715465358754996567737108346812259} a^{14} + \frac{39767329741301591572303112197991521}{3980825658968612221459497135188180502} a^{13} + \frac{700715453118333069042040426292870435}{11942476976905836664378491405564541506} a^{12} - \frac{329688084787400126016822812528185357}{1235428652783362413556395662644607742} a^{11} - \frac{3108177359937297517789600053461280379}{35827430930717509993135474216693624518} a^{10} + \frac{16408433620394159548793620610445969551}{35827430930717509993135474216693624518} a^{9} + \frac{3566219454190479921275315622883806184}{17913715465358754996567737108346812259} a^{8} + \frac{14864264491889894763007835803678467221}{35827430930717509993135474216693624518} a^{7} + \frac{9071207344779307705630152593379019439}{35827430930717509993135474216693624518} a^{6} + \frac{5491341739104713488888260523641969547}{11942476976905836664378491405564541506} a^{5} - \frac{191709742001459598386793747636319835}{3980825658968612221459497135188180502} a^{4} - \frac{6349340824173941377071181648799304139}{35827430930717509993135474216693624518} a^{3} - \frac{4841154749531044358566052819726424466}{17913715465358754996567737108346812259} a^{2} + \frac{1087058678861735331903145063717880786}{17913715465358754996567737108346812259} a + \frac{5269625272410163212535955886949729105}{35827430930717509993135474216693624518}$

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

Class group and class number

$C_{4}\times C_{4}$, which has order $16$ (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:  \( 67906762.1426 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1192:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 88 conjugacy class representatives for t16n1192 are not computed
Character table for t16n1192 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 4.0.2197.1, 8.0.1530373581113.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{4}$ $16$ $16$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ $16$

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
13Data not computed
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.4.3.1$x^{4} - 29$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$