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Properties

Label 16.8.46705271798...0000.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{24}\cdot 5^{14}\cdot 61^{6}\cdot 97^{4}$
Root discriminant $169.56$
Ramified primes $2, 5, 61, 97$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T852

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3809588801, 7365907158, -15160486496, 6851881008, -1406195937, 191867768, 129179506, -27974016, 13800238, -1018792, 141964, 27414, -10012, 486, -84, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 84*x^14 + 486*x^13 - 10012*x^12 + 27414*x^11 + 141964*x^10 - 1018792*x^9 + 13800238*x^8 - 27974016*x^7 + 129179506*x^6 + 191867768*x^5 - 1406195937*x^4 + 6851881008*x^3 - 15160486496*x^2 + 7365907158*x + 3809588801)
 
gp: K = bnfinit(x^16 - 6*x^15 - 84*x^14 + 486*x^13 - 10012*x^12 + 27414*x^11 + 141964*x^10 - 1018792*x^9 + 13800238*x^8 - 27974016*x^7 + 129179506*x^6 + 191867768*x^5 - 1406195937*x^4 + 6851881008*x^3 - 15160486496*x^2 + 7365907158*x + 3809588801, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 84 x^{14} + 486 x^{13} - 10012 x^{12} + 27414 x^{11} + 141964 x^{10} - 1018792 x^{9} + 13800238 x^{8} - 27974016 x^{7} + 129179506 x^{6} + 191867768 x^{5} - 1406195937 x^{4} + 6851881008 x^{3} - 15160486496 x^{2} + 7365907158 x + 3809588801 \)

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:  \(467052717980688838758400000000000000=2^{24}\cdot 5^{14}\cdot 61^{6}\cdot 97^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $169.56$
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:  $2, 5, 61, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2393410948522979979528233589867174121114497656597299129887624548135681959346083981} a^{15} - \frac{1041825879215354574196994506873140538449243841585653299150816778947903610986876324}{2393410948522979979528233589867174121114497656597299129887624548135681959346083981} a^{14} + \frac{870332756409111080595657387151713851317217949360023459755962754792174557981730311}{2393410948522979979528233589867174121114497656597299129887624548135681959346083981} a^{13} - \frac{921892053035259993610723692995568390603662612469517930199621191697919051409613990}{2393410948522979979528233589867174121114497656597299129887624548135681959346083981} a^{12} - \frac{165705888461591788351546067785306410324031437743196320421051707800096512897697913}{2393410948522979979528233589867174121114497656597299129887624548135681959346083981} a^{11} - \frac{67980829685192149657945905491381253406605698196738330060284259891015372236985635}{2393410948522979979528233589867174121114497656597299129887624548135681959346083981} a^{10} + \frac{51094790233776949391340873368735615249886295263485522287377463553609065253155403}{2393410948522979979528233589867174121114497656597299129887624548135681959346083981} a^{9} - \frac{796952524708172118103568647119772904264338708414154804496522128179987954890143275}{2393410948522979979528233589867174121114497656597299129887624548135681959346083981} a^{8} - \frac{4187419489052309225817704172164606511560230745946108919582542336849709139550382}{2393410948522979979528233589867174121114497656597299129887624548135681959346083981} a^{7} + \frac{356889403270704787664510353649413895091709559064162628168783772423214030455903206}{2393410948522979979528233589867174121114497656597299129887624548135681959346083981} a^{6} + \frac{189930948846498026607291073019012265503096348624376687072322776310806649067559635}{2393410948522979979528233589867174121114497656597299129887624548135681959346083981} a^{5} - \frac{323878468639378391676117034679329144421535459630082085982465029671589453886884225}{2393410948522979979528233589867174121114497656597299129887624548135681959346083981} a^{4} + \frac{444326408524026410761245307483288290225042182394605134668599117496948987500987916}{2393410948522979979528233589867174121114497656597299129887624548135681959346083981} a^{3} - \frac{738619038069636845876615787046342643518949455336900551952728207760261960472510704}{2393410948522979979528233589867174121114497656597299129887624548135681959346083981} a^{2} + \frac{515417195611870766186274242456136628715835129084330547356173738095616087813054653}{2393410948522979979528233589867174121114497656597299129887624548135681959346083981} a - \frac{60820304180236096437288790570892248227389172395779903346688950350816700421731055}{2393410948522979979528233589867174121114497656597299129887624548135681959346083981}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 196656781493 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T852:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 41 conjugacy class representatives for t16n852
Character table for t16n852 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 8.8.14884000000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
2Data not computed
5Data not computed
$61$$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
61.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
61.4.3.2$x^{4} - 244$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.4$x^{4} + 488$$4$$1$$3$$C_4$$[\ ]_{4}$
$97$97.8.4.2$x^{8} - 912673 x^{2} + 2036173463$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
97.8.0.1$x^{8} - x + 84$$1$$8$$0$$C_8$$[\ ]^{8}$