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Properties

Label 16.0.14313275354...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{4}\cdot 5^{12}\cdot 11^{2}\cdot 29^{4}\cdot 71^{2}$
Root discriminant $66.41$
Ramified primes $2, 3, 5, 11, 29, 71$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group 16T781

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1852321, 6005216, 8231782, 6674934, 4135411, 2249700, 934638, 244660, 27997, -15320, -10562, -1300, 331, 54, 2, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 2*x^14 + 54*x^13 + 331*x^12 - 1300*x^11 - 10562*x^10 - 15320*x^9 + 27997*x^8 + 244660*x^7 + 934638*x^6 + 2249700*x^5 + 4135411*x^4 + 6674934*x^3 + 8231782*x^2 + 6005216*x + 1852321)
 
gp: K = bnfinit(x^16 - 4*x^15 + 2*x^14 + 54*x^13 + 331*x^12 - 1300*x^11 - 10562*x^10 - 15320*x^9 + 27997*x^8 + 244660*x^7 + 934638*x^6 + 2249700*x^5 + 4135411*x^4 + 6674934*x^3 + 8231782*x^2 + 6005216*x + 1852321, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 2 x^{14} + 54 x^{13} + 331 x^{12} - 1300 x^{11} - 10562 x^{10} - 15320 x^{9} + 27997 x^{8} + 244660 x^{7} + 934638 x^{6} + 2249700 x^{5} + 4135411 x^{4} + 6674934 x^{3} + 8231782 x^{2} + 6005216 x + 1852321 \)

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:  \(143132753548578816000000000000=2^{24}\cdot 3^{4}\cdot 5^{12}\cdot 11^{2}\cdot 29^{4}\cdot 71^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $66.41$
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:  $2, 3, 5, 11, 29, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{11341} a^{14} + \frac{546}{11341} a^{13} - \frac{4456}{11341} a^{12} - \frac{346}{1031} a^{11} + \frac{1401}{11341} a^{10} - \frac{5106}{11341} a^{9} + \frac{5057}{11341} a^{8} - \frac{5436}{11341} a^{7} - \frac{45}{1031} a^{6} + \frac{323}{11341} a^{5} - \frac{1902}{11341} a^{4} + \frac{4500}{11341} a^{3} - \frac{1527}{11341} a^{2} + \frac{5235}{11341} a - \frac{4276}{11341}$, $\frac{1}{76959643321648124930797078489755516517859} a^{15} + \frac{741816175930985792649036898017687980}{76959643321648124930797078489755516517859} a^{14} - \frac{955587671446873015257134476022830255053}{76959643321648124930797078489755516517859} a^{13} - \frac{5217687763885650307683102011248347937314}{76959643321648124930797078489755516517859} a^{12} + \frac{3088456408811722174810272933318292249332}{76959643321648124930797078489755516517859} a^{11} + \frac{11366726255400472498571937787656707579563}{76959643321648124930797078489755516517859} a^{10} + \frac{8540577329473574549021753268221936034448}{76959643321648124930797078489755516517859} a^{9} - \frac{7311701195189570149368781752557168659761}{76959643321648124930797078489755516517859} a^{8} - \frac{10234246652676005201651546390647610853254}{76959643321648124930797078489755516517859} a^{7} - \frac{21284772188538044065431299055870388829284}{76959643321648124930797078489755516517859} a^{6} + \frac{12933628336601246565026333496363527595703}{76959643321648124930797078489755516517859} a^{5} - \frac{21171280414614836780413960796960309267160}{76959643321648124930797078489755516517859} a^{4} - \frac{19166863278607590269507585294558581985004}{76959643321648124930797078489755516517859} a^{3} + \frac{26160791983891556997311749657940408084359}{76959643321648124930797078489755516517859} a^{2} + \frac{9080115343761059775258297026471853147562}{76959643321648124930797078489755516517859} a + \frac{16227548365347722695219537524526773440392}{76959643321648124930797078489755516517859}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  \( -\frac{17649278037113435168931860429879036}{116429112438196860712249740529130887319} a^{15} + \frac{103334189534636651432667556432502056}{116429112438196860712249740529130887319} a^{14} - \frac{218248593754290574575985312109769416}{116429112438196860712249740529130887319} a^{13} - \frac{605832793152474554561462076084349772}{116429112438196860712249740529130887319} a^{12} - \frac{4566269038354533301402491152608079364}{116429112438196860712249740529130887319} a^{11} + \frac{2872068708590167332920988790019387604}{10584464767108805519295430957193717029} a^{10} + \frac{129955343755578678007795981625425392400}{116429112438196860712249740529130887319} a^{9} + \frac{12114432418633141886281571815977446401}{116429112438196860712249740529130887319} a^{8} - \frac{567645053239033298664709691629881734592}{116429112438196860712249740529130887319} a^{7} - \frac{3234976968716221302367262656260883889700}{116429112438196860712249740529130887319} a^{6} - \frac{10241014419562657591658180771890504918708}{116429112438196860712249740529130887319} a^{5} - \frac{19297996688814149271354788666918474945732}{116429112438196860712249740529130887319} a^{4} - \frac{33161512426760496676927190431446716499288}{116429112438196860712249740529130887319} a^{3} - \frac{4512599660633189784439270535639531882464}{10584464767108805519295430957193717029} a^{2} - \frac{41494568254619228339149114513981868671012}{116429112438196860712249740529130887319} a - \frac{12732717736162650001809500130870780999376}{116429112438196860712249740529130887319} \) (order $10$)
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:  \( 43936859.9817 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T781:

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 t16n781 are not computed
Character table for t16n781 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.4.58000.1, 4.0.11600.1, 8.0.3364000000.5

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$3.8.0.1$x^{8} - x^{3} + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
5Data not computed
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$71$$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$