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Properties

Label 16.12.3649495746...4416.1
Degree $16$
Signature $[12, 2]$
Discriminant $2^{24}\cdot 13^{8}\cdot 41^{3}\cdot 47^{4}\cdot 42961^{3}$
Root discriminant $395.95$
Ramified primes $2, 13, 41, 47, 42961$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1806

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![80523074268737, 11323231138444, -10805487081588, -1030653409038, 502985664539, 15514855228, -6328543931, 953569804, -191000463, -31268076, 5923877, 183856, -33949, 1498, -194, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 - 194*x^14 + 1498*x^13 - 33949*x^12 + 183856*x^11 + 5923877*x^10 - 31268076*x^9 - 191000463*x^8 + 953569804*x^7 - 6328543931*x^6 + 15514855228*x^5 + 502985664539*x^4 - 1030653409038*x^3 - 10805487081588*x^2 + 11323231138444*x + 80523074268737)
 
gp: K = bnfinit(x^16 - 8*x^15 - 194*x^14 + 1498*x^13 - 33949*x^12 + 183856*x^11 + 5923877*x^10 - 31268076*x^9 - 191000463*x^8 + 953569804*x^7 - 6328543931*x^6 + 15514855228*x^5 + 502985664539*x^4 - 1030653409038*x^3 - 10805487081588*x^2 + 11323231138444*x + 80523074268737, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} - 194 x^{14} + 1498 x^{13} - 33949 x^{12} + 183856 x^{11} + 5923877 x^{10} - 31268076 x^{9} - 191000463 x^{8} + 953569804 x^{7} - 6328543931 x^{6} + 15514855228 x^{5} + 502985664539 x^{4} - 1030653409038 x^{3} - 10805487081588 x^{2} + 11323231138444 x + 80523074268737 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(364949574671621174049498480790687937724416=2^{24}\cdot 13^{8}\cdot 41^{3}\cdot 47^{4}\cdot 42961^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $395.95$
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:  $2, 13, 41, 47, 42961$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{1761401} a^{12} - \frac{6}{1761401} a^{11} + \frac{880493}{1761401} a^{10} - \frac{879608}{1761401} a^{9} + \frac{518982}{1761401} a^{8} - \frac{321148}{1761401} a^{7} - \frac{725492}{1761401} a^{6} + \frac{663047}{1761401} a^{5} + \frac{351291}{1761401} a^{4} - \frac{423150}{1761401} a^{3} + \frac{494543}{1761401} a^{2} - \frac{13633}{42961} a + \frac{874092}{1761401}$, $\frac{1}{1761401} a^{13} + \frac{880457}{1761401} a^{11} + \frac{880548}{1761401} a^{10} + \frac{525537}{1761401} a^{9} - \frac{730058}{1761401} a^{8} + \frac{870422}{1761401} a^{7} - \frac{167103}{1761401} a^{6} + \frac{806771}{1761401} a^{5} - \frac{76805}{1761401} a^{4} - \frac{282956}{1761401} a^{3} + \frac{646904}{1761401} a^{2} - \frac{718225}{1761401} a - \frac{39651}{1761401}$, $\frac{1}{17750473466362995150364428733019304203924} a^{14} - \frac{1}{2535781923766142164337775533288472029132} a^{13} - \frac{89918534241001602741681540650911}{4437618366590748787591107183254826050981} a^{12} + \frac{2158044821784038465800356975621955}{17750473466362995150364428733019304203924} a^{11} - \frac{271958987409504677757051761483552464305}{1267890961883071082168887766644236014566} a^{10} + \frac{1286635870224799272276591400890425096005}{17750473466362995150364428733019304203924} a^{9} - \frac{367685877704718923233307260860169008836}{4437618366590748787591107183254826050981} a^{8} - \frac{1836817439580253237503508427653114592863}{17750473466362995150364428733019304203924} a^{7} + \frac{2238974889233549622401280603986451557961}{8875236733181497575182214366509652101962} a^{6} + \frac{1948953312853304946066041699284106129455}{17750473466362995150364428733019304203924} a^{5} - \frac{3218364901637324433746446561326875914465}{8875236733181497575182214366509652101962} a^{4} - \frac{6924658443161581451462949807591660656627}{17750473466362995150364428733019304203924} a^{3} + \frac{1328531892358409956883161833984180094263}{8875236733181497575182214366509652101962} a^{2} - \frac{7644702990427278609316696480656974138103}{17750473466362995150364428733019304203924} a + \frac{634411374269740111615669833085926477655}{17750473466362995150364428733019304203924}$, $\frac{1}{17750473466362995150364428733019304203924} a^{15} - \frac{359674136964006410966726162603693}{17750473466362995150364428733019304203924} a^{13} - \frac{359674136964006410966726162603553}{17750473466362995150364428733019304203924} a^{12} - \frac{543915816774187571475637722610129306655}{2535781923766142164337775533288472029132} a^{11} - \frac{7614871429543663997550052491478412201961}{17750473466362995150364428733019304203924} a^{10} + \frac{7535707580754719213002910762792299636691}{17750473466362995150364428733019304203924} a^{9} + \frac{5618451451050612062328317001281457363653}{17750473466362995150364428733019304203924} a^{8} - \frac{8379772298594673417721997785598899034119}{17750473466362995150364428733019304203924} a^{7} - \frac{2206345170602990641044887310944180466939}{17750473466362995150364428733019304203924} a^{6} + \frac{7205943386698485754969398772334991077255}{17750473466362995150364428733019304203924} a^{5} + \frac{1269653333004861927180084532889989152635}{17750473466362995150364428733019304203924} a^{4} + \frac{7435875081674735204618961213884648203909}{17750473466362995150364428733019304203924} a^{3} - \frac{6795729963772534363316859537897757022345}{17750473466362995150364428733019304203924} a^{2} + \frac{186455420183887648746040333772510061353}{8875236733181497575182214366509652101962} a + \frac{634411374269740111615669833085926477655}{2535781923766142164337775533288472029132}$

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:  $13$
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:  \( 2622536488780000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1806:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 24576
The 88 conjugacy class representatives for t16n1806 are not computed
Character table for t16n1806 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 8.8.258421755904.1

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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ R ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }$

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$2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.12.8.1$x^{12} - 39 x^{9} - 338 x^{6} + 10985 x^{3} + 228488$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
$41$$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.6.0.1$x^{6} - x + 7$$1$$6$$0$$C_6$$[\ ]^{6}$
41.6.3.2$x^{6} - 1681 x^{2} + 895973$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$47$47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.4.0.1$x^{4} - x + 39$$1$$4$$0$$C_4$$[\ ]^{4}$
47.8.4.1$x^{8} + 172302 x^{4} - 103823 x^{2} + 7421994801$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
42961Data not computed