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Properties

Label 16.0.50345559205...8125.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 101^{6}\cdot 181^{5}$
Root discriminant $95.80$
Ramified primes $5, 101, 181$
Class number $173408$ (GRH)
Class group $[2, 2, 2, 2, 10838]$ (GRH)
Galois group 16T994

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1503349111, -981056326, 1090515202, -497742168, 306325756, -105886889, 45602362, -12138424, 4005147, -801805, 214133, -30262, 6872, -604, 124, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 124*x^14 - 604*x^13 + 6872*x^12 - 30262*x^11 + 214133*x^10 - 801805*x^9 + 4005147*x^8 - 12138424*x^7 + 45602362*x^6 - 105886889*x^5 + 306325756*x^4 - 497742168*x^3 + 1090515202*x^2 - 981056326*x + 1503349111)
 
gp: K = bnfinit(x^16 - 5*x^15 + 124*x^14 - 604*x^13 + 6872*x^12 - 30262*x^11 + 214133*x^10 - 801805*x^9 + 4005147*x^8 - 12138424*x^7 + 45602362*x^6 - 105886889*x^5 + 306325756*x^4 - 497742168*x^3 + 1090515202*x^2 - 981056326*x + 1503349111, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 124 x^{14} - 604 x^{13} + 6872 x^{12} - 30262 x^{11} + 214133 x^{10} - 801805 x^{9} + 4005147 x^{8} - 12138424 x^{7} + 45602362 x^{6} - 105886889 x^{5} + 306325756 x^{4} - 497742168 x^{3} + 1090515202 x^{2} - 981056326 x + 1503349111 \)

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:  \(50345559205004654867065673828125=5^{12}\cdot 101^{6}\cdot 181^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $95.80$
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:  $5, 101, 181$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{11} - \frac{1}{5} a^{10} + \frac{2}{5} a^{9} + \frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{11} + \frac{1}{5} a^{10} - \frac{2}{5} a^{9} - \frac{2}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{14} - \frac{1}{5} a^{11} + \frac{1}{5} a^{10} + \frac{2}{5} a^{9} - \frac{2}{5} a^{8} - \frac{1}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{11182738881359957778059834289584179499333634660796175} a^{15} + \frac{732057641134838016485249074108568980238383129549436}{11182738881359957778059834289584179499333634660796175} a^{14} - \frac{206485425026440749515358320058182940968544913749788}{2236547776271991555611966857916835899866726932159235} a^{13} - \frac{17032563100680156693796651024208525281058464717029}{1016612625578177979823621299053107227212148605526925} a^{12} - \frac{2367161894931331562075432079926953301681647605050162}{11182738881359957778059834289584179499333634660796175} a^{11} - \frac{2827075379129831731983427483435711750072475970627359}{11182738881359957778059834289584179499333634660796175} a^{10} - \frac{4573282984172181392073160636488712962049016893466296}{11182738881359957778059834289584179499333634660796175} a^{9} + \frac{2832662107132551931443759343196681108685951065615519}{11182738881359957778059834289584179499333634660796175} a^{8} + \frac{3357407379446066672929480366583495261667585939995101}{11182738881359957778059834289584179499333634660796175} a^{7} - \frac{3486444364148335331637991182615796705222928648901878}{11182738881359957778059834289584179499333634660796175} a^{6} - \frac{511099432072920736781089107032783310414663600603101}{11182738881359957778059834289584179499333634660796175} a^{5} + \frac{3507215710867919152081573025763029864646907586244}{40664505023127119192944851962124289088485944221077} a^{4} + \frac{42409044982265587085936205918949811918971446588301}{92419329598016179983965572641191566110195327775175} a^{3} - \frac{39921800645940485010234139190423552268058081804602}{1016612625578177979823621299053107227212148605526925} a^{2} + \frac{258432420914759589016359400866287089346043835347007}{2236547776271991555611966857916835899866726932159235} a - \frac{423380853285922700546935708535496343168477109317276}{1016612625578177979823621299053107227212148605526925}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{10838}$, which has order $173408$ (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:  \( 6824.94222592 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T994:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 29 conjugacy class representatives for t16n994
Character table for t16n994 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2525.1, 8.8.1153988125.1

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
Arithmetically equvalently 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 $16$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
101Data not computed
181Data not computed