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Properties

Label 16.8.10176056674...5881.2
Degree $16$
Signature $[8, 4]$
Discriminant $53^{14}\cdot 97^{10}$
Root discriminant $562.96$
Ramified primes $53, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_2\times OD_{16}).C_2$ (as 16T123)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5550170605771, 8392420893314, 2958573104934, -190307136710, -340510468007, -106391518916, -13311677332, 1508236734, 609769156, 64766322, 827806, -480761, -41565, 432, -103, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 103*x^14 + 432*x^13 - 41565*x^12 - 480761*x^11 + 827806*x^10 + 64766322*x^9 + 609769156*x^8 + 1508236734*x^7 - 13311677332*x^6 - 106391518916*x^5 - 340510468007*x^4 - 190307136710*x^3 + 2958573104934*x^2 + 8392420893314*x + 5550170605771)
 
gp: K = bnfinit(x^16 - x^15 - 103*x^14 + 432*x^13 - 41565*x^12 - 480761*x^11 + 827806*x^10 + 64766322*x^9 + 609769156*x^8 + 1508236734*x^7 - 13311677332*x^6 - 106391518916*x^5 - 340510468007*x^4 - 190307136710*x^3 + 2958573104934*x^2 + 8392420893314*x + 5550170605771, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 103 x^{14} + 432 x^{13} - 41565 x^{12} - 480761 x^{11} + 827806 x^{10} + 64766322 x^{9} + 609769156 x^{8} + 1508236734 x^{7} - 13311677332 x^{6} - 106391518916 x^{5} - 340510468007 x^{4} - 190307136710 x^{3} + 2958573104934 x^{2} + 8392420893314 x + 5550170605771 \)

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:  \(101760566745940911333128943658372964904935881=53^{14}\cdot 97^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $562.96$
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:  $53, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{11} a^{13} + \frac{4}{11} a^{12} - \frac{2}{11} a^{11} - \frac{1}{11} a^{10} - \frac{5}{11} a^{9} - \frac{4}{11} a^{8} + \frac{4}{11} a^{7} + \frac{4}{11} a^{6} - \frac{5}{11} a^{5} - \frac{1}{11} a^{4} + \frac{4}{11} a^{3} - \frac{1}{11} a^{2} + \frac{4}{11} a$, $\frac{1}{163941217} a^{14} - \frac{4473629}{163941217} a^{13} + \frac{50697121}{163941217} a^{12} - \frac{7727833}{163941217} a^{11} + \frac{18423476}{163941217} a^{10} + \frac{14594549}{163941217} a^{9} - \frac{1719463}{14903747} a^{8} - \frac{24423622}{163941217} a^{7} - \frac{3050400}{7127879} a^{6} + \frac{16948772}{163941217} a^{5} - \frac{28167763}{163941217} a^{4} + \frac{4651917}{9643601} a^{3} - \frac{64327601}{163941217} a^{2} - \frac{28988285}{163941217} a - \frac{156986}{317101}$, $\frac{1}{335754472126996084110727873169115683479049307877394640464028106009015373932516338102878995} a^{15} - \frac{186344198238265541522800913347948387743379264041339908919718963244338466759286706}{67150894425399216822145574633823136695809861575478928092805621201803074786503267620575799} a^{14} - \frac{2375487365519982144926685493156114570956670442147298845782964214224828787916431387653313}{335754472126996084110727873169115683479049307877394640464028106009015373932516338102878995} a^{13} - \frac{130750788428476595784657092451913748136573923267227986670783291794910461592559586499038716}{335754472126996084110727873169115683479049307877394640464028106009015373932516338102878995} a^{12} - \frac{131839375434388828758037088072935883808965238837163852285139939425985751062648153761953571}{335754472126996084110727873169115683479049307877394640464028106009015373932516338102878995} a^{11} + \frac{9927910465855842595048568035221604285585446221945471914526861926898454821296546898672023}{30523133829726916737338897560828698498095391625217694587638918728092306721137848918443545} a^{10} - \frac{125782391045681931752513712460368975540695867036624525928494924934230709126035208744065421}{335754472126996084110727873169115683479049307877394640464028106009015373932516338102878995} a^{9} - \frac{10339464949855295299407602436190690458213877436593645908960816270258269915870182558911604}{30523133829726916737338897560828698498095391625217694587638918728092306721137848918443545} a^{8} - \frac{140976076335170061267679302188335082517339688188999816100098979516928285125698589514450168}{335754472126996084110727873169115683479049307877394640464028106009015373932516338102878995} a^{7} + \frac{74915019707987767186483081369097838595434909466799422848034508994391885351093460706580446}{335754472126996084110727873169115683479049307877394640464028106009015373932516338102878995} a^{6} + \frac{155114212348329965757614093174188066973096958879319139773129436495998875756060556042867419}{335754472126996084110727873169115683479049307877394640464028106009015373932516338102878995} a^{5} + \frac{52834680408906478762308206632538793505214653059394585530750300588976124075308613055807223}{335754472126996084110727873169115683479049307877394640464028106009015373932516338102878995} a^{4} - \frac{7794134008810982178489681270193130408404029724366268267454054646430686996953550670419769}{30523133829726916737338897560828698498095391625217694587638918728092306721137848918443545} a^{3} + \frac{8645992360842938213577850227713485064383179021567031645556477940058949262828922398486339}{17671288006684004426880414377321878077844700414599717919159374000474493364869280952783105} a^{2} - \frac{28290510380456519449079078564546003721405066219723656488316418438854579464173063885965162}{67150894425399216822145574633823136695809861575478928092805621201803074786503267620575799} a + \frac{13422137629858360972215079130757097899653059519834709011838764029331509227921176058103}{34180441018731149761857668041241543670879497900579725182126448743664397224118531823565}$

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:  $11$
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:  \( 27033148371000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times OD_{16}).C_2$ (as 16T123):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $(C_2\times OD_{16}).C_2$
Character table for $(C_2\times OD_{16}).C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{53}) \), \(\Q(\sqrt{5141}) \), \(\Q(\sqrt{97}) \), 4.4.14441069.1 x2, 4.4.1400783693.1 x2, \(\Q(\sqrt{53}, \sqrt{97})\), 8.8.1962194954574718249.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{16}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$53$53.8.7.2$x^{8} - 212$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
53.8.7.2$x^{8} - 212$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$97$97.8.6.2$x^{8} + 873 x^{4} + 235225$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
97.8.4.1$x^{8} + 432814 x^{4} - 912673 x^{2} + 46831989649$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$