↑

Properties

Label 16.4.87678952778...7713.5
Degree $16$
Signature $[4, 6]$
Discriminant $61^{4}\cdot 97^{15}$
Root discriminant $203.67$
Ramified primes $61, 97$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group 16T1223

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9682157008, -7816045840, 2585237336, 1495727762, -1308728873, 495788623, -55725340, 3674172, -557186, 416011, -128956, -16304, 6648, -1045, 126, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 126*x^14 - 1045*x^13 + 6648*x^12 - 16304*x^11 - 128956*x^10 + 416011*x^9 - 557186*x^8 + 3674172*x^7 - 55725340*x^6 + 495788623*x^5 - 1308728873*x^4 + 1495727762*x^3 + 2585237336*x^2 - 7816045840*x + 9682157008)
 
gp: K = bnfinit(x^16 - 6*x^15 + 126*x^14 - 1045*x^13 + 6648*x^12 - 16304*x^11 - 128956*x^10 + 416011*x^9 - 557186*x^8 + 3674172*x^7 - 55725340*x^6 + 495788623*x^5 - 1308728873*x^4 + 1495727762*x^3 + 2585237336*x^2 - 7816045840*x + 9682157008, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 126 x^{14} - 1045 x^{13} + 6648 x^{12} - 16304 x^{11} - 128956 x^{10} + 416011 x^{9} - 557186 x^{8} + 3674172 x^{7} - 55725340 x^{6} + 495788623 x^{5} - 1308728873 x^{4} + 1495727762 x^{3} + 2585237336 x^{2} - 7816045840 x + 9682157008 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(8767895277848913089642817986417897713=61^{4}\cdot 97^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $203.67$
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:  $61, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{12} - \frac{1}{8} a^{6} + \frac{1}{8} a^{5} - \frac{1}{4} a^{4} + \frac{1}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{12} - \frac{1}{8} a^{7} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{1}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{8128210122924897883869187803535284645813444713751296363745999806694545272} a^{15} - \frac{31509087963107493653658622476560283057720074048960396583586312682376091}{1016026265365612235483648475441910580726680589218912045468249975836818159} a^{14} - \frac{109257109863009219400363790221922399105547526510519843480385335698318129}{2032052530731224470967296950883821161453361178437824090936499951673636318} a^{13} - \frac{574257639595324751395058366692309786303014412131210354316130823132178811}{8128210122924897883869187803535284645813444713751296363745999806694545272} a^{12} + \frac{38164883609939683409030442295202015658245514323756399024992047890407853}{2032052530731224470967296950883821161453361178437824090936499951673636318} a^{11} - \frac{495300403750608979327079136641681399598267692702451834365325771499538693}{4064105061462448941934593901767642322906722356875648181872999903347272636} a^{10} + \frac{418949853478601666533799024026321052046883530695844655065171119430895065}{4064105061462448941934593901767642322906722356875648181872999903347272636} a^{9} + \frac{143958863587792320696280146789271504485943333088726589204113330415258931}{8128210122924897883869187803535284645813444713751296363745999806694545272} a^{8} - \frac{648829947892092535396722721822347763130199411332224047130534098078149443}{4064105061462448941934593901767642322906722356875648181872999903347272636} a^{7} + \frac{280150505108515753492123369724625295050565525497240869457540048075362415}{1016026265365612235483648475441910580726680589218912045468249975836818159} a^{6} + \frac{1146809195108026235885059787136313686545424526975788962714215006980494603}{4064105061462448941934593901767642322906722356875648181872999903347272636} a^{5} + \frac{200659465173943247251340181948459907441066832046819399621182901420467911}{8128210122924897883869187803535284645813444713751296363745999806694545272} a^{4} + \frac{2027941400036397665722244526914172004321347942073028446229244047104884803}{8128210122924897883869187803535284645813444713751296363745999806694545272} a^{3} + \frac{611435365419205092003104269383491201899254966704961846894831414329672843}{4064105061462448941934593901767642322906722356875648181872999903347272636} a^{2} + \frac{9972206412028546552296276330386931147254826679075406641511461397033887}{21617580114161962457098903732806608100567672111040681818473403741208897} a + \frac{198799056404613376453990711723446019083038255518620324554837386306539510}{1016026265365612235483648475441910580726680589218912045468249975836818159}$

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:  $9$
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:  \( 148088232777 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1223:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 40 conjugacy class representatives for t16n1223
Character table for t16n1223 is not computed

Intermediate fields

\(\Q(\sqrt{97}) \), 4.4.912673.1, 8.8.80798284478113.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ $16$ $16$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ $16$

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
61Data not computed
97Data not computed