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Properties

Label 16.8.50744262454...8441.2
Degree $16$
Signature $[8, 4]$
Discriminant $11^{10}\cdot 89^{14}$
Root discriminant $227.29$
Ramified primes $11, 89$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T817

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1278574016, 2329602032, -1588306153, 720593456, 120147984, -84195244, -3794428, 1672363, -1432142, -13701, 95140, -9224, 297, 577, -84, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 - 84*x^14 + 577*x^13 + 297*x^12 - 9224*x^11 + 95140*x^10 - 13701*x^9 - 1432142*x^8 + 1672363*x^7 - 3794428*x^6 - 84195244*x^5 + 120147984*x^4 + 720593456*x^3 - 1588306153*x^2 + 2329602032*x + 1278574016)
 
gp: K = bnfinit(x^16 - 7*x^15 - 84*x^14 + 577*x^13 + 297*x^12 - 9224*x^11 + 95140*x^10 - 13701*x^9 - 1432142*x^8 + 1672363*x^7 - 3794428*x^6 - 84195244*x^5 + 120147984*x^4 + 720593456*x^3 - 1588306153*x^2 + 2329602032*x + 1278574016, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} - 84 x^{14} + 577 x^{13} + 297 x^{12} - 9224 x^{11} + 95140 x^{10} - 13701 x^{9} - 1432142 x^{8} + 1672363 x^{7} - 3794428 x^{6} - 84195244 x^{5} + 120147984 x^{4} + 720593456 x^{3} - 1588306153 x^{2} + 2329602032 x + 1278574016 \)

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:  \(50744262454554467455972799262367678441=11^{10}\cdot 89^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $227.29$
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:  $11, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{44} a^{14} - \frac{1}{11} a^{13} + \frac{9}{44} a^{12} + \frac{2}{11} a^{11} - \frac{5}{22} a^{10} - \frac{5}{22} a^{9} - \frac{3}{11} a^{8} - \frac{3}{44} a^{7} + \frac{21}{44} a^{6} - \frac{21}{44} a^{5} - \frac{7}{22} a^{4} - \frac{15}{44} a^{3} - \frac{19}{44} a^{2} - \frac{1}{11} a + \frac{3}{11}$, $\frac{1}{83603655674742263739228332124236549812906456720697945614370006748736} a^{15} + \frac{606539529952492999448851724604053741078992455846174571631789676321}{83603655674742263739228332124236549812906456720697945614370006748736} a^{14} - \frac{2030054720545955909992098457062950095838554826260385545838943971523}{20900913918685565934807083031059137453226614180174486403592501687184} a^{13} + \frac{718404789364848004671793371986248974842794343422943121726235310897}{83603655674742263739228332124236549812906456720697945614370006748736} a^{12} + \frac{9226723344947309853587803760870168519860859322070319071030776817809}{83603655674742263739228332124236549812906456720697945614370006748736} a^{11} + \frac{92663647411464620143254935337174010599643825518437609569517033235}{475020770879217407609251887069525851209695776822147418263465947436} a^{10} - \frac{10273123268642438522314229798076796731099520596234916159725468410671}{20900913918685565934807083031059137453226614180174486403592501687184} a^{9} + \frac{28853248014764900840036618925925165691070046410384798974670550127867}{83603655674742263739228332124236549812906456720697945614370006748736} a^{8} - \frac{5645446937051131089711836239950106400565907895527207373540428804107}{41801827837371131869614166062118274906453228360348972807185003374368} a^{7} - \frac{2446515013637950267947697831625827732969307272314163653272923533637}{83603655674742263739228332124236549812906456720697945614370006748736} a^{6} + \frac{9253779831784648241361503034342799737154623057408118004895187557595}{20900913918685565934807083031059137453226614180174486403592501687184} a^{5} - \frac{559403773619169661870326922360423462426168705755073264186899713399}{20900913918685565934807083031059137453226614180174486403592501687184} a^{4} - \frac{104179590326395613748066842450500425035841456371655501836035170829}{1306307119917847870925442689441196090826663386260905400224531355449} a^{3} - \frac{806615161232752444204974345812305442741532452588623660938991550673}{5225228479671391483701770757764784363306653545043621600898125421796} a^{2} + \frac{2742160863799715536888233074953019293353174754975073575862191884263}{83603655674742263739228332124236549812906456720697945614370006748736} a - \frac{1414387284962310788867813083807078575239372503267809779785442058917}{10450456959342782967403541515529568726613307090087243201796250843592}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 16121598056300 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T817:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n817
Character table for t16n817 is not computed

Intermediate fields

\(\Q(\sqrt{89}) \), 4.4.704969.1, 8.8.647590974205440089.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$11$11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.3.2$x^{4} - 11$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.4.3.2$x^{4} - 11$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
89Data not computed