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Properties

Label 16.16.1348633586...0000.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{16}\cdot 5^{8}\cdot 29^{6}\cdot 89^{4}\cdot 109^{4}$
Root discriminant $156.90$
Ramified primes $2, 5, 29, 89, 109$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T839

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![92714201, -582812100, -528337915, 593654550, 284721366, -174144326, -67034348, 21219480, 7745512, -1206728, -459200, 31384, 13895, -294, -197, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 197*x^14 - 294*x^13 + 13895*x^12 + 31384*x^11 - 459200*x^10 - 1206728*x^9 + 7745512*x^8 + 21219480*x^7 - 67034348*x^6 - 174144326*x^5 + 284721366*x^4 + 593654550*x^3 - 528337915*x^2 - 582812100*x + 92714201)
 
gp: K = bnfinit(x^16 - 197*x^14 - 294*x^13 + 13895*x^12 + 31384*x^11 - 459200*x^10 - 1206728*x^9 + 7745512*x^8 + 21219480*x^7 - 67034348*x^6 - 174144326*x^5 + 284721366*x^4 + 593654550*x^3 - 528337915*x^2 - 582812100*x + 92714201, 1)
 

Normalized defining polynomial

\( x^{16} - 197 x^{14} - 294 x^{13} + 13895 x^{12} + 31384 x^{11} - 459200 x^{10} - 1206728 x^{9} + 7745512 x^{8} + 21219480 x^{7} - 67034348 x^{6} - 174144326 x^{5} + 284721366 x^{4} + 593654550 x^{3} - 528337915 x^{2} - 582812100 x + 92714201 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(134863358606854287590863897600000000=2^{16}\cdot 5^{8}\cdot 29^{6}\cdot 89^{4}\cdot 109^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $156.90$
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:  $2, 5, 29, 89, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{38577022487289881198110280813506815865005790094993947881877} a^{15} - \frac{5738002489126398365332504066670830250020291558685893214402}{38577022487289881198110280813506815865005790094993947881877} a^{14} - \frac{10284431171818964748826295149561534602368647906667803551694}{38577022487289881198110280813506815865005790094993947881877} a^{13} + \frac{1395778824795999987453053757398589419834425042246621379837}{38577022487289881198110280813506815865005790094993947881877} a^{12} + \frac{14949199105861813535619193205387912017417071240220678344097}{38577022487289881198110280813506815865005790094993947881877} a^{11} + \frac{7195305023319313314168542736989778922095849422381165182683}{38577022487289881198110280813506815865005790094993947881877} a^{10} - \frac{2848400670130510184276441242755332899225596847820490818233}{38577022487289881198110280813506815865005790094993947881877} a^{9} + \frac{4381502543040828680476950148100010761142207284367215757444}{38577022487289881198110280813506815865005790094993947881877} a^{8} - \frac{18786489176264318629926274386610497478791396898787461582705}{38577022487289881198110280813506815865005790094993947881877} a^{7} + \frac{12224174797340972260950549742426534058104796993555001904048}{38577022487289881198110280813506815865005790094993947881877} a^{6} + \frac{1978511856625366653736730638557930727572408709963458430040}{38577022487289881198110280813506815865005790094993947881877} a^{5} - \frac{5456077493701114948698904384722979075848325820320797767321}{38577022487289881198110280813506815865005790094993947881877} a^{4} + \frac{12241918900895875489496225823381182234997509506638329548384}{38577022487289881198110280813506815865005790094993947881877} a^{3} + \frac{18854014717446857535191163729273345572255535407709416155969}{38577022487289881198110280813506815865005790094993947881877} a^{2} - \frac{2562660958936782834672830864256809752497234123154583398439}{38577022487289881198110280813506815865005790094993947881877} a + \frac{5352425482494024283702471290602660011994224710539165990255}{38577022487289881198110280813506815865005790094993947881877}$

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:  $15$
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:  \( 961028616510 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T839:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 53 conjugacy class representatives for t16n839 are not computed
Character table for t16n839 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 4.4.1264400.2, 4.4.43600.1, 8.8.1598707360000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.3$x^{4} + 58$$4$$1$$3$$C_4$$[\ ]_{4}$
89Data not computed
$109$109.2.1.1$x^{2} - 109$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.1.1$x^{2} - 109$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.1$x^{2} - 109$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.1.1$x^{2} - 109$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$