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Properties

Label 16.0.26436931289...625.17
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 101^{8}$
Root discriminant $33.60$
Ramified primes $5, 101$
Class number $16$ (GRH)
Class group $[2, 2, 4]$ (GRH)
Galois group $C_2\wr C_4$ (as 16T158)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![14641, -24321, 37016, -59067, 68112, -34826, 7937, -17, -2989, 1543, 547, -331, 7, 13, -4, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 4*x^14 + 13*x^13 + 7*x^12 - 331*x^11 + 547*x^10 + 1543*x^9 - 2989*x^8 - 17*x^7 + 7937*x^6 - 34826*x^5 + 68112*x^4 - 59067*x^3 + 37016*x^2 - 24321*x + 14641)
 
gp: K = bnfinit(x^16 - x^15 - 4*x^14 + 13*x^13 + 7*x^12 - 331*x^11 + 547*x^10 + 1543*x^9 - 2989*x^8 - 17*x^7 + 7937*x^6 - 34826*x^5 + 68112*x^4 - 59067*x^3 + 37016*x^2 - 24321*x + 14641, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 4 x^{14} + 13 x^{13} + 7 x^{12} - 331 x^{11} + 547 x^{10} + 1543 x^{9} - 2989 x^{8} - 17 x^{7} + 7937 x^{6} - 34826 x^{5} + 68112 x^{4} - 59067 x^{3} + 37016 x^{2} - 24321 x + 14641 \)

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:  \(2643693128974804931640625=5^{12}\cdot 101^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.60$
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:  $5, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{9} + \frac{1}{5} a^{6} - \frac{1}{5} a^{3} + \frac{1}{5}$, $\frac{1}{5} a^{13} - \frac{1}{5} a^{10} + \frac{1}{5} a^{7} - \frac{1}{5} a^{4} + \frac{1}{5} a$, $\frac{1}{2665} a^{14} - \frac{152}{2665} a^{13} + \frac{177}{2665} a^{12} + \frac{359}{2665} a^{11} - \frac{823}{2665} a^{10} + \frac{118}{2665} a^{9} + \frac{571}{2665} a^{8} - \frac{1122}{2665} a^{7} - \frac{1168}{2665} a^{6} + \frac{1234}{2665} a^{5} + \frac{1052}{2665} a^{4} + \frac{1213}{2665} a^{3} - \frac{929}{2665} a^{2} - \frac{967}{2665} a + \frac{1302}{2665}$, $\frac{1}{17349085844052218597939334009616915} a^{15} - \frac{1820318247462295846258099847357}{17349085844052218597939334009616915} a^{14} + \frac{826317354366692095926498243365752}{17349085844052218597939334009616915} a^{13} - \frac{1457490369191101315989275953630197}{17349085844052218597939334009616915} a^{12} + \frac{489965560493171915914796636456147}{17349085844052218597939334009616915} a^{11} - \frac{2010285310021546846335396303096517}{17349085844052218597939334009616915} a^{10} - \frac{4083039314745652381293565851510868}{17349085844052218597939334009616915} a^{9} + \frac{741627081247602267057705170374198}{17349085844052218597939334009616915} a^{8} + \frac{3324778533381367306849827337197657}{17349085844052218597939334009616915} a^{7} + \frac{7259769712580130777480170654697913}{17349085844052218597939334009616915} a^{6} - \frac{8535062798474497428588157636391693}{17349085844052218597939334009616915} a^{5} - \frac{17668532138362380264266678381572}{1577189622186565327085394000874265} a^{4} - \frac{357435657674226018643846122634878}{1577189622186565327085394000874265} a^{3} + \frac{2933943656668253075706646498731308}{17349085844052218597939334009616915} a^{2} + \frac{204106136763667637907835442273099}{1334545064927093738303025693047455} a + \frac{25960637189281209630822646540379}{143380874744233211553217636443115}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}$, which has order $16$ (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:  \( \frac{5096245762055653439456216184}{423148435220785819461934975844315} a^{15} + \frac{28579301506987540856653539}{32549879632368139958610382757255} a^{14} - \frac{22337611686420350964757320362}{423148435220785819461934975844315} a^{13} + \frac{21636746212590215285951115981}{423148435220785819461934975844315} a^{12} + \frac{69545812745518403737376222508}{423148435220785819461934975844315} a^{11} - \frac{1536037915468365718991568682248}{423148435220785819461934975844315} a^{10} + \frac{902927559732341950432435972824}{423148435220785819461934975844315} a^{9} + \frac{721136225146769125718696079814}{32549879632368139958610382757255} a^{8} + \frac{1096763820820527794739034342308}{423148435220785819461934975844315} a^{7} - \frac{8409482878401610048232882803689}{423148435220785819461934975844315} a^{6} - \frac{4807254955544054714403236064117}{423148435220785819461934975844315} a^{5} - \frac{12889261682875351488657695882343}{38468039565525983587448634167665} a^{4} + \frac{16752029274958115786282650353764}{38468039565525983587448634167665} a^{3} - \frac{200309332027162039586636030449713}{423148435220785819461934975844315} a^{2} + \frac{616767244729854125179999807568898}{423148435220785819461934975844315} a - \frac{195983203371972639585787934978}{699418901191381519771793348503} \) (order $10$)
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:  \( 155135.805875 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T158):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $C_2\wr C_4$
Character table for $C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2525.1, \(\Q(\zeta_{5})\), 4.0.12625.1, 8.4.325188753125.3 x2, 8.0.16098453125.4 x2, 8.0.159390625.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 siblings: data not computed
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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
101Data not computed