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Properties

Label 16.8.66283426716...3249.1
Degree $16$
Signature $[8, 4]$
Discriminant $17^{14}\cdot 89^{8}$
Root discriminant $112.55$
Ramified primes $17, 89$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^5.C_2.C_2$ (as 16T258)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5309473, 487097, 10371672, -6092523, 2640940, -2399112, -376487, 456426, -38278, 91361, -3624, -2378, -245, -282, 44, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 44*x^14 - 282*x^13 - 245*x^12 - 2378*x^11 - 3624*x^10 + 91361*x^9 - 38278*x^8 + 456426*x^7 - 376487*x^6 - 2399112*x^5 + 2640940*x^4 - 6092523*x^3 + 10371672*x^2 + 487097*x - 5309473)
 
gp: K = bnfinit(x^16 - 2*x^15 + 44*x^14 - 282*x^13 - 245*x^12 - 2378*x^11 - 3624*x^10 + 91361*x^9 - 38278*x^8 + 456426*x^7 - 376487*x^6 - 2399112*x^5 + 2640940*x^4 - 6092523*x^3 + 10371672*x^2 + 487097*x - 5309473, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 44 x^{14} - 282 x^{13} - 245 x^{12} - 2378 x^{11} - 3624 x^{10} + 91361 x^{9} - 38278 x^{8} + 456426 x^{7} - 376487 x^{6} - 2399112 x^{5} + 2640940 x^{4} - 6092523 x^{3} + 10371672 x^{2} + 487097 x - 5309473 \)

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:  \(662834267162184237456650608633249=17^{14}\cdot 89^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $112.55$
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:  $17, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{26} a^{12} - \frac{2}{13} a^{11} + \frac{6}{13} a^{10} - \frac{5}{26} a^{9} + \frac{4}{13} a^{8} - \frac{4}{13} a^{7} - \frac{5}{26} a^{6} + \frac{6}{13} a^{5} + \frac{2}{13} a^{4} - \frac{1}{26} a^{3} + \frac{3}{13} a^{2} + \frac{3}{13} a - \frac{1}{2}$, $\frac{1}{26} a^{13} - \frac{2}{13} a^{11} - \frac{9}{26} a^{10} - \frac{6}{13} a^{9} - \frac{1}{13} a^{8} - \frac{11}{26} a^{7} - \frac{4}{13} a^{6} - \frac{11}{26} a^{4} + \frac{1}{13} a^{3} + \frac{2}{13} a^{2} + \frac{11}{26} a$, $\frac{1}{26} a^{14} + \frac{1}{26} a^{11} + \frac{5}{13} a^{10} + \frac{2}{13} a^{9} - \frac{5}{26} a^{8} + \frac{6}{13} a^{7} + \frac{3}{13} a^{6} + \frac{11}{26} a^{5} - \frac{4}{13} a^{4} + \frac{9}{26} a^{2} - \frac{1}{13} a$, $\frac{1}{106861795719979183714667608573720970553010007062442918} a^{15} - \frac{632326772165702460231112872135805426247280713884368}{53430897859989591857333804286860485276505003531221459} a^{14} - \frac{429372029797754852707646490365124399162233110072519}{53430897859989591857333804286860485276505003531221459} a^{13} + \frac{471213176347410448339116829069311377916965173282643}{53430897859989591857333804286860485276505003531221459} a^{12} + \frac{25056949236335059535608543310683265279313204356595227}{53430897859989591857333804286860485276505003531221459} a^{11} + \frac{6691721536563813204135790951964292669116321986766}{120611507584626618188112425026773104461636576819913} a^{10} - \frac{8353780035385643886388022745416747823440040807969473}{53430897859989591857333804286860485276505003531221459} a^{9} + \frac{2003793007289576572759595118773068390716199663797721}{4110069066153045527487215714373883482808077194709343} a^{8} + \frac{4706367547507580479351430202884539592443483851821978}{53430897859989591857333804286860485276505003531221459} a^{7} - \frac{2244039289515800784633429037398318780685392761644785}{53430897859989591857333804286860485276505003531221459} a^{6} + \frac{15913629820907251860826682527518599664938194672223375}{53430897859989591857333804286860485276505003531221459} a^{5} - \frac{22073149319813051675807935654955632774405156289199130}{53430897859989591857333804286860485276505003531221459} a^{4} + \frac{9376159953783760542624394324386678201178962596122956}{53430897859989591857333804286860485276505003531221459} a^{3} + \frac{1176555292729501614566010832650756209263932895833734}{4110069066153045527487215714373883482808077194709343} a^{2} - \frac{1247910494895856937530671879442215351499141808827055}{4110069066153045527487215714373883482808077194709343} a + \frac{3989134300437829837246910680689525550186948621431}{23286510289818954829955896398718886588147746145662}$

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:  \( 33415878093.8 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^5.C_2.C_2$ (as 16T258):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^5.C_2.C_2$
Character table for $C_2^5.C_2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 8.4.36520141897.1, 8.8.25745567912986193.1, 8.4.17016237880361.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{12}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
17Data not computed
$89$89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$