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Properties

Label 16.16.4776394010...0625.1
Degree $16$
Signature $[16, 0]$
Discriminant $5^{12}\cdot 89^{14}$
Root discriminant $169.80$
Ramified primes $5, 89$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_8: C_2$ (as 16T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-219439429, -256164513, 359251250, 611877670, 154809239, -119238518, -52900437, 8516080, 5957206, -234933, -336497, 226, 10289, 89, -161, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 161*x^14 + 89*x^13 + 10289*x^12 + 226*x^11 - 336497*x^10 - 234933*x^9 + 5957206*x^8 + 8516080*x^7 - 52900437*x^6 - 119238518*x^5 + 154809239*x^4 + 611877670*x^3 + 359251250*x^2 - 256164513*x - 219439429)
 
gp: K = bnfinit(x^16 - x^15 - 161*x^14 + 89*x^13 + 10289*x^12 + 226*x^11 - 336497*x^10 - 234933*x^9 + 5957206*x^8 + 8516080*x^7 - 52900437*x^6 - 119238518*x^5 + 154809239*x^4 + 611877670*x^3 + 359251250*x^2 - 256164513*x - 219439429, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 161 x^{14} + 89 x^{13} + 10289 x^{12} + 226 x^{11} - 336497 x^{10} - 234933 x^{9} + 5957206 x^{8} + 8516080 x^{7} - 52900437 x^{6} - 119238518 x^{5} + 154809239 x^{4} + 611877670 x^{3} + 359251250 x^{2} - 256164513 x - 219439429 \)

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:  \(477639401035264016929733847900390625=5^{12}\cdot 89^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $169.80$
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:  $5, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{11} a^{9} + \frac{1}{11} a^{8} + \frac{3}{11} a^{7} + \frac{5}{11} a^{6} - \frac{1}{11} a^{4} - \frac{1}{11} a^{3} - \frac{3}{11} a^{2} - \frac{5}{11} a$, $\frac{1}{22} a^{10} - \frac{1}{22} a^{9} + \frac{1}{22} a^{8} + \frac{5}{11} a^{7} + \frac{1}{22} a^{6} - \frac{1}{22} a^{5} - \frac{5}{11} a^{4} + \frac{5}{11} a^{3} + \frac{1}{22} a^{2} + \frac{5}{11} a - \frac{1}{2}$, $\frac{1}{22} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{22} a - \frac{1}{2}$, $\frac{1}{2420} a^{12} - \frac{17}{1210} a^{11} - \frac{7}{2420} a^{10} - \frac{43}{1210} a^{9} - \frac{69}{605} a^{8} + \frac{163}{605} a^{7} + \frac{60}{121} a^{6} + \frac{41}{484} a^{5} + \frac{251}{2420} a^{4} + \frac{75}{484} a^{3} - \frac{95}{242} a^{2} - \frac{1}{220} a - \frac{9}{20}$, $\frac{1}{2420} a^{13} + \frac{47}{2420} a^{11} + \frac{3}{1210} a^{10} - \frac{1}{242} a^{9} + \frac{292}{605} a^{8} - \frac{141}{1210} a^{7} + \frac{171}{484} a^{6} + \frac{841}{2420} a^{5} - \frac{331}{2420} a^{4} + \frac{69}{242} a^{3} - \frac{191}{2420} a^{2} - \frac{3}{220} a - \frac{3}{10}$, $\frac{1}{2420} a^{14} - \frac{23}{1210} a^{11} - \frac{1}{220} a^{10} + \frac{2}{121} a^{9} + \frac{81}{242} a^{8} + \frac{21}{2420} a^{7} - \frac{1109}{2420} a^{6} - \frac{53}{110} a^{5} + \frac{113}{2420} a^{4} + \frac{57}{1210} a^{3} - \frac{923}{2420} a^{2} - \frac{89}{220} a + \frac{3}{20}$, $\frac{1}{308014338216871493551058711240316820} a^{15} + \frac{15246661575961666406480106907947}{77003584554217873387764677810079205} a^{14} - \frac{51478603822225182669928348858713}{308014338216871493551058711240316820} a^{13} - \frac{2745387322528627227888399270543}{154007169108435746775529355620158410} a^{12} + \frac{348943682715131099330792957509776}{15400716910843574677552935562015841} a^{11} + \frac{1043217556545084116782971045455946}{77003584554217873387764677810079205} a^{10} - \frac{312236079851433237773321612916765}{30801433821687149355105871124031682} a^{9} + \frac{14317078453078188944084887134797317}{308014338216871493551058711240316820} a^{8} + \frac{10125115889042528768650566197009739}{61602867643374298710211742248063364} a^{7} + \frac{153296795909003466232027473973690337}{308014338216871493551058711240316820} a^{6} + \frac{2073032714345453549056141312189888}{7000325868565261217069516164552655} a^{5} + \frac{139760363248708696337378900602122421}{308014338216871493551058711240316820} a^{4} - \frac{46820767419168730631176163639503531}{308014338216871493551058711240316820} a^{3} - \frac{7259269145672759356073622216793713}{15400716910843574677552935562015841} a^{2} + \frac{1070034903444726914345600500607293}{2800130347426104486827806465821062} a - \frac{530611260036539198209041958960189}{1272786521557320221285366575373210}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 4290610616720 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}$ (as 16T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_8: C_2$
Character table for $C_8: C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{89}) \), \(\Q(\sqrt{445}) \), \(\Q(\sqrt{5}, \sqrt{89})\), 4.4.17624225.2, 4.4.704969.1, 8.8.310613306850625.1, 8.8.691114607742640625.1 x2

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

Sibling fields

Degree 8 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$89$89.8.7.3$x^{8} - 7209$$8$$1$$7$$C_8$$[\ ]_{8}$
89.8.7.3$x^{8} - 7209$$8$$1$$7$$C_8$$[\ ]_{8}$