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Properties

Label 16.0.16025640975...8497.9
Degree $16$
Signature $[0, 8]$
Discriminant $47^{8}\cdot 97^{13}$
Root discriminant $282.03$
Ramified primes $47, 97$
Class number $80$ (GRH)
Class group $[4, 20]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T260)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![321671395988, 14829382806, 12836643115, 8350714193, 4917795919, -322810806, 359119764, -9476184, 11234971, -237490, 364196, -32407, 10885, -976, 161, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 161*x^14 - 976*x^13 + 10885*x^12 - 32407*x^11 + 364196*x^10 - 237490*x^9 + 11234971*x^8 - 9476184*x^7 + 359119764*x^6 - 322810806*x^5 + 4917795919*x^4 + 8350714193*x^3 + 12836643115*x^2 + 14829382806*x + 321671395988)
 
gp: K = bnfinit(x^16 - 8*x^15 + 161*x^14 - 976*x^13 + 10885*x^12 - 32407*x^11 + 364196*x^10 - 237490*x^9 + 11234971*x^8 - 9476184*x^7 + 359119764*x^6 - 322810806*x^5 + 4917795919*x^4 + 8350714193*x^3 + 12836643115*x^2 + 14829382806*x + 321671395988, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 161 x^{14} - 976 x^{13} + 10885 x^{12} - 32407 x^{11} + 364196 x^{10} - 237490 x^{9} + 11234971 x^{8} - 9476184 x^{7} + 359119764 x^{6} - 322810806 x^{5} + 4917795919 x^{4} + 8350714193 x^{3} + 12836643115 x^{2} + 14829382806 x + 321671395988 \)

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:  \(1602564097506337056774549158159344838497=47^{8}\cdot 97^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $282.03$
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:  $47, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{2} a^{11} + \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{250689603171014137573385213854146630193976991198331145812242164433228202133096816} a^{15} + \frac{23975176092547893819149108178638299388850031210874920912839571962669703552199337}{250689603171014137573385213854146630193976991198331145812242164433228202133096816} a^{14} + \frac{28264564359241053786038166990846527600902149727920405695342455795578287335583933}{125344801585507068786692606927073315096988495599165572906121082216614101066548408} a^{13} - \frac{26487858824578554427733613986591543595629594177880511525622250764691960967784123}{125344801585507068786692606927073315096988495599165572906121082216614101066548408} a^{12} - \frac{45775593263398636115493083761618900227779543919065477680347088439795690896500753}{250689603171014137573385213854146630193976991198331145812242164433228202133096816} a^{11} - \frac{1326814777776884607393383528652599006578658766133100324478077132151855131224609}{31336200396376767196673151731768328774247123899791393226530270554153525266637102} a^{10} + \frac{31099412325568213742974424384576586586620869679474459430809550322092613375888255}{62672400792753534393346303463536657548494247799582786453060541108307050533274204} a^{9} - \frac{33443899237866095650086086559694458125044599545891034486301756996603371683764019}{125344801585507068786692606927073315096988495599165572906121082216614101066548408} a^{8} - \frac{22550214995686888707777591077823639579902346860825292617410820309655762526925803}{250689603171014137573385213854146630193976991198331145812242164433228202133096816} a^{7} + \frac{54190528743746877311150799326697253839895572998893509046279192418792649984479037}{250689603171014137573385213854146630193976991198331145812242164433228202133096816} a^{6} + \frac{95916107642670116401078209965344823093356423951625607094675871704914650212880225}{250689603171014137573385213854146630193976991198331145812242164433228202133096816} a^{5} + \frac{66488683417347927125970792325438261304786221790251800250283268123650486794919595}{250689603171014137573385213854146630193976991198331145812242164433228202133096816} a^{4} + \frac{7560934315074789872531363691325153444700467633114961837322487933016264944805813}{125344801585507068786692606927073315096988495599165572906121082216614101066548408} a^{3} - \frac{78695110744183018516920697925111681324522582203587619201779003141944705454097013}{250689603171014137573385213854146630193976991198331145812242164433228202133096816} a^{2} - \frac{31685456669610859543820578106407619540059302766825397126710101991366971115957237}{125344801585507068786692606927073315096988495599165572906121082216614101066548408} a + \frac{677750824505674509417328482687396618229747713576538072942175795450776453368239}{62672400792753534393346303463536657548494247799582786453060541108307050533274204}$

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

Class group and class number

$C_{4}\times C_{20}$, which has order $80$ (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:  \( -1 \) (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:  \( 189640310271 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T260):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-47}) \), 4.0.214273.1, 8.0.41903481092618017.3

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

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ $16$ $16$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$47$47.8.4.1$x^{8} + 172302 x^{4} - 103823 x^{2} + 7421994801$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
47.8.4.1$x^{8} + 172302 x^{4} - 103823 x^{2} + 7421994801$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$97$97.4.3.3$x^{4} + 485$$4$$1$$3$$C_4$$[\ ]_{4}$
97.4.3.4$x^{4} + 12125$$4$$1$$3$$C_4$$[\ ]_{4}$
97.8.7.2$x^{8} - 2425$$8$$1$$7$$C_8$$[\ ]_{8}$