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Properties

Label 16.16.9051792270...1248.2
Degree $16$
Signature $[16, 0]$
Discriminant $2^{79}\cdot 23^{4}\cdot 31^{8}\cdot 89^{4}$
Root discriminant $1147.61$
Ramified primes $2, 23, 31, 89$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_4.C_2^2\wr C_2$ (as 16T385)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7658348535725451218, 0, -512785191909377632, 0, 12437574187407824, 0, -142366611859328, 0, 836309669640, 0, -2581228928, 0, 4140208, 0, -3264, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3264*x^14 + 4140208*x^12 - 2581228928*x^10 + 836309669640*x^8 - 142366611859328*x^6 + 12437574187407824*x^4 - 512785191909377632*x^2 + 7658348535725451218)
 
gp: K = bnfinit(x^16 - 3264*x^14 + 4140208*x^12 - 2581228928*x^10 + 836309669640*x^8 - 142366611859328*x^6 + 12437574187407824*x^4 - 512785191909377632*x^2 + 7658348535725451218, 1)
 

Normalized defining polynomial

\( x^{16} - 3264 x^{14} + 4140208 x^{12} - 2581228928 x^{10} + 836309669640 x^{8} - 142366611859328 x^{6} + 12437574187407824 x^{4} - 512785191909377632 x^{2} + 7658348535725451218 \)

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:  \(9051792270538665113859285354790637594769776181248=2^{79}\cdot 23^{4}\cdot 31^{8}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1147.61$
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:  $2, 23, 31, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{23} a^{8} + \frac{7}{23} a^{6} - \frac{10}{23} a^{4} + \frac{3}{23} a^{2}$, $\frac{1}{23} a^{9} + \frac{7}{23} a^{7} - \frac{10}{23} a^{5} + \frac{3}{23} a^{3}$, $\frac{1}{14329} a^{10} + \frac{118}{14329} a^{8} - \frac{1763}{14329} a^{6} - \frac{1889}{14329} a^{4} + \frac{294}{2047} a^{2} + \frac{3}{7}$, $\frac{1}{14329} a^{11} + \frac{118}{14329} a^{9} - \frac{1763}{14329} a^{7} - \frac{1889}{14329} a^{5} + \frac{294}{2047} a^{3} + \frac{3}{7} a$, $\frac{1}{29331463} a^{12} + \frac{830}{29331463} a^{10} - \frac{477824}{29331463} a^{8} + \frac{11486943}{29331463} a^{6} - \frac{198459}{29331463} a^{4} - \frac{1138}{14329} a^{2} - \frac{2}{7}$, $\frac{1}{29331463} a^{13} + \frac{830}{29331463} a^{11} - \frac{477824}{29331463} a^{9} + \frac{11486943}{29331463} a^{7} - \frac{198459}{29331463} a^{5} - \frac{1138}{14329} a^{3} - \frac{2}{7} a$, $\frac{1}{321134418718370500328089684489029692223748098557} a^{14} - \frac{1345857129732627190915512536510852866078}{321134418718370500328089684489029692223748098557} a^{12} + \frac{6659247003231134122323778266213265135769272}{321134418718370500328089684489029692223748098557} a^{10} - \frac{462481278773292032786584425260132288144130054}{24702647593720807717545360345309976324903699889} a^{8} + \frac{1269408161337443863988872443501107649161477606}{45876345531195785761155669212718527460535442651} a^{6} + \frac{49335380601891967267248677857709104996867940}{156880517204870786677132234728397504750243331} a^{4} + \frac{37056851477592758022962693405454419075331}{76639236543659397497377740463310945163773} a^{2} + \frac{4781866783698770058208750121164700714}{37439783362803809231742911804255469059}$, $\frac{1}{149969773541479023653217882656376866268490362026119} a^{15} + \frac{228571852501245565301217708853421982625241}{149969773541479023653217882656376866268490362026119} a^{13} - \frac{4217575038197277676879927111050757972234712559}{149969773541479023653217882656376866268490362026119} a^{11} + \frac{183001601561596356841127560451582770497635650436}{11536136426267617204093683281259758943730027848163} a^{9} - \frac{5907530971051382971624451643866886085224076209687}{21424253363068431950459697522339552324070051718017} a^{7} + \frac{16508531382663916953827928301292769399621922127}{73263201534674657378220753618161634718363635577} a^{5} + \frac{12392222800986212608307355331721563464014390}{35790523465888938631275404796366211391481991} a^{3} - \frac{497980938373952382482338922678837312364}{17484378830429378911223939812587304050553} a$

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

Class group and class number

$C_{2}\times C_{2}$, 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:  \( 13778676273300000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.C_2^2\wr C_2$ (as 16T385):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 20 conjugacy class representatives for $C_4.C_2^2\wr C_2$
Character table for $C_4.C_2^2\wr C_2$

Intermediate fields

\(\Q(\sqrt{62}) \), 4.4.1968128.1, 8.8.1983246246084608.2

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

Sibling fields

Degree 32 siblings: data not computed
Arithmetically equvalently 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 $16$ $16$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
$23$23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
$31$31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$89$89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$