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Properties

Label 16.6.55013835857...3056.1
Degree $16$
Signature $[6, 5]$
Discriminant $-\,2^{48}\cdot 3^{11}\cdot 7^{8}\cdot 294337^{7}$
Root discriminant $11{,}124.47$
Ramified primes $2, 3, 7, 294337$
Class number $8$ (GRH)
Class group $[2, 2, 2]$ (GRH)
Galois group 16T994

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-180141748647795219, 0, 422775017521333932, 0, -36554998011899960, 0, 721181608173092, 0, -4139016974887, 0, -690444484, 0, 9542174, 0, 6392, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 6392*x^14 + 9542174*x^12 - 690444484*x^10 - 4139016974887*x^8 + 721181608173092*x^6 - 36554998011899960*x^4 + 422775017521333932*x^2 - 180141748647795219)
 
gp: K = bnfinit(x^16 + 6392*x^14 + 9542174*x^12 - 690444484*x^10 - 4139016974887*x^8 + 721181608173092*x^6 - 36554998011899960*x^4 + 422775017521333932*x^2 - 180141748647795219, 1)
 

Normalized defining polynomial

\( x^{16} + 6392 x^{14} + 9542174 x^{12} - 690444484 x^{10} - 4139016974887 x^{8} + 721181608173092 x^{6} - 36554998011899960 x^{4} + 422775017521333932 x^{2} - 180141748647795219 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-55013835857676939099088270698740000784693073140191887871554093056=-\,2^{48}\cdot 3^{11}\cdot 7^{8}\cdot 294337^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $11{,}124.47$
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:  $2, 3, 7, 294337$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{8} - \frac{1}{5} a^{6} + \frac{2}{5} a^{4} + \frac{1}{5} a^{2} + \frac{2}{5}$, $\frac{1}{13855} a^{13} - \frac{1134}{2771} a^{11} + \frac{4881}{13855} a^{9} + \frac{859}{13855} a^{7} + \frac{4202}{13855} a^{5} + \frac{766}{13855} a^{3} - \frac{3233}{13855} a$, $\frac{1}{446689292397809546514670500355739313758584219024192557332039642665} a^{14} - \frac{5900057723762684992872278992733514719886112701448543488651997498}{89337858479561909302934100071147862751716843804838511466407928533} a^{12} - \frac{53444898673673873409440198099545615681612112228873769359725483404}{446689292397809546514670500355739313758584219024192557332039642665} a^{10} - \frac{121361491932077427312678423092978065776954987680392230886526187531}{446689292397809546514670500355739313758584219024192557332039642665} a^{8} + \frac{164370335023118244824963611281861042668595718715593234090288279787}{446689292397809546514670500355739313758584219024192557332039642665} a^{6} - \frac{63391972513816458559456476606499759717518245526156090340818222554}{446689292397809546514670500355739313758584219024192557332039642665} a^{4} - \frac{176093838429131096413546120399611650987414224980625035856828006253}{446689292397809546514670500355739313758584219024192557332039642665} a^{2} - \frac{35744992207449640922226993814561827057275575582330543987962}{197793223149406560283510637277738237069111600217056391385821}$, $\frac{1}{72810354660842956081891291557985508142649227700943386845122461754395} a^{15} + \frac{119454132725063084510245280430928807335576066456418243124663612}{4282962038873115063640664209293265184861719276526081579124850691435} a^{13} - \frac{13559227033525091926853007091208369572852923205077989914575512326334}{72810354660842956081891291557985508142649227700943386845122461754395} a^{11} - \frac{21767399084715342393123593610919921987795877318982146862070894283262}{72810354660842956081891291557985508142649227700943386845122461754395} a^{9} + \frac{8935245970457755105213023200619848273998914089908268271586454910468}{72810354660842956081891291557985508142649227700943386845122461754395} a^{7} + \frac{8696973326629107236028257924487823017171421673454185004629888651829}{72810354660842956081891291557985508142649227700943386845122461754395} a^{5} - \frac{11311795139499230261879273064562101131627884343948437141581439803984}{72810354660842956081891291557985508142649227700943386845122461754395} a^{3} + \frac{211962909225771711549786768246663708394255885511442201197410657}{2740425106735027892728039879483063274592541221007316302650549955} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:  $10$
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:  \( 160928298992000000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T994:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 29 conjugacy class representatives for t16n994
Character table for t16n994 is not computed

Intermediate fields

\(\Q(\sqrt{21}) \), 4.4.2076841872.1, 8.6.975018691690245188654137344.1

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
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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.3.2$x^{4} - 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
294337Data not computed