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Properties

Label 16.0.45080303137...4161.1
Degree $16$
Signature $[0, 8]$
Discriminant $19^{8}\cdot 61^{12}$
Root discriminant $95.14$
Ramified primes $19, 61$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_2\wr C_4$ (as 16T172)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![35559, 98172, 158589, 8619, -53641, -111968, -18882, 12964, 3655, 10249, 2681, 731, 541, 3, 39, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 39*x^14 + 3*x^13 + 541*x^12 + 731*x^11 + 2681*x^10 + 10249*x^9 + 3655*x^8 + 12964*x^7 - 18882*x^6 - 111968*x^5 - 53641*x^4 + 8619*x^3 + 158589*x^2 + 98172*x + 35559)
 
gp: K = bnfinit(x^16 - x^15 + 39*x^14 + 3*x^13 + 541*x^12 + 731*x^11 + 2681*x^10 + 10249*x^9 + 3655*x^8 + 12964*x^7 - 18882*x^6 - 111968*x^5 - 53641*x^4 + 8619*x^3 + 158589*x^2 + 98172*x + 35559, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 39 x^{14} + 3 x^{13} + 541 x^{12} + 731 x^{11} + 2681 x^{10} + 10249 x^{9} + 3655 x^{8} + 12964 x^{7} - 18882 x^{6} - 111968 x^{5} - 53641 x^{4} + 8619 x^{3} + 158589 x^{2} + 98172 x + 35559 \)

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:  \(45080303137796743213973710214161=19^{8}\cdot 61^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $95.14$
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:  $19, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{18} a^{12} - \frac{1}{6} a^{11} + \frac{1}{18} a^{10} - \frac{1}{9} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} + \frac{1}{9} a^{6} + \frac{1}{6} a^{4} + \frac{7}{18} a^{3} - \frac{1}{9} a^{2} + \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{18} a^{13} - \frac{1}{9} a^{11} + \frac{1}{18} a^{10} - \frac{1}{6} a^{9} - \frac{1}{18} a^{7} + \frac{1}{6} a^{5} - \frac{1}{9} a^{4} + \frac{1}{18} a^{3} - \frac{1}{6} a^{2} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{15354} a^{14} - \frac{197}{15354} a^{13} - \frac{155}{15354} a^{12} - \frac{491}{7677} a^{11} - \frac{191}{15354} a^{10} - \frac{167}{5118} a^{9} + \frac{226}{7677} a^{8} - \frac{245}{7677} a^{7} + \frac{509}{5118} a^{6} - \frac{4121}{15354} a^{5} - \frac{2507}{7677} a^{4} + \frac{5773}{15354} a^{3} - \frac{773}{5118} a^{2} + \frac{239}{2559} a - \frac{49}{853}$, $\frac{1}{33106595696162407573425697714074} a^{15} + \frac{261334243799426691522612214}{16553297848081203786712848857037} a^{14} + \frac{8265133038795922596306583745}{11035531898720802524475232571358} a^{13} + \frac{95921777765214473567090943287}{5517765949360401262237616285679} a^{12} - \frac{1570081641364750883054889903187}{16553297848081203786712848857037} a^{11} + \frac{2147929947548563523644113002698}{16553297848081203786712848857037} a^{10} - \frac{2011655366233185193264480092065}{16553297848081203786712848857037} a^{9} + \frac{4094911563077280911115790239211}{33106595696162407573425697714074} a^{8} + \frac{5019487042454021217777339977827}{33106595696162407573425697714074} a^{7} + \frac{1236572422901462424289627360922}{16553297848081203786712848857037} a^{6} + \frac{178944540035366412697587158428}{5517765949360401262237616285679} a^{5} - \frac{6765773193014257413418247631220}{16553297848081203786712848857037} a^{4} - \frac{401367303870431191924592366912}{16553297848081203786712848857037} a^{3} - \frac{696974748131124274896228207506}{1839255316453467087412538761893} a^{2} + \frac{310784572054928930190807261575}{3678510632906934174825077523786} a - \frac{9176967733750394575231616437}{613085105484489029137512920631}$

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:  $7$
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:  \( 580126353.601 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T172):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $C_2\wr C_4$
Character table for $C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{61}) \), 4.0.226981.1, 4.2.70699.1, 4.2.4312639.1, 8.4.110068634542621.1, 8.0.110068634542621.1, 8.0.18598855144321.1

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

Sibling fields

Degree 8 siblings: data not computed
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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.8.6.2$x^{8} - 19 x^{4} + 5776$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$61$61.8.6.1$x^{8} - 61 x^{4} + 59536$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
61.8.6.1$x^{8} - 61 x^{4} + 59536$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$