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Properties

Label 16.8.36339978058...0625.1
Degree $16$
Signature $[8, 4]$
Discriminant $3^{8}\cdot 5^{12}\cdot 3881^{4}$
Root discriminant $45.71$
Ramified primes $3, 5, 3881$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1496

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-126149, 288164, 7489, -58165, 24594, -71279, 11557, 18864, -5285, -1565, 1133, 353, -364, 39, 22, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 22*x^14 + 39*x^13 - 364*x^12 + 353*x^11 + 1133*x^10 - 1565*x^9 - 5285*x^8 + 18864*x^7 + 11557*x^6 - 71279*x^5 + 24594*x^4 - 58165*x^3 + 7489*x^2 + 288164*x - 126149)
 
gp: K = bnfinit(x^16 - 8*x^15 + 22*x^14 + 39*x^13 - 364*x^12 + 353*x^11 + 1133*x^10 - 1565*x^9 - 5285*x^8 + 18864*x^7 + 11557*x^6 - 71279*x^5 + 24594*x^4 - 58165*x^3 + 7489*x^2 + 288164*x - 126149, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 22 x^{14} + 39 x^{13} - 364 x^{12} + 353 x^{11} + 1133 x^{10} - 1565 x^{9} - 5285 x^{8} + 18864 x^{7} + 11557 x^{6} - 71279 x^{5} + 24594 x^{4} - 58165 x^{3} + 7489 x^{2} + 288164 x - 126149 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(363399780582976484619140625=3^{8}\cdot 5^{12}\cdot 3881^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.71$
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:  $3, 5, 3881$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{11070963529836555953845565191810433220478589} a^{15} + \frac{2960989630945825083866022853035685675039228}{11070963529836555953845565191810433220478589} a^{14} - \frac{2382864733436074655178263226202557214351058}{11070963529836555953845565191810433220478589} a^{13} - \frac{4457800583852062619968168547152840683476627}{11070963529836555953845565191810433220478589} a^{12} + \frac{147301878803660839395144073346893873498292}{11070963529836555953845565191810433220478589} a^{11} - \frac{244713801118079330152747807111329628815787}{651233148813915056108562658341790189439917} a^{10} - \frac{3697544550644182330774726399087635554677994}{11070963529836555953845565191810433220478589} a^{9} + \frac{1120885082229142737658301691222386584229351}{11070963529836555953845565191810433220478589} a^{8} + \frac{5461609142958564340547287386377040997402068}{11070963529836555953845565191810433220478589} a^{7} - \frac{1440898118223091610757781138914539177822125}{11070963529836555953845565191810433220478589} a^{6} - \frac{1785923778957783350607620551162700734269107}{11070963529836555953845565191810433220478589} a^{5} + \frac{1478987630727035783485898795514593568177036}{11070963529836555953845565191810433220478589} a^{4} + \frac{4198634549216946559791166199162992643158857}{11070963529836555953845565191810433220478589} a^{3} + \frac{2503284087050672602838053294399556477868706}{11070963529836555953845565191810433220478589} a^{2} - \frac{5482578970707244249364518655680015629770910}{11070963529836555953845565191810433220478589} a + \frac{1334053969277840497098791592010871483647663}{11070963529836555953845565191810433220478589}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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:  \( 11607568.5324 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1496:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2304
The 40 conjugacy class representatives for t16n1496
Character table for t16n1496 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.4366125.1, 8.4.196475625.1

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

Sibling fields

Degree 12 siblings: data not computed
Degree 16 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 siblings: data not computed
Degree 36 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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }$ R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
3Data not computed
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
3881Data not computed