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Properties

Label 16.8.90390672967...1729.2
Degree $16$
Signature $[8, 4]$
Discriminant $61^{4}\cdot 97^{14}$
Root discriminant $153.02$
Ramified primes $61, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T817

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-47627, 43844, -211266, -61110, 200402, -256662, -33562, 243276, 47565, -66886, -19628, 5510, 2035, -110, -71, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 71*x^14 - 110*x^13 + 2035*x^12 + 5510*x^11 - 19628*x^10 - 66886*x^9 + 47565*x^8 + 243276*x^7 - 33562*x^6 - 256662*x^5 + 200402*x^4 - 61110*x^3 - 211266*x^2 + 43844*x - 47627)
 
gp: K = bnfinit(x^16 - 2*x^15 - 71*x^14 - 110*x^13 + 2035*x^12 + 5510*x^11 - 19628*x^10 - 66886*x^9 + 47565*x^8 + 243276*x^7 - 33562*x^6 - 256662*x^5 + 200402*x^4 - 61110*x^3 - 211266*x^2 + 43844*x - 47627, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 71 x^{14} - 110 x^{13} + 2035 x^{12} + 5510 x^{11} - 19628 x^{10} - 66886 x^{9} + 47565 x^{8} + 243276 x^{7} - 33562 x^{6} - 256662 x^{5} + 200402 x^{4} - 61110 x^{3} - 211266 x^{2} + 43844 x - 47627 \)

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:  \(90390672967514567934462041097091729=61^{4}\cdot 97^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $153.02$
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:  $61, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{5285895665842930153626394964919606275454394} a^{15} + \frac{374771728311452069596326040629573836010069}{2642947832921465076813197482459803137727197} a^{14} - \frac{154338850873735379018958467857944103392622}{2642947832921465076813197482459803137727197} a^{13} - \frac{448020029527358885975881137160378014996524}{2642947832921465076813197482459803137727197} a^{12} - \frac{280039568208057374223567996194913987119800}{2642947832921465076813197482459803137727197} a^{11} - \frac{1040511924959999687885415851523537060585879}{5285895665842930153626394964919606275454394} a^{10} + \frac{656175196425363898568418659074756369817062}{2642947832921465076813197482459803137727197} a^{9} + \frac{174682797482645912189656881930576662485017}{2642947832921465076813197482459803137727197} a^{8} + \frac{860297206487615602453081287777400454726121}{5285895665842930153626394964919606275454394} a^{7} - \frac{1587910792089365195700086545015877663930359}{5285895665842930153626394964919606275454394} a^{6} - \frac{2064575922996735199462375788260271191135795}{5285895665842930153626394964919606275454394} a^{5} - \frac{808190376158784097990956664462009745473732}{2642947832921465076813197482459803137727197} a^{4} - \frac{66598940006216893971691229977494578157769}{2642947832921465076813197482459803137727197} a^{3} - \frac{939397932695170828957467500600361890643567}{5285895665842930153626394964919606275454394} a^{2} + \frac{1187741666338105873295295107469870545769885}{5285895665842930153626394964919606275454394} a + \frac{2453762732897333463533135452207903112721443}{5285895665842930153626394964919606275454394}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 327749514382 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T817:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n817
Character table for t16n817 is not computed

Intermediate fields

\(\Q(\sqrt{97}) \), 4.4.912673.1, 8.8.80798284478113.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
61Data not computed
$97$97.8.7.1$x^{8} - 97$$8$$1$$7$$C_8$$[\ ]_{8}$
97.8.7.1$x^{8} - 97$$8$$1$$7$$C_8$$[\ ]_{8}$