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Properties

Label 16.8.90390672967...1729.4
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 16T1194

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2366739, -2315682, 1010790, 1297305, 269222, -241942, -228695, 114284, 76987, -54627, -33298, 2081, 3080, 101, -96, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 96*x^14 + 101*x^13 + 3080*x^12 + 2081*x^11 - 33298*x^10 - 54627*x^9 + 76987*x^8 + 114284*x^7 - 228695*x^6 - 241942*x^5 + 269222*x^4 + 1297305*x^3 + 1010790*x^2 - 2315682*x + 2366739)
 
gp: K = bnfinit(x^16 - 3*x^15 - 96*x^14 + 101*x^13 + 3080*x^12 + 2081*x^11 - 33298*x^10 - 54627*x^9 + 76987*x^8 + 114284*x^7 - 228695*x^6 - 241942*x^5 + 269222*x^4 + 1297305*x^3 + 1010790*x^2 - 2315682*x + 2366739, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 96 x^{14} + 101 x^{13} + 3080 x^{12} + 2081 x^{11} - 33298 x^{10} - 54627 x^{9} + 76987 x^{8} + 114284 x^{7} - 228695 x^{6} - 241942 x^{5} + 269222 x^{4} + 1297305 x^{3} + 1010790 x^{2} - 2315682 x + 2366739 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{11} + \frac{2}{9} a^{10} - \frac{4}{9} a^{9} + \frac{2}{9} a^{8} + \frac{4}{9} a^{6} - \frac{1}{9} a^{5} + \frac{4}{9} a^{4} - \frac{4}{9} a^{3} - \frac{1}{9} a^{2}$, $\frac{1}{1040225317715859772642137147237447531020524262793} a^{15} - \frac{404104961705516697799601055077647513775523139}{38526863619105917505264338786572130778537935659} a^{14} + \frac{34096204394786390073777213540694077249052314799}{346741772571953257547379049079149177006841420931} a^{13} + \frac{79118786882732511400712415697283131905240748202}{1040225317715859772642137147237447531020524262793} a^{12} - \frac{165494193748507320217451763737179077825091166089}{1040225317715859772642137147237447531020524262793} a^{11} + \frac{132461015018493161655100766300169014642666086388}{1040225317715859772642137147237447531020524262793} a^{10} - \frac{285742792090535294829133979530778670787668550366}{1040225317715859772642137147237447531020524262793} a^{9} - \frac{3282619202677628486542454904237892453185330595}{38526863619105917505264338786572130778537935659} a^{8} - \frac{90291136588329504072065654995293641531869709513}{1040225317715859772642137147237447531020524262793} a^{7} - \frac{288912021338830010931695078007438474484456788229}{1040225317715859772642137147237447531020524262793} a^{6} + \frac{92498472635765242542927328623572320477740870637}{1040225317715859772642137147237447531020524262793} a^{5} + \frac{53992013659592409261470448714050833875281193959}{1040225317715859772642137147237447531020524262793} a^{4} + \frac{486452400648855875914208006071997516835869800763}{1040225317715859772642137147237447531020524262793} a^{3} + \frac{121380258757152599485569252342596656778318991986}{346741772571953257547379049079149177006841420931} a^{2} - \frac{52602494060061193257996333643849508704309441357}{115580590857317752515793016359716392335613806977} a - \frac{3197514258713833776926309959928770454907965739}{38526863619105917505264338786572130778537935659}$

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:  \( 282182598113 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1194:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 40 conjugacy class representatives for t16n1194
Character table for t16n1194 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$61$$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
$97$97.8.7.1$x^{8} - 97$$8$$1$$7$$C_8$$[\ ]_{8}$
97.8.7.1$x^{8} - 97$$8$$1$$7$$C_8$$[\ ]_{8}$