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Properties

Label 16.0.68681261305...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{8}\cdot 11^{4}\cdot 29^{4}\cdot 509^{2}$
Root discriminant $41.19$
Ramified primes $2, 5, 11, 29, 509$
Class number $60$ (GRH)
Class group $[2, 30]$ (GRH)
Galois group 16T919

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![227821, 207856, 93272, -215170, 9355, 79098, -15085, -19298, 6324, 2208, -882, -348, 174, 22, -15, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 15*x^14 + 22*x^13 + 174*x^12 - 348*x^11 - 882*x^10 + 2208*x^9 + 6324*x^8 - 19298*x^7 - 15085*x^6 + 79098*x^5 + 9355*x^4 - 215170*x^3 + 93272*x^2 + 207856*x + 227821)
 
gp: K = bnfinit(x^16 - 2*x^15 - 15*x^14 + 22*x^13 + 174*x^12 - 348*x^11 - 882*x^10 + 2208*x^9 + 6324*x^8 - 19298*x^7 - 15085*x^6 + 79098*x^5 + 9355*x^4 - 215170*x^3 + 93272*x^2 + 207856*x + 227821, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 15 x^{14} + 22 x^{13} + 174 x^{12} - 348 x^{11} - 882 x^{10} + 2208 x^{9} + 6324 x^{8} - 19298 x^{7} - 15085 x^{6} + 79098 x^{5} + 9355 x^{4} - 215170 x^{3} + 93272 x^{2} + 207856 x + 227821 \)

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:  \(68681261305082905600000000=2^{16}\cdot 5^{8}\cdot 11^{4}\cdot 29^{4}\cdot 509^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.19$
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:  $2, 5, 11, 29, 509$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{11} a^{14} - \frac{4}{11} a^{13} - \frac{1}{11} a^{12} + \frac{3}{11} a^{10} - \frac{2}{11} a^{9} - \frac{2}{11} a^{8} - \frac{2}{11} a^{6} - \frac{5}{11} a^{4} - \frac{4}{11} a^{3} + \frac{5}{11} a^{2}$, $\frac{1}{446683080399763260466603710454166153603} a^{15} - \frac{13964318424796563851877560355000603427}{446683080399763260466603710454166153603} a^{14} - \frac{71443441231072976200907252965788906636}{446683080399763260466603710454166153603} a^{13} + \frac{8586779596629673701001048189982866308}{40607552763614841860600337314015104873} a^{12} + \frac{214300707195995411110741264432381020311}{446683080399763260466603710454166153603} a^{11} + \frac{139768153672327795544904478751564399256}{446683080399763260466603710454166153603} a^{10} - \frac{44453430694636363827536114531116050099}{446683080399763260466603710454166153603} a^{9} + \frac{17429121681363072830724742585475762708}{40607552763614841860600337314015104873} a^{8} + \frac{163449084371071881124186080919317281442}{446683080399763260466603710454166153603} a^{7} - \frac{11714493060376179626210754736291079777}{40607552763614841860600337314015104873} a^{6} + \frac{36707023517693258550144869396095613342}{446683080399763260466603710454166153603} a^{5} + \frac{175138956197080993222605236996376352286}{446683080399763260466603710454166153603} a^{4} + \frac{35171433920282962143809332606835600407}{446683080399763260466603710454166153603} a^{3} - \frac{4998388369170822099003089360609277592}{40607552763614841860600337314015104873} a^{2} + \frac{2026616449210769181518559054472666242}{40607552763614841860600337314015104873} a - \frac{2064088663369929577523542894435588468}{40607552763614841860600337314015104873}$

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

Class group and class number

$C_{2}\times C_{30}$, which has order $60$ (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:  \( 29340.4724529 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T919:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 65 conjugacy class representatives for t16n919 are not computed
Character table for t16n919 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.4400.1, 4.4.127600.1, 4.4.725.1, 8.0.267543125.1, 8.0.8287415840000.1, 8.8.16281760000.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
$29$29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
509Data not computed