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Properties

Label 16.0.21289295932...4769.1
Degree $16$
Signature $[0, 8]$
Discriminant $37^{2}\cdot 41^{15}$
Root discriminant $51.05$
Ramified primes $37, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T841

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![857, 1838, -4247, -6357, 21474, -22650, 15258, -6372, 1117, 567, -673, 431, -119, -18, 22, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 22*x^14 - 18*x^13 - 119*x^12 + 431*x^11 - 673*x^10 + 567*x^9 + 1117*x^8 - 6372*x^7 + 15258*x^6 - 22650*x^5 + 21474*x^4 - 6357*x^3 - 4247*x^2 + 1838*x + 857)
 
gp: K = bnfinit(x^16 - 6*x^15 + 22*x^14 - 18*x^13 - 119*x^12 + 431*x^11 - 673*x^10 + 567*x^9 + 1117*x^8 - 6372*x^7 + 15258*x^6 - 22650*x^5 + 21474*x^4 - 6357*x^3 - 4247*x^2 + 1838*x + 857, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 22 x^{14} - 18 x^{13} - 119 x^{12} + 431 x^{11} - 673 x^{10} + 567 x^{9} + 1117 x^{8} - 6372 x^{7} + 15258 x^{6} - 22650 x^{5} + 21474 x^{4} - 6357 x^{3} - 4247 x^{2} + 1838 x + 857 \)

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:  \(2128929593223415400006494769=37^{2}\cdot 41^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.05$
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:  $37, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{118} a^{14} + \frac{14}{59} a^{13} - \frac{23}{118} a^{12} + \frac{17}{118} a^{11} - \frac{33}{118} a^{10} + \frac{1}{118} a^{9} - \frac{11}{118} a^{8} - \frac{37}{118} a^{7} + \frac{29}{118} a^{6} + \frac{28}{59} a^{5} - \frac{5}{59} a^{4} - \frac{57}{118} a^{3} + \frac{3}{59} a^{2} - \frac{5}{118} a - \frac{15}{118}$, $\frac{1}{31215732542464748022194131954} a^{15} + \frac{14667164748989277532536792}{15607866271232374011097065977} a^{14} - \frac{5906098112690965360487177443}{31215732542464748022194131954} a^{13} + \frac{7720241019044751035475002423}{31215732542464748022194131954} a^{12} - \frac{10547892373831408061411553107}{31215732542464748022194131954} a^{11} - \frac{2571784438810524175081506707}{31215732542464748022194131954} a^{10} + \frac{451347814291195255987400081}{1006959114273056387812713934} a^{9} - \frac{9144982035586952089786694321}{31215732542464748022194131954} a^{8} + \frac{10172851436378086935119934695}{31215732542464748022194131954} a^{7} + \frac{4772353339487923558958879473}{15607866271232374011097065977} a^{6} - \frac{3301336591563431974465685466}{15607866271232374011097065977} a^{5} - \frac{10946649778525172106311227627}{31215732542464748022194131954} a^{4} + \frac{530726541569699314732749154}{15607866271232374011097065977} a^{3} - \frac{14148789437039364701986962909}{31215732542464748022194131954} a^{2} - \frac{1394841612144934810640562349}{31215732542464748022194131954} a + \frac{7358536685271163302791204173}{15607866271232374011097065977}$

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:  $7$
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:  \( 6156962.26717 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T841:

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 t16n841
Character table for t16n841 is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.0.194754273881.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ R R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$37$$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
41Data not computed