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Properties

Label 16.8.22833576610...4921.1
Degree $16$
Signature $[8, 4]$
Discriminant $41^{15}\cdot 59^{8}$
Root discriminant $249.70$
Ramified primes $41, 59$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_2\times OD_{16}).D_4$ (as 16T591)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1985183, 4815730, 10625946, -43863073, 56092254, -36196089, 13506349, -3318762, 486106, 19837, -15679, 4777, -1226, 146, -19, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 19*x^14 + 146*x^13 - 1226*x^12 + 4777*x^11 - 15679*x^10 + 19837*x^9 + 486106*x^8 - 3318762*x^7 + 13506349*x^6 - 36196089*x^5 + 56092254*x^4 - 43863073*x^3 + 10625946*x^2 + 4815730*x - 1985183)
 
gp: K = bnfinit(x^16 - 6*x^15 - 19*x^14 + 146*x^13 - 1226*x^12 + 4777*x^11 - 15679*x^10 + 19837*x^9 + 486106*x^8 - 3318762*x^7 + 13506349*x^6 - 36196089*x^5 + 56092254*x^4 - 43863073*x^3 + 10625946*x^2 + 4815730*x - 1985183, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 19 x^{14} + 146 x^{13} - 1226 x^{12} + 4777 x^{11} - 15679 x^{10} + 19837 x^{9} + 486106 x^{8} - 3318762 x^{7} + 13506349 x^{6} - 36196089 x^{5} + 56092254 x^{4} - 43863073 x^{3} + 10625946 x^{2} + 4815730 x - 1985183 \)

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:  \(228335766107949731571841524994747244921=41^{15}\cdot 59^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $249.70$
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:  $41, 59$
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}{12393624195792456850524556462947612295172742144713644441} a^{15} + \frac{3233212349402973442149537150657963277041898379626309653}{12393624195792456850524556462947612295172742144713644441} a^{14} - \frac{5770346557218303418805626187087884654699080772105294291}{12393624195792456850524556462947612295172742144713644441} a^{13} - \frac{3544595572447928612189822918875611625878077911900419742}{12393624195792456850524556462947612295172742144713644441} a^{12} + \frac{4457905105589615777225313217545706748884528897736857706}{12393624195792456850524556462947612295172742144713644441} a^{11} - \frac{1283911364034211456736548825368853872130074576861761503}{12393624195792456850524556462947612295172742144713644441} a^{10} - \frac{4622924586694314148587286611384169070172294839828713963}{12393624195792456850524556462947612295172742144713644441} a^{9} - \frac{6018779885761119753052409219546658593723924193306307328}{12393624195792456850524556462947612295172742144713644441} a^{8} + \frac{290986132294341522675232606967744779824267926016423271}{12393624195792456850524556462947612295172742144713644441} a^{7} - \frac{182957528871224260785020904607585313316255481884911909}{12393624195792456850524556462947612295172742144713644441} a^{6} - \frac{2627905094324722957317979967169922157079511138328485343}{12393624195792456850524556462947612295172742144713644441} a^{5} + \frac{647252872768569246643753050009033493424967947029142711}{12393624195792456850524556462947612295172742144713644441} a^{4} - \frac{5897800543248737185507601776102195073149652738238742789}{12393624195792456850524556462947612295172742144713644441} a^{3} - \frac{782471696874216179772729524894168569320899889454622828}{12393624195792456850524556462947612295172742144713644441} a^{2} - \frac{2089314746788830526203109806064017020336340744433800002}{12393624195792456850524556462947612295172742144713644441} a + \frac{5160055836467392163087326233678704719176463152085707459}{12393624195792456850524556462947612295172742144713644441}$

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:  \( 16501604898900 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times OD_{16}).D_4$ (as 16T591):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 28 conjugacy class representatives for $(C_2\times OD_{16}).D_4$
Character table for $(C_2\times OD_{16}).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.8.2359907842908948041.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ $16$ $16$ R

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
41Data not computed
$59$59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$