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Properties

Label 16.8.22833576610...4921.2
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![6522823, -7613151, -1512478, 4204490, -180276, -573657, -308471, 37183, 115392, -10191, -10858, -1618, 1453, 186, -95, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 95*x^14 + 186*x^13 + 1453*x^12 - 1618*x^11 - 10858*x^10 - 10191*x^9 + 115392*x^8 + 37183*x^7 - 308471*x^6 - 573657*x^5 - 180276*x^4 + 4204490*x^3 - 1512478*x^2 - 7613151*x + 6522823)
 
gp: K = bnfinit(x^16 - 4*x^15 - 95*x^14 + 186*x^13 + 1453*x^12 - 1618*x^11 - 10858*x^10 - 10191*x^9 + 115392*x^8 + 37183*x^7 - 308471*x^6 - 573657*x^5 - 180276*x^4 + 4204490*x^3 - 1512478*x^2 - 7613151*x + 6522823, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 95 x^{14} + 186 x^{13} + 1453 x^{12} - 1618 x^{11} - 10858 x^{10} - 10191 x^{9} + 115392 x^{8} + 37183 x^{7} - 308471 x^{6} - 573657 x^{5} - 180276 x^{4} + 4204490 x^{3} - 1512478 x^{2} - 7613151 x + 6522823 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{31} a^{14} + \frac{14}{31} a^{13} + \frac{1}{31} a^{12} + \frac{4}{31} a^{11} + \frac{5}{31} a^{10} - \frac{13}{31} a^{9} + \frac{1}{31} a^{8} + \frac{8}{31} a^{7} - \frac{2}{31} a^{6} + \frac{1}{31} a^{5} - \frac{1}{31} a^{4} + \frac{10}{31} a^{3} + \frac{15}{31} a^{2} + \frac{3}{31} a - \frac{11}{31}$, $\frac{1}{5336978968963281261361482751471780934124938179039} a^{15} + \frac{65749309521097278559253001786222174393715125297}{5336978968963281261361482751471780934124938179039} a^{14} + \frac{1895948686751230056520995907212055403504905840305}{5336978968963281261361482751471780934124938179039} a^{13} - \frac{546244274052700244039962257732477222392937001030}{5336978968963281261361482751471780934124938179039} a^{12} + \frac{862642281176783892638448120528828104430785078762}{5336978968963281261361482751471780934124938179039} a^{11} + \frac{1096314229414394135761608137892474754395814608123}{5336978968963281261361482751471780934124938179039} a^{10} + \frac{1607194874771506507606491077076560634585540136000}{5336978968963281261361482751471780934124938179039} a^{9} + \frac{1218531958072174822185315736525279673920725813920}{5336978968963281261361482751471780934124938179039} a^{8} + \frac{2218061319222052988234827664861433703144321425470}{5336978968963281261361482751471780934124938179039} a^{7} - \frac{1832001848670331042665325146164094514735135964044}{5336978968963281261361482751471780934124938179039} a^{6} - \frac{906781189414137942148714481332315272029052980552}{5336978968963281261361482751471780934124938179039} a^{5} + \frac{1124435959473641522986830699801264458514959519237}{5336978968963281261361482751471780934124938179039} a^{4} - \frac{1130583596865714321679943851203208783174131504091}{5336978968963281261361482751471780934124938179039} a^{3} - \frac{2383172682417120320655530251870301338827447562296}{5336978968963281261361482751471780934124938179039} a^{2} + \frac{1008111281218353469866097969360645736061346048854}{5336978968963281261361482751471780934124938179039} a - \frac{100606282999274802775530866194044801946753278084}{232042563867968750493977510933555692788040790393}$

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:  \( 16140993152900 \) (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}$