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Properties

Label 16.0.18843687673...1961.1
Degree $16$
Signature $[0, 8]$
Discriminant $41^{15}\cdot 59^{4}$
Root discriminant $90.09$
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![1848071, -1939389, -26592, -9473, 514868, -312848, 212632, -147125, 63519, -21450, 4944, -266, -365, 187, -19, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 19*x^14 + 187*x^13 - 365*x^12 - 266*x^11 + 4944*x^10 - 21450*x^9 + 63519*x^8 - 147125*x^7 + 212632*x^6 - 312848*x^5 + 514868*x^4 - 9473*x^3 - 26592*x^2 - 1939389*x + 1848071)
 
gp: K = bnfinit(x^16 - 6*x^15 - 19*x^14 + 187*x^13 - 365*x^12 - 266*x^11 + 4944*x^10 - 21450*x^9 + 63519*x^8 - 147125*x^7 + 212632*x^6 - 312848*x^5 + 514868*x^4 - 9473*x^3 - 26592*x^2 - 1939389*x + 1848071, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 19 x^{14} + 187 x^{13} - 365 x^{12} - 266 x^{11} + 4944 x^{10} - 21450 x^{9} + 63519 x^{8} - 147125 x^{7} + 212632 x^{6} - 312848 x^{5} + 514868 x^{4} - 9473 x^{3} - 26592 x^{2} - 1939389 x + 1848071 \)

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:  \(18843687673244176811423009101961=41^{15}\cdot 59^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $90.09$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{201952560275600236253972715903493641056458729} a^{15} - \frac{245528697289403705614668995013809149337465}{905616862222422584098532358311630677383223} a^{14} - \frac{56126158749620846438691033701777519534672316}{201952560275600236253972715903493641056458729} a^{13} + \frac{73654383714796109292671687096712380610492574}{201952560275600236253972715903493641056458729} a^{12} + \frac{15631950383783342642300514872885230163478853}{201952560275600236253972715903493641056458729} a^{11} - \frac{59948161190092018477835385289327531794281641}{201952560275600236253972715903493641056458729} a^{10} - \frac{35064129916199033174388272793019018710096568}{201952560275600236253972715903493641056458729} a^{9} - \frac{64584327559976864175003470307231906226031677}{201952560275600236253972715903493641056458729} a^{8} - \frac{74285952581975708576034747165458165268156234}{201952560275600236253972715903493641056458729} a^{7} + \frac{74539314280271424920747154243842240265260310}{201952560275600236253972715903493641056458729} a^{6} + \frac{34917900244540996052285507734520471035893192}{201952560275600236253972715903493641056458729} a^{5} - \frac{53694023879237790240473165763365979538119905}{201952560275600236253972715903493641056458729} a^{4} + \frac{99490142747363368502980439595751198529536037}{201952560275600236253972715903493641056458729} a^{3} + \frac{71308372424785235915721724445902501243578165}{201952560275600236253972715903493641056458729} a^{2} - \frac{16415756805300143575818271328042790370722992}{201952560275600236253972715903493641056458729} a + \frac{9445014148931831556267055158683828984628524}{201952560275600236253972715903493641056458729}$

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:  \( 562417570.815 \) (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, 8.0.194754273881.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ 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$59.2.1.1$x^{2} - 59$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.1.1$x^{2} - 59$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.1$x^{2} - 59$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.1$x^{2} - 59$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$