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Properties

Label 16.0.73802355019...8921.1
Degree $16$
Signature $[0, 8]$
Discriminant $41^{15}\cdot 83^{4}$
Root discriminant $98.12$
Ramified primes $41, 83$
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![5381, -14170, 19651, -158935, 559692, -816623, 571184, -201478, 102969, -43934, 15792, -2725, 1576, -183, 69, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 69*x^14 - 183*x^13 + 1576*x^12 - 2725*x^11 + 15792*x^10 - 43934*x^9 + 102969*x^8 - 201478*x^7 + 571184*x^6 - 816623*x^5 + 559692*x^4 - 158935*x^3 + 19651*x^2 - 14170*x + 5381)
 
gp: K = bnfinit(x^16 - 4*x^15 + 69*x^14 - 183*x^13 + 1576*x^12 - 2725*x^11 + 15792*x^10 - 43934*x^9 + 102969*x^8 - 201478*x^7 + 571184*x^6 - 816623*x^5 + 559692*x^4 - 158935*x^3 + 19651*x^2 - 14170*x + 5381, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 69 x^{14} - 183 x^{13} + 1576 x^{12} - 2725 x^{11} + 15792 x^{10} - 43934 x^{9} + 102969 x^{8} - 201478 x^{7} + 571184 x^{6} - 816623 x^{5} + 559692 x^{4} - 158935 x^{3} + 19651 x^{2} - 14170 x + 5381 \)

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:  \(73802355019427518458620621498921=41^{15}\cdot 83^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $98.12$
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:  $41, 83$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{83} a^{14} - \frac{5}{83} a^{13} + \frac{34}{83} a^{12} - \frac{17}{83} a^{11} - \frac{16}{83} a^{10} - \frac{37}{83} a^{9} + \frac{35}{83} a^{8} + \frac{7}{83} a^{7} - \frac{30}{83} a^{6} - \frac{38}{83} a^{5} - \frac{29}{83} a^{4} - \frac{14}{83} a^{3} + \frac{35}{83} a^{2} + \frac{37}{83} a + \frac{37}{83}$, $\frac{1}{5901517308600484229433286572100413734557843} a^{15} + \frac{6707711036574448289779394414013118084395}{5901517308600484229433286572100413734557843} a^{14} - \frac{568245157884673214610913925787570312269832}{5901517308600484229433286572100413734557843} a^{13} + \frac{2753754702855352868324722317853642806223162}{5901517308600484229433286572100413734557843} a^{12} - \frac{78491430974525916129995491471097461999190}{5901517308600484229433286572100413734557843} a^{11} + \frac{1323303762581586079183414451443092114499923}{5901517308600484229433286572100413734557843} a^{10} + \frac{595344940043040556618226928089118706271008}{5901517308600484229433286572100413734557843} a^{9} + \frac{1663948289089977052059523351026477375237842}{5901517308600484229433286572100413734557843} a^{8} + \frac{1330831715473922229359528528341272170535457}{5901517308600484229433286572100413734557843} a^{7} + \frac{1697490920743944359208410615362903402926959}{5901517308600484229433286572100413734557843} a^{6} + \frac{1055467790087459105529431498369623748550652}{5901517308600484229433286572100413734557843} a^{5} + \frac{1998722717873167513207097899490068401541621}{5901517308600484229433286572100413734557843} a^{4} + \frac{2297306137312853343964174523364613595224021}{5901517308600484229433286572100413734557843} a^{3} + \frac{2079390320033092344116283825500041670250259}{5901517308600484229433286572100413734557843} a^{2} + \frac{67292538617470676090523510129520085032371}{5901517308600484229433286572100413734557843} a - \frac{1846033377343154042387647076908868532683713}{5901517308600484229433286572100413734557843}$

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:  \( 1206166793.77 \) (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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{12}$

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
$83$$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$