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Properties

Label 16.0.73802355019...8921.2
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![422911, 191982, -155647, 453939, 306321, 47655, 244921, 77252, 16943, 25891, 7063, 567, 365, 13, -34, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 34*x^14 + 13*x^13 + 365*x^12 + 567*x^11 + 7063*x^10 + 25891*x^9 + 16943*x^8 + 77252*x^7 + 244921*x^6 + 47655*x^5 + 306321*x^4 + 453939*x^3 - 155647*x^2 + 191982*x + 422911)
 
gp: K = bnfinit(x^16 - 2*x^15 - 34*x^14 + 13*x^13 + 365*x^12 + 567*x^11 + 7063*x^10 + 25891*x^9 + 16943*x^8 + 77252*x^7 + 244921*x^6 + 47655*x^5 + 306321*x^4 + 453939*x^3 - 155647*x^2 + 191982*x + 422911, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 34 x^{14} + 13 x^{13} + 365 x^{12} + 567 x^{11} + 7063 x^{10} + 25891 x^{9} + 16943 x^{8} + 77252 x^{7} + 244921 x^{6} + 47655 x^{5} + 306321 x^{4} + 453939 x^{3} - 155647 x^{2} + 191982 x + 422911 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{900241956392009119656580957205482911902879308421} a^{15} - \frac{415008949947925002705819114619574146467949597497}{900241956392009119656580957205482911902879308421} a^{14} + \frac{407770282902619226433055900805416623277968317373}{900241956392009119656580957205482911902879308421} a^{13} + \frac{65090226284642462874923428555188650563997126330}{900241956392009119656580957205482911902879308421} a^{12} + \frac{71455433680774156814450822508149300812995230398}{900241956392009119656580957205482911902879308421} a^{11} - \frac{359861513860676840165902262013059617469855234585}{900241956392009119656580957205482911902879308421} a^{10} + \frac{198709726366168831674258502269489464881488057340}{900241956392009119656580957205482911902879308421} a^{9} + \frac{203398070437872075710289851093463328733613501234}{900241956392009119656580957205482911902879308421} a^{8} - \frac{364303353757107652805650925713416282770793172852}{900241956392009119656580957205482911902879308421} a^{7} - \frac{400879217469970177505775657802730509427494182209}{900241956392009119656580957205482911902879308421} a^{6} + \frac{389605923502885050472199518352131172660565082142}{900241956392009119656580957205482911902879308421} a^{5} + \frac{307154517860101543659576504886246947349179124696}{900241956392009119656580957205482911902879308421} a^{4} + \frac{312926728018983359480552115824440147191318240220}{900241956392009119656580957205482911902879308421} a^{3} + \frac{147612146377327229419058914513027083206292721876}{900241956392009119656580957205482911902879308421} a^{2} - \frac{410493855807547307333351876740755746245823634157}{900241956392009119656580957205482911902879308421} a - \frac{434866711515826952051914911234795050631353141059}{900241956392009119656580957205482911902879308421}$

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:  \( 1280887073.7 \) (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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{12}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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}$