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Properties

Label 16.8.35025358550...1641.1
Degree $16$
Signature $[8, 4]$
Discriminant $41^{15}\cdot 83^{8}$
Root discriminant $296.16$
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![-3392863, 8942321, -3919178, -2183884, -510408, 747363, 526043, -90245, -38030, 13097, 3656, -8752, 305, 104, -95, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 95*x^14 + 104*x^13 + 305*x^12 - 8752*x^11 + 3656*x^10 + 13097*x^9 - 38030*x^8 - 90245*x^7 + 526043*x^6 + 747363*x^5 - 510408*x^4 - 2183884*x^3 - 3919178*x^2 + 8942321*x - 3392863)
 
gp: K = bnfinit(x^16 - 4*x^15 - 95*x^14 + 104*x^13 + 305*x^12 - 8752*x^11 + 3656*x^10 + 13097*x^9 - 38030*x^8 - 90245*x^7 + 526043*x^6 + 747363*x^5 - 510408*x^4 - 2183884*x^3 - 3919178*x^2 + 8942321*x - 3392863, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 95 x^{14} + 104 x^{13} + 305 x^{12} - 8752 x^{11} + 3656 x^{10} + 13097 x^{9} - 38030 x^{8} - 90245 x^{7} + 526043 x^{6} + 747363 x^{5} - 510408 x^{4} - 2183884 x^{3} - 3919178 x^{2} + 8942321 x - 3392863 \)

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:  \(3502535855067952407242642672315293971641=41^{15}\cdot 83^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $296.16$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{37} a^{14} - \frac{11}{37} a^{13} + \frac{5}{37} a^{12} + \frac{1}{37} a^{11} + \frac{6}{37} a^{10} - \frac{2}{37} a^{9} - \frac{3}{37} a^{8} + \frac{11}{37} a^{7} + \frac{8}{37} a^{6} + \frac{10}{37} a^{5} + \frac{17}{37} a^{4} - \frac{9}{37} a^{2} - \frac{7}{37} a$, $\frac{1}{8176573294021418854647214736725929919775474973335179} a^{15} + \frac{775582432095455029497812422655835068519204252350}{220988467405984293368843641533133241075012837117167} a^{14} + \frac{2536126760652802089987619384714223504726224064149574}{8176573294021418854647214736725929919775474973335179} a^{13} + \frac{2180720250777116005849712225661822200185796216799607}{8176573294021418854647214736725929919775474973335179} a^{12} - \frac{446694535364729824677457080305374163823634385530140}{8176573294021418854647214736725929919775474973335179} a^{11} + \frac{1816987212781960894293923227449362118541131258166344}{8176573294021418854647214736725929919775474973335179} a^{10} - \frac{3613915165053081178407335705558157662685155414524778}{8176573294021418854647214736725929919775474973335179} a^{9} - \frac{100807211419806538776973289244814426307409490164981}{8176573294021418854647214736725929919775474973335179} a^{8} + \frac{1888834949666332072779132766279039581801459651653756}{8176573294021418854647214736725929919775474973335179} a^{7} - \frac{3663773189502527246201525038425385501180370082828383}{8176573294021418854647214736725929919775474973335179} a^{6} + \frac{125092230150366564848727738951403627743727370726065}{8176573294021418854647214736725929919775474973335179} a^{5} - \frac{409733657025483745911784334620838690249536359565865}{8176573294021418854647214736725929919775474973335179} a^{4} - \frac{676475113378868523925583330250650744493243442504729}{8176573294021418854647214736725929919775474973335179} a^{3} + \frac{806260879027185507396231931344120213783271314999805}{8176573294021418854647214736725929919775474973335179} a^{2} - \frac{2032724374394757569217347376412580122008487512075934}{8176573294021418854647214736725929919775474973335179} a + \frac{59497782381747406135141241384890968958043947408221}{220988467405984293368843641533133241075012837117167}$

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:  \( 55152918821800 \) (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, 8.8.9242710845966413801.1

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$83.2.1.2$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}$
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}$