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Properties

Label 16.0.10627476673...0625.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 89^{15}$
Root discriminant $274.89$
Ramified primes $5, 89$
Class number $103504$ (GRH)
Class group $[4, 25876]$ (GRH)
Galois group $C_2^3.C_8$ (as 16T104)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![22137835, -20284665, -657200, 6342040, -3011335, 3926065, -997465, -600295, 410366, -129737, 110498, 4931, -2450, 881, 48, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 48*x^14 + 881*x^13 - 2450*x^12 + 4931*x^11 + 110498*x^10 - 129737*x^9 + 410366*x^8 - 600295*x^7 - 997465*x^6 + 3926065*x^5 - 3011335*x^4 + 6342040*x^3 - 657200*x^2 - 20284665*x + 22137835)
 
gp: K = bnfinit(x^16 - 7*x^15 + 48*x^14 + 881*x^13 - 2450*x^12 + 4931*x^11 + 110498*x^10 - 129737*x^9 + 410366*x^8 - 600295*x^7 - 997465*x^6 + 3926065*x^5 - 3011335*x^4 + 6342040*x^3 - 657200*x^2 - 20284665*x + 22137835, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 48 x^{14} + 881 x^{13} - 2450 x^{12} + 4931 x^{11} + 110498 x^{10} - 129737 x^{9} + 410366 x^{8} - 600295 x^{7} - 997465 x^{6} + 3926065 x^{5} - 3011335 x^{4} + 6342040 x^{3} - 657200 x^{2} - 20284665 x + 22137835 \)

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:  \(1062747667303462437668657811578369140625=5^{14}\cdot 89^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $274.89$
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:  $5, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is 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}{85687665714421969920688592203439302833375923667755873308481} a^{15} + \frac{25346584811060065175387386970834579348978674805172857934133}{85687665714421969920688592203439302833375923667755873308481} a^{14} + \frac{31851166036261362364914431418115144458491245616804724126647}{85687665714421969920688592203439302833375923667755873308481} a^{13} - \frac{2893946361668322115701838390864788888255923463042108532728}{85687665714421969920688592203439302833375923667755873308481} a^{12} + \frac{2604905951404326850658257160605750875410500438293958426342}{85687665714421969920688592203439302833375923667755873308481} a^{11} + \frac{815356609268536878588663803699202821149104511886872580649}{85687665714421969920688592203439302833375923667755873308481} a^{10} + \frac{616013855515592652136311555573593934090944240501232660168}{85687665714421969920688592203439302833375923667755873308481} a^{9} + \frac{17751028222111771372050539925236330562035511223315961200088}{85687665714421969920688592203439302833375923667755873308481} a^{8} + \frac{7858289165133466904992037202340967696086093561680958729241}{85687665714421969920688592203439302833375923667755873308481} a^{7} + \frac{17411105954878465662101463732995175735553332652298662124022}{85687665714421969920688592203439302833375923667755873308481} a^{6} - \frac{30743426864214033519623167028216506363101648785290914337363}{85687665714421969920688592203439302833375923667755873308481} a^{5} + \frac{25386064633989490838396204887378080302072245846092565305295}{85687665714421969920688592203439302833375923667755873308481} a^{4} + \frac{1027029654569602261544644777129175731455112817500657606803}{85687665714421969920688592203439302833375923667755873308481} a^{3} + \frac{22398015636607164729149078595482466809494547881246457676338}{85687665714421969920688592203439302833375923667755873308481} a^{2} - \frac{22780988026738907818363068161691550964047125879048907504183}{85687665714421969920688592203439302833375923667755873308481} a - \frac{33436136921323137930160892557432864612222011604661622967779}{85687665714421969920688592203439302833375923667755873308481}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{4}\times C_{25876}$, which has order $103504$ (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:  \( 54766314.0729 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_8$ (as 16T104):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $C_2^3.C_8$
Character table for $C_2^3.C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{89}) \), 4.4.17624225.2, 8.8.691114607742640625.3

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ R $16$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ $16$ $16$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ $16$

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
$5$5.8.7.3$x^{8} + 10$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.3$x^{8} + 10$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
89Data not computed